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<span class="kw">reconsider </span><font color="Maroon" title="c1">F</font> =  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  as   <a href="turing_1.html#NM3" title="TURING_1:NM.3">Symbol</a> of <a href="turing_1.html#K16" title="TURING_1:func.16">SuccTuring</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="turing_1.html#E53"><span class="lab"><font color="Green" title="E41">Lm6</font></span></a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c2">s</font> be   <a href="turing_1.html#NM4" title="TURING_1:NM.4">All-State</a> of <a href="turing_1.html#K16" title="TURING_1:func.16">SuccTuring</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">t</font> being   <a href="turing_1.html#NM2" title="TURING_1:NM.2">Tape</a> of <a href="turing_1.html#K16" title="TURING_1:func.16">SuccTuring</a><br/>  for <font color="Olive" title="b2">head</font>, <font color="Olive" title="b3">n</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon" title="c2">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,<font color="Olive" title="b2">head</font>,<font color="Olive" title="b1">t</font></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span> &amp; <font color="Olive" title="b1">t</font> <a href="turing_1.html#R2" title="TURING_1:pred.2">storeData</a> <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><font color="Olive" title="b2">head</font>,<font color="Olive" title="b3">n</font></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> holds <br/>( <font color="Maroon" title="c2">s</font> is  <a href="turing_1.html#V2" title="TURING_1:attr.2">Accept-Halt</a>  &amp; <span class="p1">(<span class="default"><a href="turing_1.html#K11" title="TURING_1:func.11">Result</a> <font color="Maroon" title="c2">s</font></span>)</span> <a href="mcart_1.html#K6" title="MCART_1:func.6">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">head</font> &amp; <span class="p1">(<span class="default"><a href="turing_1.html#K11" title="TURING_1:func.11">Result</a> <font color="Maroon" title="c2">s</font></span>)</span> <a href="mcart_1.html#K7" title="MCART_1:func.7">`3</a>  <a href="turing_1.html#R2" title="TURING_1:pred.2">storeData</a> <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><font color="Olive" title="b2">head</font>,<span class="p2">(<span class="default"><font color="Olive" title="b3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">t</font> be   <a href="turing_1.html#NM2" title="TURING_1:NM.2">Tape</a> of <a href="turing_1.html#K16" title="TURING_1:func.16">SuccTuring</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">head</font>, <font color="Olive" title="b2">n</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon" title="c2">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,<font color="Olive" title="b1">head</font>,<font color="Maroon" title="c3">t</font></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span> &amp; <font color="Maroon" title="c3">t</font> <a href="turing_1.html#R2" title="TURING_1:pred.2">storeData</a> <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><font color="Olive" title="b1">head</font>,<font color="Olive" title="b2">n</font></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> holds <br/>( <font color="Maroon" title="c2">s</font> is  <a href="turing_1.html#V2" title="TURING_1:attr.2">Accept-Halt</a>  &amp; <span class="p1">(<span class="default"><a href="turing_1.html#K11" title="TURING_1:func.11">Result</a> <font color="Maroon" title="c2">s</font></span>)</span> <a href="mcart_1.html#K6" title="MCART_1:func.6">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">head</font> &amp; <span class="p1">(<span class="default"><a href="turing_1.html#K11" title="TURING_1:func.11">Result</a> <font color="Maroon" title="c2">s</font></span>)</span> <a href="mcart_1.html#K7" title="MCART_1:func.7">`3</a>  <a href="turing_1.html#R2" title="TURING_1:pred.2">storeData</a> <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><font color="Olive" title="b1">head</font>,<span class="p2">(<span class="default"><font color="Olive" title="b2">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> )</span><br/><span class="kw">let </span><font color="Maroon" title="c4">h</font>, <font color="Maroon" title="c5">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,<font color="Maroon" title="c4">h</font>,<font color="Maroon" title="c3">t</font></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span> &amp; <font color="Maroon" title="c3">t</font> <a href="turing_1.html#R2" title="TURING_1:pred.2">storeData</a> <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c4">h</font>,<font color="Maroon" title="c5">n</font></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> implies ( <font color="Maroon" title="c2">s</font> is  <a href="turing_1.html#V2" title="TURING_1:attr.2">Accept-Halt</a>  &amp; <span class="p1">(<span class="default"><a href="turing_1.html#K11" title="TURING_1:func.11">Result</a> <font color="Maroon" title="c2">s</font></span>)</span> <a href="mcart_1.html#K6" title="MCART_1:func.6">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">h</font> &amp; <span class="p1">(<span class="default"><a href="turing_1.html#K11" title="TURING_1:func.11">Result</a> <font color="Maroon" title="c2">s</font></span>)</span> <a href="mcart_1.html#K7" title="MCART_1:func.7">`3</a>  <a href="turing_1.html#R2" title="TURING_1:pred.2">storeData</a> <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c4">h</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> ) )</span><br/>





<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E2:61"/><span class="lab"><font color="Green" title="E44">A1</font></span>: 
<font color="Maroon" title="c2">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,<font color="Maroon" title="c4">h</font>,<font color="Maroon" title="c3">t</font></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span>
 <span class="kw">and </span><br/><a NAME="E3:61"/><span class="lab"><font color="Green" title="E45">A2</font></span>: 
<font color="Maroon" title="c3">t</font> <a href="turing_1.html#R2" title="TURING_1:pred.2">storeData</a> <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c4">h</font>,<font color="Maroon" title="c5">n</font></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">s</font> is  <a href="turing_1.html#V2" title="TURING_1:attr.2">Accept-Halt</a>  &amp; <span class="p1">(<span class="default"><a href="turing_1.html#K11" title="TURING_1:func.11">Result</a> <font color="Maroon" title="c2">s</font></span>)</span> <a href="mcart_1.html#K6" title="MCART_1:func.6">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">h</font> &amp; <span class="p1">(<span class="default"><a href="turing_1.html#K11" title="TURING_1:func.11">Result</a> <font color="Maroon" title="c2">s</font></span>)</span> <a href="mcart_1.html#K7" title="MCART_1:func.7">`3</a>  <a href="turing_1.html#R2" title="TURING_1:pred.2">storeData</a> <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c4">h</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> )</span><br/>

<a NAME="E4:61"/><span class="lab"><font color="Green" title="E46">A3</font></span>: 
<font color="Maroon" title="c3">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:61"><span class="lab"><font color="Green" title="E45">A2</font></span></a>, <a class="ref" href="turing_1.html#T17" target="_self" title="TURING_1:th.17">Th20</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c6">i3</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">h</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c7">h1</font> = <font color="Maroon" title="c4">h</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#D2" title="INT_1:def.2">INT_1:def 2</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c8">p1</font> = 1 as   <a href="turing_1.html#NM1" title="TURING_1:NM.1">State</a> of <a href="turing_1.html#K16" title="TURING_1:func.16">SuccTuring</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="turing_1.html#E52"><span class="lab"><font color="Green" title="E40">Lm5</font></span></a></span>;<br/>
<a NAME="E7:61"/><span class="lab"><font color="Green" title="E47">A4</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">h1</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c5">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">h</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T29" title="XREAL_1:th.29">XREAL_1:29</a></span>;<br/>
<a NAME="E8:61"/>
<font color="Maroon" title="c4">h</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">h</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c5">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></span>;<br/>
<a NAME="E9:61"/><span class="kw">then </span><span class="lab"><font color="Green" title="E48">A5</font></span>: 
<font color="Maroon" title="c4">h</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">h</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T7" title="XREAL_1:th.7">XREAL_1:7</a></span>;<br/>
<a NAME="E10:61"/><span class="lab"><font color="Green" title="E49">A6</font></span>: 
<font color="Maroon" title="c7">h1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">h</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></span>;<br/>
<a NAME="E11:61"/><span class="kw">then </span><span class="lab"><font color="Green" title="E50">A7</font></span>: 
<font color="Maroon" title="c7">h1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">h</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:61"><span class="lab"><font color="Green" title="E48">A5</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c9">p2</font> = 2 as   <a href="turing_1.html#NM1" title="TURING_1:NM.1">State</a> of <a href="turing_1.html#K16" title="TURING_1:func.16">SuccTuring</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="turing_1.html#E52"><span class="lab"><font color="Green" title="E40">Lm5</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c10">i2</font> = <font color="Maroon" title="c7">h1</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1 as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#D2" title="INT_1:def.2">INT_1:def 2</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c11">nk</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c7">h1</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c5">n</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#D2" title="INT_1:def.2">INT_1:def 2</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c12">hh</font> = <font color="Maroon" title="c4">h</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#D2" title="INT_1:def.2">INT_1:def 2</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c13">n3</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">h</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#D2" title="INT_1:def.2">INT_1:def 2</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c14">T</font> = 1 as   <a href="turing_1.html#NM3" title="TURING_1:NM.3">Symbol</a> of <a href="turing_1.html#K16" title="TURING_1:func.16">SuccTuring</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="turing_1.html#E53"><span class="lab"><font color="Green" title="E41">Lm6</font></span></a></span>;<br/>
<a NAME="E18:61"/><span class="lab"><font color="Green" title="E51">A8</font></span>: 
<font color="Maroon" title="c13">n3</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">h</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c5">n</font>
 
;<br/>
<span class="kw">set </span><font color="Maroon" title="c15">t1</font> =  <a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<font color="Maroon" title="c3">t</font>,<font color="Maroon" title="c7">h1</font>,<font color="Maroon" title="c14">T</font>);<br/>
<a NAME="E19:61"/><span class="lab"><font color="Green" title="E52">A9</font></span>: 
<font color="Maroon" title="c4">h</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">h1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T29" title="XREAL_1:th.29">XREAL_1:29</a></span>;<br/>
<a NAME="E20:61"/><span class="lab"><font color="Green" title="E53">A10</font></span>: 
<font color="Maroon" title="c3">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">h</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:61"><span class="lab"><font color="Green" title="E45">A2</font></span></a>, <a class="ref" href="turing_1.html#T17" target="_self" title="TURING_1:th.17">Th20</a></span>;<br/>
<a NAME="E21:61"/><span class="lab"><font color="Green" title="E54">A11</font></span>: 
( <span class="p1">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<font color="Maroon" title="c3">t</font>,<font color="Maroon" title="c7">h1</font>,<font color="Maroon" title="c14">T</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <span class="p1">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<font color="Maroon" title="c3">t</font>,<font color="Maroon" title="c7">h1</font>,<font color="Maroon" title="c14">T</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">h</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; (  for <font color="Olive" title="b1">i</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>  st <font color="Maroon" title="c4">h</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">h</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 holds <br/><span class="p1">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<font color="Maroon" title="c3">t</font>,<font color="Maroon" title="c7">h1</font>,<font color="Maroon" title="c14">T</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">thus </span><a NAME="E1:61_1"/>
<span class="p1">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<font color="Maroon" title="c3">t</font>,<font color="Maroon" title="c7">h1</font>,<font color="Maroon" title="c14">T</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:61"><span class="lab"><font color="Green" title="E46">A3</font></span></a>, <a class="txt" href="#E19:61"><span class="lab"><font color="Green" title="E52">A9</font></span></a>, <a class="ref" href="turing_1.html#T26" target="_self" title="TURING_1:th.26">Th30</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <span class="p1">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<font color="Maroon" title="c3">t</font>,<font color="Maroon" title="c7">h1</font>,<font color="Maroon" title="c14">T</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">h</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; (  for <font color="Olive" title="b1">i</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>  st <font color="Maroon" title="c4">h</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">h</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 holds <br/><span class="p1">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<font color="Maroon" title="c3">t</font>,<font color="Maroon" title="c7">h1</font>,<font color="Maroon" title="c14">T</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) )</span><br/>

<span class="kw">thus </span><a NAME="E2:61_1"/>
<span class="p1">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<font color="Maroon" title="c3">t</font>,<font color="Maroon" title="c7">h1</font>,<font color="Maroon" title="c14">T</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">h</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:61"><span class="lab"><font color="Green" title="E53">A10</font></span></a>, <a class="txt" href="#E9:61"><span class="lab"><font color="Green" title="E48">A5</font></span></a>, <a class="txt" href="#E10:61"><span class="lab"><font color="Green" title="E49">A6</font></span></a>, <a class="ref" href="turing_1.html#T26" target="_self" title="TURING_1:th.26">Th30</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">i</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>  st <font color="Maroon" title="c4">h</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">h</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 holds <br/><span class="p1">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<font color="Maroon" title="c3">t</font>,<font color="Maroon" title="c7">h1</font>,<font color="Maroon" title="c14">T</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1</span><br/>

<div><span class="kw">hereby </span> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span>
<div class="add"><span class="kw">let </span><font color="Maroon" title="c16">i</font> be   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">h</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c16">i</font> &amp; <font color="Maroon" title="c16">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">h</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 implies <span class="p1">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<font color="Maroon" title="c3">t</font>,<font color="Maroon" title="c7">h1</font>,<font color="Maroon" title="c14">T</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )</span><br/><span class="kw">assume </span><a NAME="E1:61_1_1"/><span class="lab"><font color="Green" title="E55">A12</font></span>: 
( <font color="Maroon" title="c4">h</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c16">i</font> &amp; <font color="Maroon" title="c16">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">h</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<font color="Maroon" title="c3">t</font>,<font color="Maroon" title="c7">h1</font>,<font color="Maroon" title="c14">T</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1</span><br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_1_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:61_1_1_1"/>
( <font color="Maroon" title="c7">h1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">i</font> or <font color="Maroon" title="c7">h1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c16">i</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:61_1_1_1_1"/>
<font color="Maroon" title="c7">h1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">i</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<font color="Maroon" title="c3">t</font>,<font color="Maroon" title="c7">h1</font>,<font color="Maroon" title="c14">T</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:61_1_1_1_1"/>
<span class="p1">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<font color="Maroon" title="c3">t</font>,<font color="Maroon" title="c7">h1</font>,<font color="Maroon" title="c14">T</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="turing_1.html#T26" target="_self" title="TURING_1:th.26">Th30</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:61_1_1_1_2"/>
<font color="Maroon" title="c7">h1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c16">i</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<font color="Maroon" title="c3">t</font>,<font color="Maroon" title="c7">h1</font>,<font color="Maroon" title="c14">T</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1</span><br/><div class="add"><span class="kw">hence </span><span class="p1">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<font color="Maroon" title="c3">t</font>,<font color="Maroon" title="c7">h1</font>,<font color="Maroon" title="c14">T</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">i</font> = 
<font color="Maroon" title="c3">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">i</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="turing_1.html#T26" target="_self" title="TURING_1:th.26">Th30</a></span>
<br/>.= 
1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:61"><span class="lab"><font color="Green" title="E45">A2</font></span></a>, <a class="txt" href="#E1:61_1_1"><span class="lab"><font color="Green" title="E55">A12</font></span></a>, <a class="ref" href="turing_1.html#T17" target="_self" title="TURING_1:th.17">Th20</a></span>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div>
<span class="kw">end;</span></div>


</div><span class="kw">end;</span></div>
<a NAME="E22:61"/><span class="lab"><font color="Green" title="E55">A13</font></span>: 
 for <font color="Olive" title="b1">i</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>  st <span class="p1">(<span class="default"><font color="Maroon" title="c4">h</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">h</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c5">n</font> holds <br/><span class="p1">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<font color="Maroon" title="c3">t</font>,<font color="Maroon" title="c7">h1</font>,<font color="Maroon" title="c14">T</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c16">i</font> be   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <span class="p1">(<span class="default"><font color="Maroon" title="c4">h</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">i</font> &amp; <font color="Maroon" title="c16">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">h</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c5">n</font> implies <span class="p1">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<font color="Maroon" title="c3">t</font>,<font color="Maroon" title="c7">h1</font>,<font color="Maroon" title="c14">T</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:61_2"/><span class="lab"><font color="Green" title="E56">A14</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">h</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">i</font>
 <span class="kw">and </span><br/><a NAME="E2:61_2"/><span class="lab"><font color="Green" title="E57">A15</font></span>: 
<font color="Maroon" title="c16">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">h</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c5">n</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<font color="Maroon" title="c3">t</font>,<font color="Maroon" title="c7">h1</font>,<font color="Maroon" title="c14">T</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1</span><br/>

<a NAME="E3:61_2"/>
<font color="Maroon" title="c7">h1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">h1</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T29" title="XREAL_1:th.29">XREAL_1:29</a></span>;<br/>
<a NAME="E4:61_2"/><span class="kw">then </span>
<font color="Maroon" title="c7">h1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c16">i</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:61_2"><span class="lab"><font color="Green" title="E56">A14</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E5:61_2"/><span class="kw">then </span>
<font color="Maroon" title="c4">h</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c16">i</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:61"><span class="lab"><font color="Green" title="E52">A9</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:61_2"/>
<span class="p1">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<font color="Maroon" title="c3">t</font>,<font color="Maroon" title="c7">h1</font>,<font color="Maroon" title="c14">T</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:61"><span class="lab"><font color="Green" title="E54">A11</font></span></a>, <a class="txt" href="#E2:61_2"><span class="lab"><font color="Green" title="E57">A15</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">reconsider </span><font color="Maroon" title="c16">s3</font> = <font color="Maroon" title="c2">s</font> <a href="mcart_1.html#K7" title="MCART_1:func.7">`3</a>  as   <a href="turing_1.html#NM2" title="TURING_1:NM.2">Tape</a> of <a href="turing_1.html#K16" title="TURING_1:func.16">SuccTuring</a> ;<br/>
<span class="lab"><font color="Green" title="E56">A16</font></span>:  <a href="turing_1.html#K8" title="TURING_1:func.8">TRAN</a> <font color="Maroon" title="c2">s</font> = 
<a href="turing_1.html#K15" title="TURING_1:func.15">Succ_Tran</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">s</font> <a href="mcart_1.html#K5" title="MCART_1:func.5">`1</a></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c16">s3</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><a href="turing_1.html#K7" title="TURING_1:func.7">Head</a> <font color="Maroon" title="c2">s</font></span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="turing_1.html#D17" target="_self" title="TURING_1:def.17">Def18</a></span>
<br/>.= 
<a href="turing_1.html#K15" title="TURING_1:func.15">Succ_Tran</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,<span class="p2">(<span class="default"><font color="Maroon" title="c16">s3</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><a href="turing_1.html#K7" title="TURING_1:func.7">Head</a> <font color="Maroon" title="c2">s</font></span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:61"><span class="lab"><font color="Green" title="E44">A1</font></span></a>, <a class="ref" href="mcart_1.html#T64" title="MCART_1:th.64">MCART_1:64</a></span>
<br/>.= 
<a href="turing_1.html#K15" title="TURING_1:func.15">Succ_Tran</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><a href="turing_1.html#K7" title="TURING_1:func.7">Head</a> <font color="Maroon" title="c2">s</font></span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:61"><span class="lab"><font color="Green" title="E44">A1</font></span></a>, <a class="ref" href="mcart_1.html#T64" title="MCART_1:th.64">MCART_1:64</a></span>
<br/>.= 
<span class="p1"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default">1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1</span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:61"><span class="lab"><font color="Green" title="E44">A1</font></span></a>, <a class="txt" href="#E4:61"><span class="lab"><font color="Green" title="E46">A3</font></span></a>, <a class="ref" href="turing_1.html#T30" target="_self" title="TURING_1:th.30">Th34</a>, <a class="ref" href="mcart_1.html#T64" title="MCART_1:th.64">MCART_1:64</a></span>
;<br/>
<a NAME="E25:61"/><span class="kw">then </span><span class="lab"><font color="Green" title="E57">A17</font></span>: 
 <a href="turing_1.html#K6" title="TURING_1:func.6">offset</a> <span class="p1">(<span class="default"><a href="turing_1.html#K8" title="TURING_1:func.8">TRAN</a> <font color="Maroon" title="c2">s</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="mcart_1.html#T64" title="MCART_1:th.64">MCART_1:64</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c17">s1</font> = <span class="p1"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c8">p1</font>,<font color="Maroon" title="c7">h1</font>,<font color="Maroon" title="c3">t</font></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c18">s3</font> = <span class="p1"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c8">p1</font>,<font color="Maroon" title="c7">h1</font>,<font color="Maroon" title="c3">t</font></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span> <a href="mcart_1.html#K7" title="MCART_1:func.7">`3</a>  as   <a href="turing_1.html#NM2" title="TURING_1:NM.2">Tape</a> of <a href="turing_1.html#K16" title="TURING_1:func.16">SuccTuring</a> ;<br/>
 <a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c2">s</font> <a href="mcart_1.html#K7" title="MCART_1:func.7">`3</a></span>)</span>,<span class="p1">(<span class="default"><a href="turing_1.html#K7" title="TURING_1:func.7">Head</a> <font color="Maroon" title="c2">s</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K8" title="TURING_1:func.8">TRAN</a> <font color="Maroon" title="c2">s</font></span>)</span> <a href="mcart_1.html#K6" title="MCART_1:func.6">`2</a></span>)</span>) = 
 <a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<font color="Maroon" title="c3">t</font>,<span class="p1">(<span class="default"><a href="turing_1.html#K7" title="TURING_1:func.7">Head</a> <font color="Maroon" title="c2">s</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K8" title="TURING_1:func.8">TRAN</a> <font color="Maroon" title="c2">s</font></span>)</span> <a href="mcart_1.html#K6" title="MCART_1:func.6">`2</a></span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:61"><span class="lab"><font color="Green" title="E44">A1</font></span></a>, <a class="ref" href="mcart_1.html#T64" title="MCART_1:th.64">MCART_1:64</a></span>
<br/>.= 
 <a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<font color="Maroon" title="c3">t</font>,<font color="Maroon" title="c4">h</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K8" title="TURING_1:func.8">TRAN</a> <font color="Maroon" title="c2">s</font></span>)</span> <a href="mcart_1.html#K6" title="MCART_1:func.6">`2</a></span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:61"><span class="lab"><font color="Green" title="E44">A1</font></span></a>, <a class="ref" href="mcart_1.html#T64" title="MCART_1:th.64">MCART_1:64</a></span>
<br/>.= 
 <a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<font color="Maroon" title="c3">t</font>,<font color="Maroon" title="c4">h</font>,<font color="Maroon" title="c1">F</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:61"><span class="lab"><font color="Green" title="E56">A16</font></span></a>, <a class="ref" href="mcart_1.html#T64" title="MCART_1:th.64">MCART_1:64</a></span>
<br/>.= 
<font color="Maroon" title="c3">t</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:61"><span class="lab"><font color="Green" title="E46">A3</font></span></a>, <a class="ref" href="turing_1.html#T24" target="_self" title="TURING_1:th.24">Th28</a></span>
;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E58">A18</font></span>:  <a href="turing_1.html#K9" title="TURING_1:func.9">Following</a> <font color="Maroon" title="c2">s</font> = 
<span class="p1"><a href="mcart_1.html#K3" title="MCART_1:func.3">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="turing_1.html#K8" title="TURING_1:func.8">TRAN</a> <font color="Maroon" title="c2">s</font></span>)</span> <a href="mcart_1.html#K5" title="MCART_1:func.5">`1</a></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="turing_1.html#K7" title="TURING_1:func.7">Head</a> <font color="Maroon" title="c2">s</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p3">(<span class="default"><a href="turing_1.html#K6" title="TURING_1:func.6">offset</a> <span class="p4">(<span class="default"><a href="turing_1.html#K8" title="TURING_1:func.8">TRAN</a> <font color="Maroon" title="c2">s</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c3">t</font></span><a href="mcart_1.html#K3" title="MCART_1:func.3">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:61"><span class="lab"><font color="Green" title="E44">A1</font></span></a>, <a class="txt" href="turing_1.html#E54"><span class="lab"><font color="Green" title="E42">Lm7</font></span></a></span>
<br/>.= 
<span class="p1"><a href="mcart_1.html#K3" title="MCART_1:func.3">[</a><span class="default">1,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="turing_1.html#K7" title="TURING_1:func.7">Head</a> <font color="Maroon" title="c2">s</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p3">(<span class="default"><a href="turing_1.html#K6" title="TURING_1:func.6">offset</a> <span class="p4">(<span class="default"><a href="turing_1.html#K8" title="TURING_1:func.8">TRAN</a> <font color="Maroon" title="c2">s</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c3">t</font></span><a href="mcart_1.html#K3" title="MCART_1:func.3">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:61"><span class="lab"><font color="Green" title="E56">A16</font></span></a>, <a class="ref" href="mcart_1.html#T64" title="MCART_1:th.64">MCART_1:64</a></span>
<br/>.= 
<span class="p1"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c8">p1</font>,<font color="Maroon" title="c7">h1</font>,<font color="Maroon" title="c3">t</font></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:61"><span class="lab"><font color="Green" title="E44">A1</font></span></a>, <a class="txt" href="#E25:61"><span class="lab"><font color="Green" title="E57">A17</font></span></a>, <a class="ref" href="mcart_1.html#T64" title="MCART_1:th.64">MCART_1:64</a></span>
;<br/>
<span class="lab"><font color="Green" title="E59">A19</font></span>:  <a href="turing_1.html#K8" title="TURING_1:func.8">TRAN</a> <span class="p1"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c8">p1</font>,<font color="Maroon" title="c7">h1</font>,<font color="Maroon" title="c3">t</font></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span> = 
<a href="turing_1.html#K15" title="TURING_1:func.15">Succ_Tran</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c8">p1</font>,<font color="Maroon" title="c7">h1</font>,<font color="Maroon" title="c3">t</font></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span> <a href="mcart_1.html#K5" title="MCART_1:func.5">`1</a></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c18">s3</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><a href="turing_1.html#K7" title="TURING_1:func.7">Head</a> <span class="p4"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c8">p1</font>,<font color="Maroon" title="c7">h1</font>,<font color="Maroon" title="c3">t</font></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span></span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="turing_1.html#D17" target="_self" title="TURING_1:def.17">Def18</a></span>
<br/>.= 
<a href="turing_1.html#K15" title="TURING_1:func.15">Succ_Tran</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">p1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c18">s3</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><a href="turing_1.html#K7" title="TURING_1:func.7">Head</a> <span class="p4"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c8">p1</font>,<font color="Maroon" title="c7">h1</font>,<font color="Maroon" title="c3">t</font></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span></span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="mcart_1.html#T64" title="MCART_1:th.64">MCART_1:64</a></span>
<br/>.= 
<a href="turing_1.html#K15" title="TURING_1:func.15">Succ_Tran</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">p1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><a href="turing_1.html#K7" title="TURING_1:func.7">Head</a> <span class="p4"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c8">p1</font>,<font color="Maroon" title="c7">h1</font>,<font color="Maroon" title="c3">t</font></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span></span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="mcart_1.html#T64" title="MCART_1:th.64">MCART_1:64</a></span>
<br/>.= 
<a href="turing_1.html#K15" title="TURING_1:func.15">Succ_Tran</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default">1,<span class="p2">(<span class="default"><font color="Maroon" title="c3">t</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">h1</font></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="mcart_1.html#T64" title="MCART_1:th.64">MCART_1:64</a></span>
<br/>.= 
<span class="p1"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default">1,1,1</span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:61"><span class="lab"><font color="Green" title="E45">A2</font></span></a>, <a class="txt" href="#E19:61"><span class="lab"><font color="Green" title="E52">A9</font></span></a>, <a class="txt" href="#E11:61"><span class="lab"><font color="Green" title="E50">A7</font></span></a>, <a class="ref" href="turing_1.html#T17" target="_self" title="TURING_1:th.17">Th20</a>, <a class="ref" href="turing_1.html#T30" target="_self" title="TURING_1:th.30">Th34</a></span>
;<br/>
<a NAME="E30:61"/><span class="kw">then </span><span class="lab"><font color="Green" title="E60">A20</font></span>: 
 <a href="turing_1.html#K6" title="TURING_1:func.6">offset</a> <span class="p1">(<span class="default"><a href="turing_1.html#K8" title="TURING_1:func.8">TRAN</a> <span class="p2"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c8">p1</font>,<font color="Maroon" title="c7">h1</font>,<font color="Maroon" title="c3">t</font></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="mcart_1.html#T64" title="MCART_1:th.64">MCART_1:64</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c19">p1</font> = 1 as   <a href="turing_1.html#NM1" title="TURING_1:NM.1">State</a> of <a href="turing_1.html#K16" title="TURING_1:func.16">SuccTuring</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="turing_1.html#E52"><span class="lab"><font color="Green" title="E40">Lm5</font></span></a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c20">s2</font> = <span class="p1"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c19">p1</font>,<font color="Maroon" title="c10">i2</font>,<span class="p2">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<font color="Maroon" title="c3">t</font>,<font color="Maroon" title="c7">h1</font>,<font color="Maroon" title="c14">T</font>)</span>)</span></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span>;<br/>
 <a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p1">(<span class="default"><span class="p2"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c8">p1</font>,<font color="Maroon" title="c7">h1</font>,<font color="Maroon" title="c3">t</font></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span> <a href="mcart_1.html#K7" title="MCART_1:func.7">`3</a></span>)</span>,<span class="p1">(<span class="default"><a href="turing_1.html#K7" title="TURING_1:func.7">Head</a> <span class="p2"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c8">p1</font>,<font color="Maroon" title="c7">h1</font>,<font color="Maroon" title="c3">t</font></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K8" title="TURING_1:func.8">TRAN</a> <span class="p3"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c8">p1</font>,<font color="Maroon" title="c7">h1</font>,<font color="Maroon" title="c3">t</font></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span></span>)</span> <a href="mcart_1.html#K6" title="MCART_1:func.6">`2</a></span>)</span>) = 
 <a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<font color="Maroon" title="c3">t</font>,<span class="p1">(<span class="default"><a href="turing_1.html#K7" title="TURING_1:func.7">Head</a> <span class="p2"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c8">p1</font>,<font color="Maroon" title="c7">h1</font>,<font color="Maroon" title="c3">t</font></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K8" title="TURING_1:func.8">TRAN</a> <span class="p3"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c8">p1</font>,<font color="Maroon" title="c7">h1</font>,<font color="Maroon" title="c3">t</font></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span></span>)</span> <a href="mcart_1.html#K6" title="MCART_1:func.6">`2</a></span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="mcart_1.html#T64" title="MCART_1:th.64">MCART_1:64</a></span>
<br/>.= 
 <a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<font color="Maroon" title="c3">t</font>,<font color="Maroon" title="c7">h1</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K8" title="TURING_1:func.8">TRAN</a> <span class="p3"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c8">p1</font>,<font color="Maroon" title="c7">h1</font>,<font color="Maroon" title="c3">t</font></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span></span>)</span> <a href="mcart_1.html#K6" title="MCART_1:func.6">`2</a></span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="mcart_1.html#T64" title="MCART_1:th.64">MCART_1:64</a></span>
<br/>.= 
 <a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<font color="Maroon" title="c3">t</font>,<font color="Maroon" title="c7">h1</font>,<font color="Maroon" title="c14">T</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:61"><span class="lab"><font color="Green" title="E59">A19</font></span></a>, <a class="ref" href="mcart_1.html#T64" title="MCART_1:th.64">MCART_1:64</a></span>
;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E61">A21</font></span>:  <a href="turing_1.html#K9" title="TURING_1:func.9">Following</a> <span class="p1"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c8">p1</font>,<font color="Maroon" title="c7">h1</font>,<font color="Maroon" title="c3">t</font></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span> = 
<span class="p1"><a href="mcart_1.html#K3" title="MCART_1:func.3">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="turing_1.html#K8" title="TURING_1:func.8">TRAN</a> <span class="p4"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c8">p1</font>,<font color="Maroon" title="c7">h1</font>,<font color="Maroon" title="c3">t</font></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span></span>)</span> <a href="mcart_1.html#K5" title="MCART_1:func.5">`1</a></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="turing_1.html#K7" title="TURING_1:func.7">Head</a> <span class="p4"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c8">p1</font>,<font color="Maroon" title="c7">h1</font>,<font color="Maroon" title="c3">t</font></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p3">(<span class="default"><a href="turing_1.html#K6" title="TURING_1:func.6">offset</a> <span class="p4">(<span class="default"><a href="turing_1.html#K8" title="TURING_1:func.8">TRAN</a> <span class="p5"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c8">p1</font>,<font color="Maroon" title="c7">h1</font>,<font color="Maroon" title="c3">t</font></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<font color="Maroon" title="c3">t</font>,<font color="Maroon" title="c7">h1</font>,<font color="Maroon" title="c14">T</font>)</span>)</span></span><a href="mcart_1.html#K3" title="MCART_1:func.3">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="turing_1.html#E54"><span class="lab"><font color="Green" title="E42">Lm7</font></span></a></span>
<br/>.= 
<span class="p1"><a href="mcart_1.html#K3" title="MCART_1:func.3">[</a><span class="default">1,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="turing_1.html#K7" title="TURING_1:func.7">Head</a> <span class="p4"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c8">p1</font>,<font color="Maroon" title="c7">h1</font>,<font color="Maroon" title="c3">t</font></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p3">(<span class="default"><a href="turing_1.html#K6" title="TURING_1:func.6">offset</a> <span class="p4">(<span class="default"><a href="turing_1.html#K8" title="TURING_1:func.8">TRAN</a> <span class="p5"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c8">p1</font>,<font color="Maroon" title="c7">h1</font>,<font color="Maroon" title="c3">t</font></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<font color="Maroon" title="c3">t</font>,<font color="Maroon" title="c7">h1</font>,<font color="Maroon" title="c14">T</font>)</span>)</span></span><a href="mcart_1.html#K3" title="MCART_1:func.3">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:61"><span class="lab"><font color="Green" title="E59">A19</font></span></a>, <a class="ref" href="mcart_1.html#T64" title="MCART_1:th.64">MCART_1:64</a></span>
<br/>.= 
<span class="p1"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c19">p1</font>,<font color="Maroon" title="c10">i2</font>,<span class="p2">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<font color="Maroon" title="c3">t</font>,<font color="Maroon" title="c7">h1</font>,<font color="Maroon" title="c14">T</font>)</span>)</span></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E30:61"><span class="lab"><font color="Green" title="E60">A20</font></span></a>, <a class="ref" href="mcart_1.html#T64" title="MCART_1:th.64">MCART_1:64</a></span>
;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c21">p3</font> = 3 as   <a href="turing_1.html#NM1" title="TURING_1:NM.1">State</a> of <a href="turing_1.html#K16" title="TURING_1:func.16">SuccTuring</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="turing_1.html#E52"><span class="lab"><font color="Green" title="E40">Lm5</font></span></a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c22">sn</font> = <span class="p1"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c19">p1</font>,<font color="Maroon" title="c11">nk</font>,<span class="p2">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<font color="Maroon" title="c3">t</font>,<font color="Maroon" title="c7">h1</font>,<font color="Maroon" title="c14">T</font>)</span>)</span></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c23">t2</font> =  <a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p1">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<font color="Maroon" title="c3">t</font>,<font color="Maroon" title="c7">h1</font>,<font color="Maroon" title="c14">T</font>)</span>)</span>,<font color="Maroon" title="c11">nk</font>,<font color="Maroon" title="c14">T</font>);<br/>
<span class="kw">set </span><font color="Maroon" title="c24">t3</font> =  <a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p1">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p2">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<font color="Maroon" title="c3">t</font>,<font color="Maroon" title="c7">h1</font>,<font color="Maroon" title="c14">T</font>)</span>)</span>,<font color="Maroon" title="c11">nk</font>,<font color="Maroon" title="c14">T</font>)</span>)</span>,<font color="Maroon" title="c13">n3</font>,<font color="Maroon" title="c1">F</font>);<br/>
<a NAME="E35:61"/>
(  the <a href="turing_1.html#U3" title="TURING_1:sel.3">Tran</a> of <a href="turing_1.html#K16" title="TURING_1:func.16">SuccTuring</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c19">p1</font>,1</span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c19">p1</font>,1,1</span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span> &amp; <font color="Maroon" title="c19">p1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  the <a href="turing_1.html#U5" title="TURING_1:sel.5">AcceptS</a> of <a href="turing_1.html#K16" title="TURING_1:func.16">SuccTuring</a> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="turing_1.html#D17" target="_self" title="TURING_1:def.17">Def18</a>, <a class="ref" href="turing_1.html#T30" target="_self" title="TURING_1:th.30">Th34</a></span>;<br/>
<a NAME="E36:61"/><span class="kw">then </span><span class="lab"><font color="Green" title="E62">A22</font></span>: 
<span class="p1">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <span class="p2"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c19">p1</font>,<font color="Maroon" title="c10">i2</font>,<span class="p3">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<font color="Maroon" title="c3">t</font>,<font color="Maroon" title="c7">h1</font>,<font color="Maroon" title="c14">T</font>)</span>)</span></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c5">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c19">p1</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">h</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c5">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<font color="Maroon" title="c3">t</font>,<font color="Maroon" title="c7">h1</font>,<font color="Maroon" title="c14">T</font>)</span>)</span></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:61"><span class="lab"><font color="Green" title="E55">A13</font></span></a>, <a class="txt" href="turing_1.html#E44"><span class="lab"><font color="Green" title="E34">Lm4</font></span></a></span>;<br/>
<a NAME="E37:61"/>
( <font color="Maroon" title="c7">h1</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">h</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c5">n</font> &amp; <font color="Maroon" title="c7">h1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">h1</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1 )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a>, <a class="ref" href="xreal_1.html#T29" title="XREAL_1:th.29">XREAL_1:29</a></span>;<br/>
<a NAME="E38:61"/><span class="kw">then </span><span class="lab"><font color="Green" title="E63">A23</font></span>: 
<font color="Maroon" title="c7">h1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">h1</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c5">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E39:61"/><span class="lab"><font color="Green" title="E64">A24</font></span>: 
( <span class="p1">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p2">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<font color="Maroon" title="c3">t</font>,<font color="Maroon" title="c7">h1</font>,<font color="Maroon" title="c14">T</font>)</span>)</span>,<font color="Maroon" title="c11">nk</font>,<font color="Maroon" title="c14">T</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; (  for <font color="Olive" title="b1">i</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>  st <font color="Maroon" title="c4">h</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">h</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 holds <br/><span class="p1">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p2">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<font color="Maroon" title="c3">t</font>,<font color="Maroon" title="c7">h1</font>,<font color="Maroon" title="c14">T</font>)</span>)</span>,<font color="Maroon" title="c11">nk</font>,<font color="Maroon" title="c14">T</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_9"><span class="kw">proof </span></a><div class="add">

<span class="kw">thus </span><a NAME="E1:61_9"/>
<span class="p1">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p2">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<font color="Maroon" title="c3">t</font>,<font color="Maroon" title="c7">h1</font>,<font color="Maroon" title="c14">T</font>)</span>)</span>,<font color="Maroon" title="c11">nk</font>,<font color="Maroon" title="c14">T</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:61"><span class="lab"><font color="Green" title="E52">A9</font></span></a>, <a class="txt" href="#E21:61"><span class="lab"><font color="Green" title="E54">A11</font></span></a>, <a class="txt" href="#E38:61"><span class="lab"><font color="Green" title="E63">A23</font></span></a>, <a class="ref" href="turing_1.html#T26" target="_self" title="TURING_1:th.26">Th30</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">i</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>  st <font color="Maroon" title="c4">h</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">h</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 holds <br/><span class="p1">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p2">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<font color="Maroon" title="c3">t</font>,<font color="Maroon" title="c7">h1</font>,<font color="Maroon" title="c14">T</font>)</span>)</span>,<font color="Maroon" title="c11">nk</font>,<font color="Maroon" title="c14">T</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1</span><br/>

<div><span class="kw">hereby </span> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span>
<div class="add"><span class="kw">let </span><font color="Maroon" title="c25">i</font> be   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">h</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c25">i</font> &amp; <font color="Maroon" title="c25">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">h</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 implies <span class="p1">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p2">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<font color="Maroon" title="c3">t</font>,<font color="Maroon" title="c7">h1</font>,<font color="Maroon" title="c14">T</font>)</span>)</span>,<font color="Maroon" title="c11">nk</font>,<font color="Maroon" title="c14">T</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:61_9_1"/><span class="lab"><font color="Green" title="E65">A25</font></span>: 
<font color="Maroon" title="c4">h</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c25">i</font>
 <span class="kw">and </span><br/><a NAME="E2:61_9_1"/><span class="lab"><font color="Green" title="E66">A26</font></span>: 
<font color="Maroon" title="c25">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">h</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p2">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<font color="Maroon" title="c3">t</font>,<font color="Maroon" title="c7">h1</font>,<font color="Maroon" title="c14">T</font>)</span>)</span>,<font color="Maroon" title="c11">nk</font>,<font color="Maroon" title="c14">T</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1</span><br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_9_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:61_9_1_1"/>
( <font color="Maroon" title="c25">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">h</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 or <font color="Maroon" title="c25">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">h</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_9_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:61_9_1_1_1"/><span class="lab"><font color="Green" title="E67">A27</font></span>: 
<font color="Maroon" title="c25">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">h</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p2">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<font color="Maroon" title="c3">t</font>,<font color="Maroon" title="c7">h1</font>,<font color="Maroon" title="c14">T</font>)</span>)</span>,<font color="Maroon" title="c11">nk</font>,<font color="Maroon" title="c14">T</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1</span><br/><div class="add"><a NAME="E2:61_9_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E68">A28</font></span>: 
<font color="Maroon" title="c25">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">h</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:61_9_1"><span class="lab"><font color="Green" title="E66">A26</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><span class="kw">thus </span><span class="p1">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p2">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<font color="Maroon" title="c3">t</font>,<font color="Maroon" title="c7">h1</font>,<font color="Maroon" title="c14">T</font>)</span>)</span>,<font color="Maroon" title="c11">nk</font>,<font color="Maroon" title="c14">T</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">i</font> = 
<span class="p1">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<font color="Maroon" title="c3">t</font>,<font color="Maroon" title="c7">h1</font>,<font color="Maroon" title="c14">T</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">i</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:61_9_1_1_1"><span class="lab"><font color="Green" title="E67">A27</font></span></a>, <a class="ref" href="turing_1.html#T26" target="_self" title="TURING_1:th.26">Th30</a></span>
<br/>.= 
1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:61"><span class="lab"><font color="Green" title="E54">A11</font></span></a>, <a class="txt" href="#E1:61_9_1"><span class="lab"><font color="Green" title="E65">A25</font></span></a>, <a class="txt" href="#E2:61_9_1_1_1"><span class="lab"><font color="Green" title="E68">A28</font></span></a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_9_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:61_9_1_1_2"/>
<font color="Maroon" title="c25">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">h</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p2">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<font color="Maroon" title="c3">t</font>,<font color="Maroon" title="c7">h1</font>,<font color="Maroon" title="c14">T</font>)</span>)</span>,<font color="Maroon" title="c11">nk</font>,<font color="Maroon" title="c14">T</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:61_9_1_1_2"/>
<span class="p1">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p2">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<font color="Maroon" title="c3">t</font>,<font color="Maroon" title="c7">h1</font>,<font color="Maroon" title="c14">T</font>)</span>)</span>,<font color="Maroon" title="c11">nk</font>,<font color="Maroon" title="c14">T</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="turing_1.html#T26" target="_self" title="TURING_1:th.26">Th30</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div>
<span class="kw">end;</span></div>


</div><span class="kw">end;</span></div>
<span class="kw">set </span><font color="Maroon" title="c25">sp3</font> = <span class="p1"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c21">p3</font>,<font color="Maroon" title="c11">nk</font>,<span class="p2">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p3">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p4">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<font color="Maroon" title="c3">t</font>,<font color="Maroon" title="c7">h1</font>,<font color="Maroon" title="c14">T</font>)</span>)</span>,<font color="Maroon" title="c11">nk</font>,<font color="Maroon" title="c14">T</font>)</span>)</span>,<font color="Maroon" title="c13">n3</font>,<font color="Maroon" title="c1">F</font>)</span>)</span></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c26">sm</font> = <span class="p1"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c9">p2</font>,<font color="Maroon" title="c13">n3</font>,<span class="p2">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p3">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<font color="Maroon" title="c3">t</font>,<font color="Maroon" title="c7">h1</font>,<font color="Maroon" title="c14">T</font>)</span>)</span>,<font color="Maroon" title="c11">nk</font>,<font color="Maroon" title="c14">T</font>)</span>)</span></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c27">sm3</font> = <span class="p1"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c9">p2</font>,<font color="Maroon" title="c13">n3</font>,<span class="p2">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p3">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<font color="Maroon" title="c3">t</font>,<font color="Maroon" title="c7">h1</font>,<font color="Maroon" title="c14">T</font>)</span>)</span>,<font color="Maroon" title="c11">nk</font>,<font color="Maroon" title="c14">T</font>)</span>)</span></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span> <a href="mcart_1.html#K7" title="MCART_1:func.7">`3</a>  as   <a href="turing_1.html#NM2" title="TURING_1:NM.2">Tape</a> of <a href="turing_1.html#K16" title="TURING_1:func.16">SuccTuring</a> ;<br/>
<a NAME="E41:61"/><span class="lab"><font color="Green" title="E65">A29</font></span>: 
 the <a href="turing_1.html#U1" title="TURING_1:sel.1">Symbols</a> of <a href="turing_1.html#K16" title="TURING_1:func.16">SuccTuring</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1</span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="turing_1.html#D17" target="_self" title="TURING_1:def.17">Def18</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E66">A30</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="61_10"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_10_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:61_10_1"/>
( <span class="p1">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p2">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<font color="Maroon" title="c3">t</font>,<font color="Maroon" title="c7">h1</font>,<font color="Maroon" title="c14">T</font>)</span>)</span>,<font color="Maroon" title="c11">nk</font>,<font color="Maroon" title="c14">T</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">n3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <span class="p1">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p2">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<font color="Maroon" title="c3">t</font>,<font color="Maroon" title="c7">h1</font>,<font color="Maroon" title="c14">T</font>)</span>)</span>,<font color="Maroon" title="c11">nk</font>,<font color="Maroon" title="c14">T</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">n3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E41:61"><span class="lab"><font color="Green" title="E65">A29</font></span></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_10_1_1"><span class="kw">suppose </span></a><a NAME="E1:61_10_1_1"/>
<span class="p1">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p2">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<font color="Maroon" title="c3">t</font>,<font color="Maroon" title="c7">h1</font>,<font color="Maroon" title="c14">T</font>)</span>)</span>,<font color="Maroon" title="c11">nk</font>,<font color="Maroon" title="c14">T</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">n3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <a href="turing_1.html#K15" title="TURING_1:func.15">Succ_Tran</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default">2,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p4">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<font color="Maroon" title="c3">t</font>,<font color="Maroon" title="c7">h1</font>,<font color="Maroon" title="c14">T</font>)</span>)</span>,<font color="Maroon" title="c11">nk</font>,<font color="Maroon" title="c14">T</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">n3</font></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c21">p3</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:61_10_1_1"/>
<a href="turing_1.html#K15" title="TURING_1:func.15">Succ_Tran</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default">2,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p4">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<font color="Maroon" title="c3">t</font>,<font color="Maroon" title="c7">h1</font>,<font color="Maroon" title="c14">T</font>)</span>)</span>,<font color="Maroon" title="c11">nk</font>,<font color="Maroon" title="c14">T</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">n3</font></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c21">p3</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="turing_1.html#T30" target="_self" title="TURING_1:th.30">Th34</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_10_1_2"><span class="kw">suppose </span></a><a NAME="E1:61_10_1_2"/>
<span class="p1">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p2">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<font color="Maroon" title="c3">t</font>,<font color="Maroon" title="c7">h1</font>,<font color="Maroon" title="c14">T</font>)</span>)</span>,<font color="Maroon" title="c11">nk</font>,<font color="Maroon" title="c14">T</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">n3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <a href="turing_1.html#K15" title="TURING_1:func.15">Succ_Tran</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default">2,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p4">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<font color="Maroon" title="c3">t</font>,<font color="Maroon" title="c7">h1</font>,<font color="Maroon" title="c14">T</font>)</span>)</span>,<font color="Maroon" title="c11">nk</font>,<font color="Maroon" title="c14">T</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">n3</font></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c21">p3</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:61_10_1_2"/>
<a href="turing_1.html#K15" title="TURING_1:func.15">Succ_Tran</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default">2,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p4">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<font color="Maroon" title="c3">t</font>,<font color="Maroon" title="c7">h1</font>,<font color="Maroon" title="c14">T</font>)</span>)</span>,<font color="Maroon" title="c11">nk</font>,<font color="Maroon" title="c14">T</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">n3</font></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c21">p3</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="turing_1.html#T30" target="_self" title="TURING_1:th.30">Th34</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<span class="lab"><font color="Green" title="E67">A31</font></span>:  <a href="turing_1.html#K8" title="TURING_1:func.8">TRAN</a> <span class="p1"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c9">p2</font>,<font color="Maroon" title="c13">n3</font>,<span class="p2">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p3">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<font color="Maroon" title="c3">t</font>,<font color="Maroon" title="c7">h1</font>,<font color="Maroon" title="c14">T</font>)</span>)</span>,<font color="Maroon" title="c11">nk</font>,<font color="Maroon" title="c14">T</font>)</span>)</span></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span> = 
<a href="turing_1.html#K15" title="TURING_1:func.15">Succ_Tran</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c9">p2</font>,<font color="Maroon" title="c13">n3</font>,<span class="p4">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p5">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<font color="Maroon" title="c3">t</font>,<font color="Maroon" title="c7">h1</font>,<font color="Maroon" title="c14">T</font>)</span>)</span>,<font color="Maroon" title="c11">nk</font>,<font color="Maroon" title="c14">T</font>)</span>)</span></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span> <a href="mcart_1.html#K5" title="MCART_1:func.5">`1</a></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c27">sm3</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><a href="turing_1.html#K7" title="TURING_1:func.7">Head</a> <span class="p4"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c9">p2</font>,<font color="Maroon" title="c13">n3</font>,<span class="p5">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p0">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<font color="Maroon" title="c3">t</font>,<font color="Maroon" title="c7">h1</font>,<font color="Maroon" title="c14">T</font>)</span>)</span>,<font color="Maroon" title="c11">nk</font>,<font color="Maroon" title="c14">T</font>)</span>)</span></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span></span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="turing_1.html#D17" target="_self" title="TURING_1:def.17">Def18</a></span>
<br/>.= 
<a href="turing_1.html#K15" title="TURING_1:func.15">Succ_Tran</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">2,<span class="p2">(<span class="default"><font color="Maroon" title="c27">sm3</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><a href="turing_1.html#K7" title="TURING_1:func.7">Head</a> <span class="p4"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c9">p2</font>,<font color="Maroon" title="c13">n3</font>,<span class="p5">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p0">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<font color="Maroon" title="c3">t</font>,<font color="Maroon" title="c7">h1</font>,<font color="Maroon" title="c14">T</font>)</span>)</span>,<font color="Maroon" title="c11">nk</font>,<font color="Maroon" title="c14">T</font>)</span>)</span></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span></span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="mcart_1.html#T64" title="MCART_1:th.64">MCART_1:64</a></span>
<br/>.= 
<a href="turing_1.html#K15" title="TURING_1:func.15">Succ_Tran</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">2,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p4">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<font color="Maroon" title="c3">t</font>,<font color="Maroon" title="c7">h1</font>,<font color="Maroon" title="c14">T</font>)</span>)</span>,<font color="Maroon" title="c11">nk</font>,<font color="Maroon" title="c14">T</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><a href="turing_1.html#K7" title="TURING_1:func.7">Head</a> <span class="p4"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c9">p2</font>,<font color="Maroon" title="c13">n3</font>,<span class="p5">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p0">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<font color="Maroon" title="c3">t</font>,<font color="Maroon" title="c7">h1</font>,<font color="Maroon" title="c14">T</font>)</span>)</span>,<font color="Maroon" title="c11">nk</font>,<font color="Maroon" title="c14">T</font>)</span>)</span></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span></span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="mcart_1.html#T64" title="MCART_1:th.64">MCART_1:64</a></span>
<br/>.= 
<span class="p1"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c21">p3</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E42:61"><span class="lab"><font color="Green" title="E66">A30</font></span></a>, <a class="ref" href="mcart_1.html#T64" title="MCART_1:th.64">MCART_1:64</a></span>
;<br/>
<a NAME="E44:61"/><span class="kw">then </span><span class="lab"><font color="Green" title="E68">A32</font></span>: 
 <a href="turing_1.html#K6" title="TURING_1:func.6">offset</a> <span class="p1">(<span class="default"><a href="turing_1.html#K8" title="TURING_1:func.8">TRAN</a> <span class="p2"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c9">p2</font>,<font color="Maroon" title="c13">n3</font>,<span class="p3">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p4">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<font color="Maroon" title="c3">t</font>,<font color="Maroon" title="c7">h1</font>,<font color="Maroon" title="c14">T</font>)</span>)</span>,<font color="Maroon" title="c11">nk</font>,<font color="Maroon" title="c14">T</font>)</span>)</span></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="mcart_1.html#T64" title="MCART_1:th.64">MCART_1:64</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c28">j1</font> = <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c5">n</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c29">sp5</font> = <span class="p1"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c21">p3</font>,<font color="Maroon" title="c12">hh</font>,<span class="p2">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p3">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p4">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<font color="Maroon" title="c3">t</font>,<font color="Maroon" title="c7">h1</font>,<font color="Maroon" title="c14">T</font>)</span>)</span>,<font color="Maroon" title="c11">nk</font>,<font color="Maroon" title="c14">T</font>)</span>)</span>,<font color="Maroon" title="c13">n3</font>,<font color="Maroon" title="c1">F</font>)</span>)</span></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span>;<br/>
<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ] <span class="kw">means </span>( <font color="Maroon" title="c4">h</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> $1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">nk</font> implies <span class="p1">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <span class="p2"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c21">p3</font>,<font color="Maroon" title="c11">nk</font>,<span class="p3">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p4">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p5">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<font color="Maroon" title="c3">t</font>,<font color="Maroon" title="c7">h1</font>,<font color="Maroon" title="c14">T</font>)</span>)</span>,<font color="Maroon" title="c11">nk</font>,<font color="Maroon" title="c14">T</font>)</span>)</span>,<font color="Maroon" title="c13">n3</font>,<font color="Maroon" title="c1">F</font>)</span>)</span></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default">3,<span class="p2">(<span class="default"><font color="Maroon" title="c11">nk</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> $1</span>)</span>,<span class="p2">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p3">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p4">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<font color="Maroon" title="c3">t</font>,<font color="Maroon" title="c7">h1</font>,<font color="Maroon" title="c14">T</font>)</span>)</span>,<font color="Maroon" title="c11">nk</font>,<font color="Maroon" title="c14">T</font>)</span>)</span>,<font color="Maroon" title="c13">n3</font>,<font color="Maroon" title="c1">F</font>)</span>)</span></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span> );<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c30">sn3</font> = <span class="p1"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c19">p1</font>,<font color="Maroon" title="c11">nk</font>,<span class="p2">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<font color="Maroon" title="c3">t</font>,<font color="Maroon" title="c7">h1</font>,<font color="Maroon" title="c14">T</font>)</span>)</span></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span> <a href="mcart_1.html#K7" title="MCART_1:func.7">`3</a>  as   <a href="turing_1.html#NM2" title="TURING_1:NM.2">Tape</a> of <a href="turing_1.html#K16" title="TURING_1:func.16">SuccTuring</a> ;<br/>
<a NAME="E46:61"/><span class="lab"><font color="Green" title="E69">A33</font></span>: 
<font color="Maroon" title="c4">h</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">nk</font>
 
;<br/>
<span class="kw">set </span><font color="Maroon" title="c31">Rs</font> = <span class="p1">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <font color="Maroon" title="c2">s</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span>;<br/>
<span class="lab"><font color="Green" title="E70">A34</font></span>:  <a href="turing_1.html#K8" title="TURING_1:func.8">TRAN</a> <span class="p1"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c19">p1</font>,<font color="Maroon" title="c11">nk</font>,<span class="p2">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<font color="Maroon" title="c3">t</font>,<font color="Maroon" title="c7">h1</font>,<font color="Maroon" title="c14">T</font>)</span>)</span></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span> = 
<a href="turing_1.html#K15" title="TURING_1:func.15">Succ_Tran</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c19">p1</font>,<font color="Maroon" title="c11">nk</font>,<span class="p4">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<font color="Maroon" title="c3">t</font>,<font color="Maroon" title="c7">h1</font>,<font color="Maroon" title="c14">T</font>)</span>)</span></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span> <a href="mcart_1.html#K5" title="MCART_1:func.5">`1</a></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c30">sn3</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><a href="turing_1.html#K7" title="TURING_1:func.7">Head</a> <span class="p4"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c19">p1</font>,<font color="Maroon" title="c11">nk</font>,<span class="p5">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<font color="Maroon" title="c3">t</font>,<font color="Maroon" title="c7">h1</font>,<font color="Maroon" title="c14">T</font>)</span>)</span></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span></span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="turing_1.html#D17" target="_self" title="TURING_1:def.17">Def18</a></span>
<br/>.= 
<a href="turing_1.html#K15" title="TURING_1:func.15">Succ_Tran</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c19">p1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c30">sn3</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><a href="turing_1.html#K7" title="TURING_1:func.7">Head</a> <span class="p4"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c19">p1</font>,<font color="Maroon" title="c11">nk</font>,<span class="p5">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<font color="Maroon" title="c3">t</font>,<font color="Maroon" title="c7">h1</font>,<font color="Maroon" title="c14">T</font>)</span>)</span></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span></span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="mcart_1.html#T64" title="MCART_1:th.64">MCART_1:64</a></span>
<br/>.= 
<a href="turing_1.html#K15" title="TURING_1:func.15">Succ_Tran</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c19">p1</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<font color="Maroon" title="c3">t</font>,<font color="Maroon" title="c7">h1</font>,<font color="Maroon" title="c14">T</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><a href="turing_1.html#K7" title="TURING_1:func.7">Head</a> <span class="p4"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c19">p1</font>,<font color="Maroon" title="c11">nk</font>,<span class="p5">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<font color="Maroon" title="c3">t</font>,<font color="Maroon" title="c7">h1</font>,<font color="Maroon" title="c14">T</font>)</span>)</span></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span></span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="mcart_1.html#T64" title="MCART_1:th.64">MCART_1:64</a></span>
<br/>.= 
<span class="p1"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default">2,1,1</span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:61"><span class="lab"><font color="Green" title="E54">A11</font></span></a>, <a class="ref" href="turing_1.html#T30" target="_self" title="TURING_1:th.30">Th34</a>, <a class="ref" href="mcart_1.html#T64" title="MCART_1:th.64">MCART_1:64</a></span>
;<br/>
<a NAME="E48:61"/><span class="kw">then </span><span class="lab"><font color="Green" title="E71">A35</font></span>: 
 <a href="turing_1.html#K6" title="TURING_1:func.6">offset</a> <span class="p1">(<span class="default"><a href="turing_1.html#K8" title="TURING_1:func.8">TRAN</a> <span class="p2"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c19">p1</font>,<font color="Maroon" title="c11">nk</font>,<span class="p3">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<font color="Maroon" title="c3">t</font>,<font color="Maroon" title="c7">h1</font>,<font color="Maroon" title="c14">T</font>)</span>)</span></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="mcart_1.html#T64" title="MCART_1:th.64">MCART_1:64</a></span>;<br/>
<a NAME="E49:61"/><span class="lab"><font color="Green" title="E72">A36</font></span>: 
<font color="Maroon" title="c4">h</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">h1</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c5">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:61"><span class="lab"><font color="Green" title="E52">A9</font></span></a>, <a class="txt" href="#E38:61"><span class="lab"><font color="Green" title="E63">A23</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E50:61"/><span class="lab"><font color="Green" title="E73">A37</font></span>: 
( <span class="p1">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p2">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p3">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<font color="Maroon" title="c3">t</font>,<font color="Maroon" title="c7">h1</font>,<font color="Maroon" title="c14">T</font>)</span>)</span>,<font color="Maroon" title="c11">nk</font>,<font color="Maroon" title="c14">T</font>)</span>)</span>,<font color="Maroon" title="c13">n3</font>,<font color="Maroon" title="c1">F</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <span class="p1">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p2">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p3">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<font color="Maroon" title="c3">t</font>,<font color="Maroon" title="c7">h1</font>,<font color="Maroon" title="c14">T</font>)</span>)</span>,<font color="Maroon" title="c11">nk</font>,<font color="Maroon" title="c14">T</font>)</span>)</span>,<font color="Maroon" title="c13">n3</font>,<font color="Maroon" title="c1">F</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">h</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; (  for <font color="Olive" title="b1">k</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>  st <font color="Maroon" title="c4">h</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">k</font> &amp; <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">h</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 holds <br/><span class="p1">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p2">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p3">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<font color="Maroon" title="c3">t</font>,<font color="Maroon" title="c7">h1</font>,<font color="Maroon" title="c14">T</font>)</span>)</span>,<font color="Maroon" title="c11">nk</font>,<font color="Maroon" title="c14">T</font>)</span>)</span>,<font color="Maroon" title="c13">n3</font>,<font color="Maroon" title="c1">F</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_13"><span class="kw">proof </span></a><div class="add">

<span class="kw">thus </span><a NAME="E1:61_13"/>
<span class="p1">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p2">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p3">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<font color="Maroon" title="c3">t</font>,<font color="Maroon" title="c7">h1</font>,<font color="Maroon" title="c14">T</font>)</span>)</span>,<font color="Maroon" title="c11">nk</font>,<font color="Maroon" title="c14">T</font>)</span>)</span>,<font color="Maroon" title="c13">n3</font>,<font color="Maroon" title="c1">F</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E49:61"><span class="lab"><font color="Green" title="E72">A36</font></span></a>, <a class="txt" href="#E39:61"><span class="lab"><font color="Green" title="E64">A24</font></span></a>, <a class="txt" href="#E7:61"><span class="lab"><font color="Green" title="E47">A4</font></span></a>, <a class="ref" href="turing_1.html#T26" target="_self" title="TURING_1:th.26">Th30</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <span class="p1">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p2">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p3">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<font color="Maroon" title="c3">t</font>,<font color="Maroon" title="c7">h1</font>,<font color="Maroon" title="c14">T</font>)</span>)</span>,<font color="Maroon" title="c11">nk</font>,<font color="Maroon" title="c14">T</font>)</span>)</span>,<font color="Maroon" title="c13">n3</font>,<font color="Maroon" title="c1">F</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">h</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; (  for <font color="Olive" title="b1">k</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>  st <font color="Maroon" title="c4">h</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">k</font> &amp; <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">h</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 holds <br/><span class="p1">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p2">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p3">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<font color="Maroon" title="c3">t</font>,<font color="Maroon" title="c7">h1</font>,<font color="Maroon" title="c14">T</font>)</span>)</span>,<font color="Maroon" title="c11">nk</font>,<font color="Maroon" title="c14">T</font>)</span>)</span>,<font color="Maroon" title="c13">n3</font>,<font color="Maroon" title="c1">F</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) )</span><br/>

<span class="kw">thus </span><a NAME="E2:61_13"/>
<span class="p1">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p2">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p3">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<font color="Maroon" title="c3">t</font>,<font color="Maroon" title="c7">h1</font>,<font color="Maroon" title="c14">T</font>)</span>)</span>,<font color="Maroon" title="c11">nk</font>,<font color="Maroon" title="c14">T</font>)</span>)</span>,<font color="Maroon" title="c13">n3</font>,<font color="Maroon" title="c1">F</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">h</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="turing_1.html#T26" target="_self" title="TURING_1:th.26">Th30</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">k</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>  st <font color="Maroon" title="c4">h</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">k</font> &amp; <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">h</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 holds <br/><span class="p1">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p2">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p3">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<font color="Maroon" title="c3">t</font>,<font color="Maroon" title="c7">h1</font>,<font color="Maroon" title="c14">T</font>)</span>)</span>,<font color="Maroon" title="c11">nk</font>,<font color="Maroon" title="c14">T</font>)</span>)</span>,<font color="Maroon" title="c13">n3</font>,<font color="Maroon" title="c1">F</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1</span><br/>

<div><span class="kw">hereby </span> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span>
<div class="add"><span class="kw">let </span><font color="Maroon" title="c32">i</font> be   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">h</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c32">i</font> &amp; <font color="Maroon" title="c32">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">h</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 implies <span class="p1">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p2">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p3">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<font color="Maroon" title="c3">t</font>,<font color="Maroon" title="c7">h1</font>,<font color="Maroon" title="c14">T</font>)</span>)</span>,<font color="Maroon" title="c11">nk</font>,<font color="Maroon" title="c14">T</font>)</span>)</span>,<font color="Maroon" title="c13">n3</font>,<font color="Maroon" title="c1">F</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c32">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:61_13_1"/><span class="lab"><font color="Green" title="E74">A38</font></span>: 
<font color="Maroon" title="c4">h</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c32">i</font>
 <span class="kw">and </span><br/><a NAME="E2:61_13_1"/><span class="lab"><font color="Green" title="E75">A39</font></span>: 
<font color="Maroon" title="c32">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">h</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p2">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p3">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<font color="Maroon" title="c3">t</font>,<font color="Maroon" title="c7">h1</font>,<font color="Maroon" title="c14">T</font>)</span>)</span>,<font color="Maroon" title="c11">nk</font>,<font color="Maroon" title="c14">T</font>)</span>)</span>,<font color="Maroon" title="c13">n3</font>,<font color="Maroon" title="c1">F</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c32">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1</span><br/><a NAME="E3:61_13_1"/>
<font color="Maroon" title="c32">i</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">h</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:61_13_1"><span class="lab"><font color="Green" title="E75">A39</font></span></a>, <a class="ref" href="int_1.html#T7" title="INT_1:th.7">INT_1:7</a></span>;<br/><a NAME="E4:61_13_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E76">A40</font></span>: 
<font color="Maroon" title="c32">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">h</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T19" title="XREAL_1:th.19">XREAL_1:19</a></span>;<br/><span class="kw">thus </span><span class="p1">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p2">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p3">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<font color="Maroon" title="c3">t</font>,<font color="Maroon" title="c7">h1</font>,<font color="Maroon" title="c14">T</font>)</span>)</span>,<font color="Maroon" title="c11">nk</font>,<font color="Maroon" title="c14">T</font>)</span>)</span>,<font color="Maroon" title="c13">n3</font>,<font color="Maroon" title="c1">F</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c32">i</font> = 
<span class="p1">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p2">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<font color="Maroon" title="c3">t</font>,<font color="Maroon" title="c7">h1</font>,<font color="Maroon" title="c14">T</font>)</span>)</span>,<font color="Maroon" title="c11">nk</font>,<font color="Maroon" title="c14">T</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c32">i</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:61_13_1"><span class="lab"><font color="Green" title="E75">A39</font></span></a>, <a class="ref" href="turing_1.html#T26" target="_self" title="TURING_1:th.26">Th30</a></span>
<br/>.= 
1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E39:61"><span class="lab"><font color="Green" title="E64">A24</font></span></a>, <a class="txt" href="#E1:61_13_1"><span class="lab"><font color="Green" title="E74">A38</font></span></a>, <a class="txt" href="#E4:61_13_1"><span class="lab"><font color="Green" title="E76">A40</font></span></a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div>
<span class="kw">end;</span></div>


</div><span class="kw">end;</span></div>
<a NAME="E51:61"/><span class="kw">then </span><span class="lab"><font color="Green" title="E74">A41</font></span>: 
 <a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p1">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p2">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<font color="Maroon" title="c3">t</font>,<font color="Maroon" title="c7">h1</font>,<font color="Maroon" title="c14">T</font>)</span>)</span>,<font color="Maroon" title="c11">nk</font>,<font color="Maroon" title="c14">T</font>)</span>)</span>,<font color="Maroon" title="c13">n3</font>,<font color="Maroon" title="c1">F</font>) <a href="turing_1.html#R1" title="TURING_1:pred.1">is_1_between</a> <font color="Maroon" title="c4">h</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">h</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="turing_1.html#D12" target="_self" title="TURING_1:def.12">Def13</a></span>;<br/>
 <a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p1">(<span class="default"><span class="p2"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c9">p2</font>,<font color="Maroon" title="c13">n3</font>,<span class="p3">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p4">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<font color="Maroon" title="c3">t</font>,<font color="Maroon" title="c7">h1</font>,<font color="Maroon" title="c14">T</font>)</span>)</span>,<font color="Maroon" title="c11">nk</font>,<font color="Maroon" title="c14">T</font>)</span>)</span></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span> <a href="mcart_1.html#K7" title="MCART_1:func.7">`3</a></span>)</span>,<span class="p1">(<span class="default"><a href="turing_1.html#K7" title="TURING_1:func.7">Head</a> <span class="p2"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c9">p2</font>,<font color="Maroon" title="c13">n3</font>,<span class="p3">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p4">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<font color="Maroon" title="c3">t</font>,<font color="Maroon" title="c7">h1</font>,<font color="Maroon" title="c14">T</font>)</span>)</span>,<font color="Maroon" title="c11">nk</font>,<font color="Maroon" title="c14">T</font>)</span>)</span></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K8" title="TURING_1:func.8">TRAN</a> <span class="p3"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c9">p2</font>,<font color="Maroon" title="c13">n3</font>,<span class="p4">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p5">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<font color="Maroon" title="c3">t</font>,<font color="Maroon" title="c7">h1</font>,<font color="Maroon" title="c14">T</font>)</span>)</span>,<font color="Maroon" title="c11">nk</font>,<font color="Maroon" title="c14">T</font>)</span>)</span></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span></span>)</span> <a href="mcart_1.html#K6" title="MCART_1:func.6">`2</a></span>)</span>) = 
 <a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p1">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p2">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<font color="Maroon" title="c3">t</font>,<font color="Maroon" title="c7">h1</font>,<font color="Maroon" title="c14">T</font>)</span>)</span>,<font color="Maroon" title="c11">nk</font>,<font color="Maroon" title="c14">T</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="turing_1.html#K7" title="TURING_1:func.7">Head</a> <span class="p2"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c9">p2</font>,<font color="Maroon" title="c13">n3</font>,<span class="p3">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p4">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<font color="Maroon" title="c3">t</font>,<font color="Maroon" title="c7">h1</font>,<font color="Maroon" title="c14">T</font>)</span>)</span>,<font color="Maroon" title="c11">nk</font>,<font color="Maroon" title="c14">T</font>)</span>)</span></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K8" title="TURING_1:func.8">TRAN</a> <span class="p3"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c9">p2</font>,<font color="Maroon" title="c13">n3</font>,<span class="p4">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p5">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<font color="Maroon" title="c3">t</font>,<font color="Maroon" title="c7">h1</font>,<font color="Maroon" title="c14">T</font>)</span>)</span>,<font color="Maroon" title="c11">nk</font>,<font color="Maroon" title="c14">T</font>)</span>)</span></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span></span>)</span> <a href="mcart_1.html#K6" title="MCART_1:func.6">`2</a></span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="mcart_1.html#T64" title="MCART_1:th.64">MCART_1:64</a></span>
<br/>.= 
 <a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p1">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p2">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<font color="Maroon" title="c3">t</font>,<font color="Maroon" title="c7">h1</font>,<font color="Maroon" title="c14">T</font>)</span>)</span>,<font color="Maroon" title="c11">nk</font>,<font color="Maroon" title="c14">T</font>)</span>)</span>,<font color="Maroon" title="c13">n3</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K8" title="TURING_1:func.8">TRAN</a> <span class="p3"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c9">p2</font>,<font color="Maroon" title="c13">n3</font>,<span class="p4">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p5">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<font color="Maroon" title="c3">t</font>,<font color="Maroon" title="c7">h1</font>,<font color="Maroon" title="c14">T</font>)</span>)</span>,<font color="Maroon" title="c11">nk</font>,<font color="Maroon" title="c14">T</font>)</span>)</span></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span></span>)</span> <a href="mcart_1.html#K6" title="MCART_1:func.6">`2</a></span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="mcart_1.html#T64" title="MCART_1:th.64">MCART_1:64</a></span>
<br/>.= 
 <a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p1">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p2">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<font color="Maroon" title="c3">t</font>,<font color="Maroon" title="c7">h1</font>,<font color="Maroon" title="c14">T</font>)</span>)</span>,<font color="Maroon" title="c11">nk</font>,<font color="Maroon" title="c14">T</font>)</span>)</span>,<font color="Maroon" title="c13">n3</font>,<font color="Maroon" title="c1">F</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E43:61"><span class="lab"><font color="Green" title="E67">A31</font></span></a>, <a class="ref" href="mcart_1.html#T64" title="MCART_1:th.64">MCART_1:64</a></span>
;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E75">A42</font></span>:  <a href="turing_1.html#K9" title="TURING_1:func.9">Following</a> <span class="p1"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c9">p2</font>,<font color="Maroon" title="c13">n3</font>,<span class="p2">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p3">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<font color="Maroon" title="c3">t</font>,<font color="Maroon" title="c7">h1</font>,<font color="Maroon" title="c14">T</font>)</span>)</span>,<font color="Maroon" title="c11">nk</font>,<font color="Maroon" title="c14">T</font>)</span>)</span></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span> = 
<span class="p1"><a href="mcart_1.html#K3" title="MCART_1:func.3">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="turing_1.html#K8" title="TURING_1:func.8">TRAN</a> <span class="p4"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c9">p2</font>,<font color="Maroon" title="c13">n3</font>,<span class="p5">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p0">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<font color="Maroon" title="c3">t</font>,<font color="Maroon" title="c7">h1</font>,<font color="Maroon" title="c14">T</font>)</span>)</span>,<font color="Maroon" title="c11">nk</font>,<font color="Maroon" title="c14">T</font>)</span>)</span></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span></span>)</span> <a href="mcart_1.html#K5" title="MCART_1:func.5">`1</a></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="turing_1.html#K7" title="TURING_1:func.7">Head</a> <span class="p4"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c9">p2</font>,<font color="Maroon" title="c13">n3</font>,<span class="p5">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p0">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<font color="Maroon" title="c3">t</font>,<font color="Maroon" title="c7">h1</font>,<font color="Maroon" title="c14">T</font>)</span>)</span>,<font color="Maroon" title="c11">nk</font>,<font color="Maroon" title="c14">T</font>)</span>)</span></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p3">(<span class="default"><a href="turing_1.html#K6" title="TURING_1:func.6">offset</a> <span class="p4">(<span class="default"><a href="turing_1.html#K8" title="TURING_1:func.8">TRAN</a> <span class="p5"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c9">p2</font>,<font color="Maroon" title="c13">n3</font>,<span class="p0">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p1">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<font color="Maroon" title="c3">t</font>,<font color="Maroon" title="c7">h1</font>,<font color="Maroon" title="c14">T</font>)</span>)</span>,<font color="Maroon" title="c11">nk</font>,<font color="Maroon" title="c14">T</font>)</span>)</span></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p3">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p4">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<font color="Maroon" title="c3">t</font>,<font color="Maroon" title="c7">h1</font>,<font color="Maroon" title="c14">T</font>)</span>)</span>,<font color="Maroon" title="c11">nk</font>,<font color="Maroon" title="c14">T</font>)</span>)</span>,<font color="Maroon" title="c13">n3</font>,<font color="Maroon" title="c1">F</font>)</span>)</span></span><a href="mcart_1.html#K3" title="MCART_1:func.3">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="turing_1.html#E54"><span class="lab"><font color="Green" title="E42">Lm7</font></span></a></span>
<br/>.= 
<span class="p1"><a href="mcart_1.html#K3" title="MCART_1:func.3">[</a><span class="default"><font color="Maroon" title="c21">p3</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="turing_1.html#K7" title="TURING_1:func.7">Head</a> <span class="p4"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c9">p2</font>,<font color="Maroon" title="c13">n3</font>,<span class="p5">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p0">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<font color="Maroon" title="c3">t</font>,<font color="Maroon" title="c7">h1</font>,<font color="Maroon" title="c14">T</font>)</span>)</span>,<font color="Maroon" title="c11">nk</font>,<font color="Maroon" title="c14">T</font>)</span>)</span></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p3">(<span class="default"><a href="turing_1.html#K6" title="TURING_1:func.6">offset</a> <span class="p4">(<span class="default"><a href="turing_1.html#K8" title="TURING_1:func.8">TRAN</a> <span class="p5"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c9">p2</font>,<font color="Maroon" title="c13">n3</font>,<span class="p0">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p1">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<font color="Maroon" title="c3">t</font>,<font color="Maroon" title="c7">h1</font>,<font color="Maroon" title="c14">T</font>)</span>)</span>,<font color="Maroon" title="c11">nk</font>,<font color="Maroon" title="c14">T</font>)</span>)</span></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p3">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p4">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<font color="Maroon" title="c3">t</font>,<font color="Maroon" title="c7">h1</font>,<font color="Maroon" title="c14">T</font>)</span>)</span>,<font color="Maroon" title="c11">nk</font>,<font color="Maroon" title="c14">T</font>)</span>)</span>,<font color="Maroon" title="c13">n3</font>,<font color="Maroon" title="c1">F</font>)</span>)</span></span><a href="mcart_1.html#K3" title="MCART_1:func.3">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E43:61"><span class="lab"><font color="Green" title="E67">A31</font></span></a>, <a class="ref" href="mcart_1.html#T64" title="MCART_1:th.64">MCART_1:64</a></span>
<br/>.= 
<span class="p1"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c21">p3</font>,<font color="Maroon" title="c11">nk</font>,<span class="p2">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p3">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p4">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<font color="Maroon" title="c3">t</font>,<font color="Maroon" title="c7">h1</font>,<font color="Maroon" title="c14">T</font>)</span>)</span>,<font color="Maroon" title="c11">nk</font>,<font color="Maroon" title="c14">T</font>)</span>)</span>,<font color="Maroon" title="c13">n3</font>,<font color="Maroon" title="c1">F</font>)</span>)</span></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E44:61"><span class="lab"><font color="Green" title="E68">A32</font></span></a>, <a class="txt" href="#E18:61"><span class="lab"><font color="Green" title="E51">A8</font></span></a>, <a class="ref" href="mcart_1.html#T64" title="MCART_1:th.64">MCART_1:64</a></span>
;<br/>
 <a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p1">(<span class="default"><span class="p2"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c19">p1</font>,<font color="Maroon" title="c11">nk</font>,<span class="p3">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<font color="Maroon" title="c3">t</font>,<font color="Maroon" title="c7">h1</font>,<font color="Maroon" title="c14">T</font>)</span>)</span></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span> <a href="mcart_1.html#K7" title="MCART_1:func.7">`3</a></span>)</span>,<span class="p1">(<span class="default"><a href="turing_1.html#K7" title="TURING_1:func.7">Head</a> <span class="p2"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c19">p1</font>,<font color="Maroon" title="c11">nk</font>,<span class="p3">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<font color="Maroon" title="c3">t</font>,<font color="Maroon" title="c7">h1</font>,<font color="Maroon" title="c14">T</font>)</span>)</span></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K8" title="TURING_1:func.8">TRAN</a> <span class="p3"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c19">p1</font>,<font color="Maroon" title="c11">nk</font>,<span class="p4">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<font color="Maroon" title="c3">t</font>,<font color="Maroon" title="c7">h1</font>,<font color="Maroon" title="c14">T</font>)</span>)</span></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span></span>)</span> <a href="mcart_1.html#K6" title="MCART_1:func.6">`2</a></span>)</span>) = 
 <a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p1">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<font color="Maroon" title="c3">t</font>,<font color="Maroon" title="c7">h1</font>,<font color="Maroon" title="c14">T</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="turing_1.html#K7" title="TURING_1:func.7">Head</a> <span class="p2"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c19">p1</font>,<font color="Maroon" title="c11">nk</font>,<span class="p3">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<font color="Maroon" title="c3">t</font>,<font color="Maroon" title="c7">h1</font>,<font color="Maroon" title="c14">T</font>)</span>)</span></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K8" title="TURING_1:func.8">TRAN</a> <span class="p3"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c19">p1</font>,<font color="Maroon" title="c11">nk</font>,<span class="p4">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<font color="Maroon" title="c3">t</font>,<font color="Maroon" title="c7">h1</font>,<font color="Maroon" title="c14">T</font>)</span>)</span></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span></span>)</span> <a href="mcart_1.html#K6" title="MCART_1:func.6">`2</a></span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="mcart_1.html#T64" title="MCART_1:th.64">MCART_1:64</a></span>
<br/>.= 
 <a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p1">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<font color="Maroon" title="c3">t</font>,<font color="Maroon" title="c7">h1</font>,<font color="Maroon" title="c14">T</font>)</span>)</span>,<font color="Maroon" title="c11">nk</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K8" title="TURING_1:func.8">TRAN</a> <span class="p3"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c19">p1</font>,<font color="Maroon" title="c11">nk</font>,<span class="p4">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<font color="Maroon" title="c3">t</font>,<font color="Maroon" title="c7">h1</font>,<font color="Maroon" title="c14">T</font>)</span>)</span></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span></span>)</span> <a href="mcart_1.html#K6" title="MCART_1:func.6">`2</a></span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="mcart_1.html#T64" title="MCART_1:th.64">MCART_1:64</a></span>
<br/>.= 
 <a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p1">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<font color="Maroon" title="c3">t</font>,<font color="Maroon" title="c7">h1</font>,<font color="Maroon" title="c14">T</font>)</span>)</span>,<font color="Maroon" title="c11">nk</font>,<font color="Maroon" title="c14">T</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E47:61"><span class="lab"><font color="Green" title="E70">A34</font></span></a>, <a class="ref" href="mcart_1.html#T64" title="MCART_1:th.64">MCART_1:64</a></span>
;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E76">A43</font></span>:  <a href="turing_1.html#K9" title="TURING_1:func.9">Following</a> <span class="p1"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c19">p1</font>,<font color="Maroon" title="c11">nk</font>,<span class="p2">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<font color="Maroon" title="c3">t</font>,<font color="Maroon" title="c7">h1</font>,<font color="Maroon" title="c14">T</font>)</span>)</span></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span> = 
<span class="p1"><a href="mcart_1.html#K3" title="MCART_1:func.3">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="turing_1.html#K8" title="TURING_1:func.8">TRAN</a> <span class="p4"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c19">p1</font>,<font color="Maroon" title="c11">nk</font>,<span class="p5">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<font color="Maroon" title="c3">t</font>,<font color="Maroon" title="c7">h1</font>,<font color="Maroon" title="c14">T</font>)</span>)</span></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span></span>)</span> <a href="mcart_1.html#K5" title="MCART_1:func.5">`1</a></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="turing_1.html#K7" title="TURING_1:func.7">Head</a> <span class="p4"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c19">p1</font>,<font color="Maroon" title="c11">nk</font>,<span class="p5">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<font color="Maroon" title="c3">t</font>,<font color="Maroon" title="c7">h1</font>,<font color="Maroon" title="c14">T</font>)</span>)</span></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p3">(<span class="default"><a href="turing_1.html#K6" title="TURING_1:func.6">offset</a> <span class="p4">(<span class="default"><a href="turing_1.html#K8" title="TURING_1:func.8">TRAN</a> <span class="p5"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c19">p1</font>,<font color="Maroon" title="c11">nk</font>,<span class="p0">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<font color="Maroon" title="c3">t</font>,<font color="Maroon" title="c7">h1</font>,<font color="Maroon" title="c14">T</font>)</span>)</span></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p3">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<font color="Maroon" title="c3">t</font>,<font color="Maroon" title="c7">h1</font>,<font color="Maroon" title="c14">T</font>)</span>)</span>,<font color="Maroon" title="c11">nk</font>,<font color="Maroon" title="c14">T</font>)</span>)</span></span><a href="mcart_1.html#K3" title="MCART_1:func.3">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="turing_1.html#E54"><span class="lab"><font color="Green" title="E42">Lm7</font></span></a></span>
<br/>.= 
<span class="p1"><a href="mcart_1.html#K3" title="MCART_1:func.3">[</a><span class="default">2,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="turing_1.html#K7" title="TURING_1:func.7">Head</a> <span class="p4"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c19">p1</font>,<font color="Maroon" title="c11">nk</font>,<span class="p5">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<font color="Maroon" title="c3">t</font>,<font color="Maroon" title="c7">h1</font>,<font color="Maroon" title="c14">T</font>)</span>)</span></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p3">(<span class="default"><a href="turing_1.html#K6" title="TURING_1:func.6">offset</a> <span class="p4">(<span class="default"><a href="turing_1.html#K8" title="TURING_1:func.8">TRAN</a> <span class="p5"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c19">p1</font>,<font color="Maroon" title="c11">nk</font>,<span class="p0">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<font color="Maroon" title="c3">t</font>,<font color="Maroon" title="c7">h1</font>,<font color="Maroon" title="c14">T</font>)</span>)</span></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p3">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<font color="Maroon" title="c3">t</font>,<font color="Maroon" title="c7">h1</font>,<font color="Maroon" title="c14">T</font>)</span>)</span>,<font color="Maroon" title="c11">nk</font>,<font color="Maroon" title="c14">T</font>)</span>)</span></span><a href="mcart_1.html#K3" title="MCART_1:func.3">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E47:61"><span class="lab"><font color="Green" title="E70">A34</font></span></a>, <a class="ref" href="mcart_1.html#T64" title="MCART_1:th.64">MCART_1:64</a></span>
<br/>.= 
<span class="p1"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c9">p2</font>,<font color="Maroon" title="c13">n3</font>,<span class="p2">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p3">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<font color="Maroon" title="c3">t</font>,<font color="Maroon" title="c7">h1</font>,<font color="Maroon" title="c14">T</font>)</span>)</span>,<font color="Maroon" title="c11">nk</font>,<font color="Maroon" title="c14">T</font>)</span>)</span></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E48:61"><span class="lab"><font color="Green" title="E71">A35</font></span></a>, <a class="ref" href="mcart_1.html#T64" title="MCART_1:th.64">MCART_1:64</a></span>
;<br/>
<a NAME="E56:61"/><span class="lab"><font color="Green" title="E77">A44</font></span>: 
 for <font color="Olive" title="b1">k</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_18"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c32">k</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c32">k</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c32">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1] )</span><br/>

<span class="kw">assume </span><a NAME="E1:61_18"/><span class="lab"><font color="Green" title="E78">A45</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c32">k</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c32">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]</span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_18_1"><span class="kw">now </span></a><div class="add"><span class="kw">reconsider </span><font color="Maroon" title="c33">ik</font> = <font color="Maroon" title="c11">nk</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c32">k</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#D2" title="INT_1:def.2">INT_1:def 2</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c34">k3</font> = <font color="Maroon" title="c11">nk</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c32">k</font>;<br/><a NAME="E2:61_18_1"/><span class="lab"><font color="Green" title="E79">A46</font></span>: 
<font color="Maroon" title="c4">h</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c32">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">h</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c32">k</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T29" title="XREAL_1:th.29">XREAL_1:29</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c35">sk</font> = <span class="p1"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c21">p3</font>,<font color="Maroon" title="c33">ik</font>,<span class="p2">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p3">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p4">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<font color="Maroon" title="c3">t</font>,<font color="Maroon" title="c7">h1</font>,<font color="Maroon" title="c14">T</font>)</span>)</span>,<font color="Maroon" title="c11">nk</font>,<font color="Maroon" title="c14">T</font>)</span>)</span>,<font color="Maroon" title="c13">n3</font>,<font color="Maroon" title="c1">F</font>)</span>)</span></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c36">tt</font> = <span class="p1"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c21">p3</font>,<font color="Maroon" title="c33">ik</font>,<span class="p2">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p3">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p4">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<font color="Maroon" title="c3">t</font>,<font color="Maroon" title="c7">h1</font>,<font color="Maroon" title="c14">T</font>)</span>)</span>,<font color="Maroon" title="c11">nk</font>,<font color="Maroon" title="c14">T</font>)</span>)</span>,<font color="Maroon" title="c13">n3</font>,<font color="Maroon" title="c1">F</font>)</span>)</span></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span> <a href="mcart_1.html#K7" title="MCART_1:func.7">`3</a>  as   <a href="turing_1.html#NM2" title="TURING_1:NM.2">Tape</a> of <a href="turing_1.html#K16" title="TURING_1:func.16">SuccTuring</a> ;<br/><a NAME="E4:61_18_1"/>
<font color="Maroon" title="c11">nk</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">nk</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c32">k</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#T6" title="INT_1:th.6">INT_1:6</a></span>;<br/><a NAME="E5:61_18_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E80">A47</font></span>: 
<font color="Maroon" title="c11">nk</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c32">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">nk</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T20" title="XREAL_1:th.20">XREAL_1:20</a></span>;<br/><span class="kw">assume </span><a NAME="E6:61_18_1"/><span class="lab"><font color="Green" title="E81">A48</font></span>: 
<font color="Maroon" title="c4">h</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c32">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">nk</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <span class="p2"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c21">p3</font>,<font color="Maroon" title="c11">nk</font>,<span class="p3">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p4">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p5">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<font color="Maroon" title="c3">t</font>,<font color="Maroon" title="c7">h1</font>,<font color="Maroon" title="c14">T</font>)</span>)</span>,<font color="Maroon" title="c11">nk</font>,<font color="Maroon" title="c14">T</font>)</span>)</span>,<font color="Maroon" title="c13">n3</font>,<font color="Maroon" title="c1">F</font>)</span>)</span></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c32">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default">3,<span class="p2">(<span class="default"><font color="Maroon" title="c11">nk</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c32">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p3">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p4">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<font color="Maroon" title="c3">t</font>,<font color="Maroon" title="c7">h1</font>,<font color="Maroon" title="c14">T</font>)</span>)</span>,<font color="Maroon" title="c11">nk</font>,<font color="Maroon" title="c14">T</font>)</span>)</span>,<font color="Maroon" title="c13">n3</font>,<font color="Maroon" title="c1">F</font>)</span>)</span></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span></span><br/><a NAME="E7:61_18_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">h</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c32">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c11">nk</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:61_18_1"><span class="lab"><font color="Green" title="E79">A46</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E8:61_18_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E82">A49</font></span>: 
<font color="Maroon" title="c4">h</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c11">nk</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c32">k</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T21" title="XREAL_1:th.21">XREAL_1:21</a></span>;<br/><a NAME="E9:61_18_1"/>
<font color="Maroon" title="c4">h</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T2" title="NAT_1:th.2">NAT_1:2</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c37">ii</font> = <font color="Maroon" title="c11">nk</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c32">k</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:61_18_1"><span class="lab"><font color="Green" title="E82">A49</font></span></a>, <a class="ref" href="int_1.html#T3" title="INT_1:th.3">INT_1:3</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E11:61_18_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">h</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">h</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T29" title="XREAL_1:th.29">XREAL_1:29</a></span>;<br/><a NAME="E12:61_18_1"/><span class="kw">then </span>
<font color="Maroon" title="c37">ii</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">h</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:61_18_1"><span class="lab"><font color="Green" title="E80">A47</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E13:61_18_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E83">A50</font></span>: 
<span class="p1">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p2">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p3">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<font color="Maroon" title="c3">t</font>,<font color="Maroon" title="c7">h1</font>,<font color="Maroon" title="c14">T</font>)</span>)</span>,<font color="Maroon" title="c11">nk</font>,<font color="Maroon" title="c14">T</font>)</span>)</span>,<font color="Maroon" title="c13">n3</font>,<font color="Maroon" title="c1">F</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c37">ii</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E50:61"><span class="lab"><font color="Green" title="E73">A37</font></span></a>, <a class="txt" href="#E8:61_18_1"><span class="lab"><font color="Green" title="E82">A49</font></span></a></span>;<br/><span class="lab"><font color="Green" title="E84">A51</font></span>:  <a href="turing_1.html#K8" title="TURING_1:func.8">TRAN</a> <span class="p1"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c21">p3</font>,<font color="Maroon" title="c33">ik</font>,<span class="p2">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p3">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p4">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<font color="Maroon" title="c3">t</font>,<font color="Maroon" title="c7">h1</font>,<font color="Maroon" title="c14">T</font>)</span>)</span>,<font color="Maroon" title="c11">nk</font>,<font color="Maroon" title="c14">T</font>)</span>)</span>,<font color="Maroon" title="c13">n3</font>,<font color="Maroon" title="c1">F</font>)</span>)</span></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span> = 
<a href="turing_1.html#K15" title="TURING_1:func.15">Succ_Tran</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c21">p3</font>,<font color="Maroon" title="c33">ik</font>,<span class="p4">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p5">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p0">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<font color="Maroon" title="c3">t</font>,<font color="Maroon" title="c7">h1</font>,<font color="Maroon" title="c14">T</font>)</span>)</span>,<font color="Maroon" title="c11">nk</font>,<font color="Maroon" title="c14">T</font>)</span>)</span>,<font color="Maroon" title="c13">n3</font>,<font color="Maroon" title="c1">F</font>)</span>)</span></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span> <a href="mcart_1.html#K5" title="MCART_1:func.5">`1</a></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c36">tt</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><a href="turing_1.html#K7" title="TURING_1:func.7">Head</a> <span class="p4"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c21">p3</font>,<font color="Maroon" title="c33">ik</font>,<span class="p5">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p0">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p1">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<font color="Maroon" title="c3">t</font>,<font color="Maroon" title="c7">h1</font>,<font color="Maroon" title="c14">T</font>)</span>)</span>,<font color="Maroon" title="c11">nk</font>,<font color="Maroon" title="c14">T</font>)</span>)</span>,<font color="Maroon" title="c13">n3</font>,<font color="Maroon" title="c1">F</font>)</span>)</span></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span></span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="turing_1.html#D17" target="_self" title="TURING_1:def.17">Def18</a></span>
<br/>.= 
<a href="turing_1.html#K15" title="TURING_1:func.15">Succ_Tran</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c21">p3</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c36">tt</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><a href="turing_1.html#K7" title="TURING_1:func.7">Head</a> <span class="p4"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c21">p3</font>,<font color="Maroon" title="c33">ik</font>,<span class="p5">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p0">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p1">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<font color="Maroon" title="c3">t</font>,<font color="Maroon" title="c7">h1</font>,<font color="Maroon" title="c14">T</font>)</span>)</span>,<font color="Maroon" title="c11">nk</font>,<font color="Maroon" title="c14">T</font>)</span>)</span>,<font color="Maroon" title="c13">n3</font>,<font color="Maroon" title="c1">F</font>)</span>)</span></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span></span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="mcart_1.html#T64" title="MCART_1:th.64">MCART_1:64</a></span>
<br/>.= 
<a href="turing_1.html#K15" title="TURING_1:func.15">Succ_Tran</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c21">p3</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p4">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p5">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<font color="Maroon" title="c3">t</font>,<font color="Maroon" title="c7">h1</font>,<font color="Maroon" title="c14">T</font>)</span>)</span>,<font color="Maroon" title="c11">nk</font>,<font color="Maroon" title="c14">T</font>)</span>)</span>,<font color="Maroon" title="c13">n3</font>,<font color="Maroon" title="c1">F</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><a href="turing_1.html#K7" title="TURING_1:func.7">Head</a> <span class="p4"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c21">p3</font>,<font color="Maroon" title="c33">ik</font>,<span class="p5">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p0">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p1">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<font color="Maroon" title="c3">t</font>,<font color="Maroon" title="c7">h1</font>,<font color="Maroon" title="c14">T</font>)</span>)</span>,<font color="Maroon" title="c11">nk</font>,<font color="Maroon" title="c14">T</font>)</span>)</span>,<font color="Maroon" title="c13">n3</font>,<font color="Maroon" title="c1">F</font>)</span>)</span></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span></span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="mcart_1.html#T64" title="MCART_1:th.64">MCART_1:64</a></span>
<br/>.= 
<span class="p1"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default">3,1,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:61_18_1"><span class="lab"><font color="Green" title="E83">A50</font></span></a>, <a class="ref" href="turing_1.html#T30" target="_self" title="TURING_1:th.30">Th34</a>, <a class="ref" href="mcart_1.html#T64" title="MCART_1:th.64">MCART_1:64</a></span>
;<br/><a NAME="E15:61_18_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E85">A52</font></span>: 
 <a href="turing_1.html#K6" title="TURING_1:func.6">offset</a> <span class="p1">(<span class="default"><a href="turing_1.html#K8" title="TURING_1:func.8">TRAN</a> <span class="p2"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c21">p3</font>,<font color="Maroon" title="c33">ik</font>,<span class="p3">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p4">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p5">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<font color="Maroon" title="c3">t</font>,<font color="Maroon" title="c7">h1</font>,<font color="Maroon" title="c14">T</font>)</span>)</span>,<font color="Maroon" title="c11">nk</font>,<font color="Maroon" title="c14">T</font>)</span>)</span>,<font color="Maroon" title="c13">n3</font>,<font color="Maroon" title="c1">F</font>)</span>)</span></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="mcart_1.html#T64" title="MCART_1:th.64">MCART_1:64</a></span>;<br/><span class="lab"><font color="Green" title="E86">A53</font></span>:  <a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p1">(<span class="default"><span class="p2"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c21">p3</font>,<font color="Maroon" title="c33">ik</font>,<span class="p3">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p4">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p5">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<font color="Maroon" title="c3">t</font>,<font color="Maroon" title="c7">h1</font>,<font color="Maroon" title="c14">T</font>)</span>)</span>,<font color="Maroon" title="c11">nk</font>,<font color="Maroon" title="c14">T</font>)</span>)</span>,<font color="Maroon" title="c13">n3</font>,<font color="Maroon" title="c1">F</font>)</span>)</span></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span> <a href="mcart_1.html#K7" title="MCART_1:func.7">`3</a></span>)</span>,<span class="p1">(<span class="default"><a href="turing_1.html#K7" title="TURING_1:func.7">Head</a> <span class="p2"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c21">p3</font>,<font color="Maroon" title="c33">ik</font>,<span class="p3">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p4">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p5">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<font color="Maroon" title="c3">t</font>,<font color="Maroon" title="c7">h1</font>,<font color="Maroon" title="c14">T</font>)</span>)</span>,<font color="Maroon" title="c11">nk</font>,<font color="Maroon" title="c14">T</font>)</span>)</span>,<font color="Maroon" title="c13">n3</font>,<font color="Maroon" title="c1">F</font>)</span>)</span></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K8" title="TURING_1:func.8">TRAN</a> <span class="p3"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c21">p3</font>,<font color="Maroon" title="c33">ik</font>,<span class="p4">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p5">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p0">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<font color="Maroon" title="c3">t</font>,<font color="Maroon" title="c7">h1</font>,<font color="Maroon" title="c14">T</font>)</span>)</span>,<font color="Maroon" title="c11">nk</font>,<font color="Maroon" title="c14">T</font>)</span>)</span>,<font color="Maroon" title="c13">n3</font>,<font color="Maroon" title="c1">F</font>)</span>)</span></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span></span>)</span> <a href="mcart_1.html#K6" title="MCART_1:func.6">`2</a></span>)</span>) = 
 <a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p1">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p2">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p3">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<font color="Maroon" title="c3">t</font>,<font color="Maroon" title="c7">h1</font>,<font color="Maroon" title="c14">T</font>)</span>)</span>,<font color="Maroon" title="c11">nk</font>,<font color="Maroon" title="c14">T</font>)</span>)</span>,<font color="Maroon" title="c13">n3</font>,<font color="Maroon" title="c1">F</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="turing_1.html#K7" title="TURING_1:func.7">Head</a> <span class="p2"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c21">p3</font>,<font color="Maroon" title="c33">ik</font>,<span class="p3">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p4">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p5">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<font color="Maroon" title="c3">t</font>,<font color="Maroon" title="c7">h1</font>,<font color="Maroon" title="c14">T</font>)</span>)</span>,<font color="Maroon" title="c11">nk</font>,<font color="Maroon" title="c14">T</font>)</span>)</span>,<font color="Maroon" title="c13">n3</font>,<font color="Maroon" title="c1">F</font>)</span>)</span></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K8" title="TURING_1:func.8">TRAN</a> <span class="p3"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c21">p3</font>,<font color="Maroon" title="c33">ik</font>,<span class="p4">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p5">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p0">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<font color="Maroon" title="c3">t</font>,<font color="Maroon" title="c7">h1</font>,<font color="Maroon" title="c14">T</font>)</span>)</span>,<font color="Maroon" title="c11">nk</font>,<font color="Maroon" title="c14">T</font>)</span>)</span>,<font color="Maroon" title="c13">n3</font>,<font color="Maroon" title="c1">F</font>)</span>)</span></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span></span>)</span> <a href="mcart_1.html#K6" title="MCART_1:func.6">`2</a></span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="mcart_1.html#T64" title="MCART_1:th.64">MCART_1:64</a></span>
<br/>.= 
 <a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p1">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p2">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p3">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<font color="Maroon" title="c3">t</font>,<font color="Maroon" title="c7">h1</font>,<font color="Maroon" title="c14">T</font>)</span>)</span>,<font color="Maroon" title="c11">nk</font>,<font color="Maroon" title="c14">T</font>)</span>)</span>,<font color="Maroon" title="c13">n3</font>,<font color="Maroon" title="c1">F</font>)</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c11">nk</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c32">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K8" title="TURING_1:func.8">TRAN</a> <span class="p3"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c21">p3</font>,<font color="Maroon" title="c33">ik</font>,<span class="p4">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p5">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p0">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<font color="Maroon" title="c3">t</font>,<font color="Maroon" title="c7">h1</font>,<font color="Maroon" title="c14">T</font>)</span>)</span>,<font color="Maroon" title="c11">nk</font>,<font color="Maroon" title="c14">T</font>)</span>)</span>,<font color="Maroon" title="c13">n3</font>,<font color="Maroon" title="c1">F</font>)</span>)</span></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span></span>)</span> <a href="mcart_1.html#K6" title="MCART_1:func.6">`2</a></span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="mcart_1.html#T64" title="MCART_1:th.64">MCART_1:64</a></span>
<br/>.= 
 <a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p1">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p2">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p3">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<font color="Maroon" title="c3">t</font>,<font color="Maroon" title="c7">h1</font>,<font color="Maroon" title="c14">T</font>)</span>)</span>,<font color="Maroon" title="c11">nk</font>,<font color="Maroon" title="c14">T</font>)</span>)</span>,<font color="Maroon" title="c13">n3</font>,<font color="Maroon" title="c1">F</font>)</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c11">nk</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c32">k</font></span>)</span>,<font color="Maroon" title="c14">T</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:61_18_1"><span class="lab"><font color="Green" title="E84">A51</font></span></a>, <a class="ref" href="mcart_1.html#T64" title="MCART_1:th.64">MCART_1:64</a></span>
<br/>.= 
 <a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p1">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p2">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<font color="Maroon" title="c3">t</font>,<font color="Maroon" title="c7">h1</font>,<font color="Maroon" title="c14">T</font>)</span>)</span>,<font color="Maroon" title="c11">nk</font>,<font color="Maroon" title="c14">T</font>)</span>)</span>,<font color="Maroon" title="c13">n3</font>,<font color="Maroon" title="c1">F</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:61_18_1"><span class="lab"><font color="Green" title="E83">A50</font></span></a>, <a class="ref" href="turing_1.html#T24" target="_self" title="TURING_1:th.24">Th28</a></span>
;<br/><div><span class="kw">hereby </span> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span>
<div class="add"><span class="kw">thus </span><span class="p1">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <span class="p2"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c21">p3</font>,<font color="Maroon" title="c11">nk</font>,<span class="p3">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p4">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p5">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<font color="Maroon" title="c3">t</font>,<font color="Maroon" title="c7">h1</font>,<font color="Maroon" title="c14">T</font>)</span>)</span>,<font color="Maroon" title="c11">nk</font>,<font color="Maroon" title="c14">T</font>)</span>)</span>,<font color="Maroon" title="c13">n3</font>,<font color="Maroon" title="c1">F</font>)</span>)</span></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c32">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> = 
 <a href="turing_1.html#K9" title="TURING_1:func.9">Following</a> <span class="p1"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c21">p3</font>,<font color="Maroon" title="c33">ik</font>,<span class="p2">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p3">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p4">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<font color="Maroon" title="c3">t</font>,<font color="Maroon" title="c7">h1</font>,<font color="Maroon" title="c14">T</font>)</span>)</span>,<font color="Maroon" title="c11">nk</font>,<font color="Maroon" title="c14">T</font>)</span>)</span>,<font color="Maroon" title="c13">n3</font>,<font color="Maroon" title="c1">F</font>)</span>)</span></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:61_18"><span class="lab"><font color="Green" title="E78">A45</font></span></a>, <a class="txt" href="#E6:61_18_1"><span class="lab"><font color="Green" title="E81">A48</font></span></a>, <a class="txt" href="#E2:61_18_1"><span class="lab"><font color="Green" title="E79">A46</font></span></a>, <a class="ref" href="turing_1.html#D7" target="_self" title="TURING_1:def.7">Def8</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>
<br/>.= 
<span class="p1"><a href="mcart_1.html#K3" title="MCART_1:func.3">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="turing_1.html#K8" title="TURING_1:func.8">TRAN</a> <span class="p4"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c21">p3</font>,<font color="Maroon" title="c33">ik</font>,<span class="p5">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p0">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p1">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<font color="Maroon" title="c3">t</font>,<font color="Maroon" title="c7">h1</font>,<font color="Maroon" title="c14">T</font>)</span>)</span>,<font color="Maroon" title="c11">nk</font>,<font color="Maroon" title="c14">T</font>)</span>)</span>,<font color="Maroon" title="c13">n3</font>,<font color="Maroon" title="c1">F</font>)</span>)</span></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span></span>)</span> <a href="mcart_1.html#K5" title="MCART_1:func.5">`1</a></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="turing_1.html#K7" title="TURING_1:func.7">Head</a> <span class="p4"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c21">p3</font>,<font color="Maroon" title="c33">ik</font>,<span class="p5">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p0">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p1">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<font color="Maroon" title="c3">t</font>,<font color="Maroon" title="c7">h1</font>,<font color="Maroon" title="c14">T</font>)</span>)</span>,<font color="Maroon" title="c11">nk</font>,<font color="Maroon" title="c14">T</font>)</span>)</span>,<font color="Maroon" title="c13">n3</font>,<font color="Maroon" title="c1">F</font>)</span>)</span></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p3">(<span class="default"><a href="turing_1.html#K6" title="TURING_1:func.6">offset</a> <span class="p4">(<span class="default"><a href="turing_1.html#K8" title="TURING_1:func.8">TRAN</a> <span class="p5"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c21">p3</font>,<font color="Maroon" title="c33">ik</font>,<span class="p0">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p1">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p2">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<font color="Maroon" title="c3">t</font>,<font color="Maroon" title="c7">h1</font>,<font color="Maroon" title="c14">T</font>)</span>)</span>,<font color="Maroon" title="c11">nk</font>,<font color="Maroon" title="c14">T</font>)</span>)</span>,<font color="Maroon" title="c13">n3</font>,<font color="Maroon" title="c1">F</font>)</span>)</span></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p3">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p4">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<font color="Maroon" title="c3">t</font>,<font color="Maroon" title="c7">h1</font>,<font color="Maroon" title="c14">T</font>)</span>)</span>,<font color="Maroon" title="c11">nk</font>,<font color="Maroon" title="c14">T</font>)</span>)</span>,<font color="Maroon" title="c13">n3</font>,<font color="Maroon" title="c1">F</font>)</span>)</span></span><a href="mcart_1.html#K3" title="MCART_1:func.3">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:61_18_1"><span class="lab"><font color="Green" title="E86">A53</font></span></a>, <a class="txt" href="turing_1.html#E54"><span class="lab"><font color="Green" title="E42">Lm7</font></span></a></span>
<br/>.= 
<span class="p1"><a href="mcart_1.html#K3" title="MCART_1:func.3">[</a><span class="default">3,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="turing_1.html#K7" title="TURING_1:func.7">Head</a> <span class="p4"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c21">p3</font>,<font color="Maroon" title="c33">ik</font>,<span class="p5">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p0">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p1">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<font color="Maroon" title="c3">t</font>,<font color="Maroon" title="c7">h1</font>,<font color="Maroon" title="c14">T</font>)</span>)</span>,<font color="Maroon" title="c11">nk</font>,<font color="Maroon" title="c14">T</font>)</span>)</span>,<font color="Maroon" title="c13">n3</font>,<font color="Maroon" title="c1">F</font>)</span>)</span></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p3">(<span class="default"><a href="turing_1.html#K6" title="TURING_1:func.6">offset</a> <span class="p4">(<span class="default"><a href="turing_1.html#K8" title="TURING_1:func.8">TRAN</a> <span class="p5"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c21">p3</font>,<font color="Maroon" title="c33">ik</font>,<span class="p0">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p1">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p2">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<font color="Maroon" title="c3">t</font>,<font color="Maroon" title="c7">h1</font>,<font color="Maroon" title="c14">T</font>)</span>)</span>,<font color="Maroon" title="c11">nk</font>,<font color="Maroon" title="c14">T</font>)</span>)</span>,<font color="Maroon" title="c13">n3</font>,<font color="Maroon" title="c1">F</font>)</span>)</span></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p3">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p4">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<font color="Maroon" title="c3">t</font>,<font color="Maroon" title="c7">h1</font>,<font color="Maroon" title="c14">T</font>)</span>)</span>,<font color="Maroon" title="c11">nk</font>,<font color="Maroon" title="c14">T</font>)</span>)</span>,<font color="Maroon" title="c13">n3</font>,<font color="Maroon" title="c1">F</font>)</span>)</span></span><a href="mcart_1.html#K3" title="MCART_1:func.3">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:61_18_1"><span class="lab"><font color="Green" title="E84">A51</font></span></a>, <a class="ref" href="mcart_1.html#T64" title="MCART_1:th.64">MCART_1:64</a></span>
<br/>.= 
<span class="p1"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default">3,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c11">nk</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c32">k</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p3">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p4">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<font color="Maroon" title="c3">t</font>,<font color="Maroon" title="c7">h1</font>,<font color="Maroon" title="c14">T</font>)</span>)</span>,<font color="Maroon" title="c11">nk</font>,<font color="Maroon" title="c14">T</font>)</span>)</span>,<font color="Maroon" title="c13">n3</font>,<font color="Maroon" title="c1">F</font>)</span>)</span></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:61_18_1"><span class="lab"><font color="Green" title="E85">A52</font></span></a>, <a class="ref" href="mcart_1.html#T64" title="MCART_1:th.64">MCART_1:64</a></span>
<br/>.= 
<span class="p1"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default">3,<span class="p2">(<span class="default"><font color="Maroon" title="c11">nk</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c32">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p3">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p4">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<font color="Maroon" title="c3">t</font>,<font color="Maroon" title="c7">h1</font>,<font color="Maroon" title="c14">T</font>)</span>)</span>,<font color="Maroon" title="c11">nk</font>,<font color="Maroon" title="c14">T</font>)</span>)</span>,<font color="Maroon" title="c13">n3</font>,<font color="Maroon" title="c1">F</font>)</span>)</span></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span>

; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div>
<span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E3:61_18"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c32">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E57:61"/><span class="lab"><font color="Green" title="E78">A54</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="turing_1.html#D7" target="_self" title="TURING_1:def.7">Def8</a></span>;<br/>
<a NAME="E58:61"/>
 for <font color="Olive" title="b1">k</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="#E57:61"><span class="lab"><font color="Green" title="E78">A54</font></span></a>, <a class="txt" href="#E56:61"><span class="lab"><font color="Green" title="E77">A44</font></span></a>);<br/></span>
<span class="kw">then </span><span class="lab"><font color="Green" title="E79">A55</font></span>: <span class="p1">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <span class="p2"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c21">p3</font>,<font color="Maroon" title="c11">nk</font>,<span class="p3">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p4">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p5">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<font color="Maroon" title="c3">t</font>,<font color="Maroon" title="c7">h1</font>,<font color="Maroon" title="c14">T</font>)</span>)</span>,<font color="Maroon" title="c11">nk</font>,<font color="Maroon" title="c14">T</font>)</span>)</span>,<font color="Maroon" title="c13">n3</font>,<font color="Maroon" title="c1">F</font>)</span>)</span></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c5">n</font></span>)</span> = 
<span class="p1"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default">3,<span class="p2">(<span class="default"><font color="Maroon" title="c11">nk</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c5">n</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p3">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p4">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<font color="Maroon" title="c3">t</font>,<font color="Maroon" title="c7">h1</font>,<font color="Maroon" title="c14">T</font>)</span>)</span>,<font color="Maroon" title="c11">nk</font>,<font color="Maroon" title="c14">T</font>)</span>)</span>,<font color="Maroon" title="c13">n3</font>,<font color="Maroon" title="c1">F</font>)</span>)</span></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E46:61"><span class="lab"><font color="Green" title="E69">A33</font></span></a></span>
<br/>.= 
<span class="p1"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c21">p3</font>,<font color="Maroon" title="c12">hh</font>,<span class="p2">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p3">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p4">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<font color="Maroon" title="c3">t</font>,<font color="Maroon" title="c7">h1</font>,<font color="Maroon" title="c14">T</font>)</span>)</span>,<font color="Maroon" title="c11">nk</font>,<font color="Maroon" title="c14">T</font>)</span>)</span>,<font color="Maroon" title="c13">n3</font>,<font color="Maroon" title="c1">F</font>)</span>)</span></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span>

;<br/>
<div><span class="lab"><font color="Green" title="E80">A56</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="61_20"><span class="kw">now </span></a><div class="add"><span class="kw">reconsider </span><font color="Maroon" title="c32">tt</font> = <span class="p1"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c21">p3</font>,<font color="Maroon" title="c12">hh</font>,<span class="p2">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p3">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p4">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<font color="Maroon" title="c3">t</font>,<font color="Maroon" title="c7">h1</font>,<font color="Maroon" title="c14">T</font>)</span>)</span>,<font color="Maroon" title="c11">nk</font>,<font color="Maroon" title="c14">T</font>)</span>)</span>,<font color="Maroon" title="c13">n3</font>,<font color="Maroon" title="c1">F</font>)</span>)</span></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span> <a href="mcart_1.html#K7" title="MCART_1:func.7">`3</a>  as   <a href="turing_1.html#NM2" title="TURING_1:NM.2">Tape</a> of <a href="turing_1.html#K16" title="TURING_1:func.16">SuccTuring</a> ;<br/><span class="lab"><font color="Green" title="E81">A57</font></span>:  <a href="turing_1.html#K8" title="TURING_1:func.8">TRAN</a> <span class="p1"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c21">p3</font>,<font color="Maroon" title="c12">hh</font>,<span class="p2">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p3">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p4">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<font color="Maroon" title="c3">t</font>,<font color="Maroon" title="c7">h1</font>,<font color="Maroon" title="c14">T</font>)</span>)</span>,<font color="Maroon" title="c11">nk</font>,<font color="Maroon" title="c14">T</font>)</span>)</span>,<font color="Maroon" title="c13">n3</font>,<font color="Maroon" title="c1">F</font>)</span>)</span></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span> = 
<a href="turing_1.html#K15" title="TURING_1:func.15">Succ_Tran</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c21">p3</font>,<font color="Maroon" title="c12">hh</font>,<span class="p4">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p5">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p0">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<font color="Maroon" title="c3">t</font>,<font color="Maroon" title="c7">h1</font>,<font color="Maroon" title="c14">T</font>)</span>)</span>,<font color="Maroon" title="c11">nk</font>,<font color="Maroon" title="c14">T</font>)</span>)</span>,<font color="Maroon" title="c13">n3</font>,<font color="Maroon" title="c1">F</font>)</span>)</span></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span> <a href="mcart_1.html#K5" title="MCART_1:func.5">`1</a></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c32">tt</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><a href="turing_1.html#K7" title="TURING_1:func.7">Head</a> <span class="p4"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c21">p3</font>,<font color="Maroon" title="c12">hh</font>,<span class="p5">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p0">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p1">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<font color="Maroon" title="c3">t</font>,<font color="Maroon" title="c7">h1</font>,<font color="Maroon" title="c14">T</font>)</span>)</span>,<font color="Maroon" title="c11">nk</font>,<font color="Maroon" title="c14">T</font>)</span>)</span>,<font color="Maroon" title="c13">n3</font>,<font color="Maroon" title="c1">F</font>)</span>)</span></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span></span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="turing_1.html#D17" target="_self" title="TURING_1:def.17">Def18</a></span>
<br/>.= 
<a href="turing_1.html#K15" title="TURING_1:func.15">Succ_Tran</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">3,<span class="p2">(<span class="default"><font color="Maroon" title="c32">tt</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><a href="turing_1.html#K7" title="TURING_1:func.7">Head</a> <span class="p4"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c21">p3</font>,<font color="Maroon" title="c12">hh</font>,<span class="p5">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p0">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p1">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<font color="Maroon" title="c3">t</font>,<font color="Maroon" title="c7">h1</font>,<font color="Maroon" title="c14">T</font>)</span>)</span>,<font color="Maroon" title="c11">nk</font>,<font color="Maroon" title="c14">T</font>)</span>)</span>,<font color="Maroon" title="c13">n3</font>,<font color="Maroon" title="c1">F</font>)</span>)</span></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span></span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="mcart_1.html#T64" title="MCART_1:th.64">MCART_1:64</a></span>
<br/>.= 
<a href="turing_1.html#K15" title="TURING_1:func.15">Succ_Tran</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">3,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p4">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p5">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<font color="Maroon" title="c3">t</font>,<font color="Maroon" title="c7">h1</font>,<font color="Maroon" title="c14">T</font>)</span>)</span>,<font color="Maroon" title="c11">nk</font>,<font color="Maroon" title="c14">T</font>)</span>)</span>,<font color="Maroon" title="c13">n3</font>,<font color="Maroon" title="c1">F</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><a href="turing_1.html#K7" title="TURING_1:func.7">Head</a> <span class="p4"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c21">p3</font>,<font color="Maroon" title="c12">hh</font>,<span class="p5">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p0">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p1">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<font color="Maroon" title="c3">t</font>,<font color="Maroon" title="c7">h1</font>,<font color="Maroon" title="c14">T</font>)</span>)</span>,<font color="Maroon" title="c11">nk</font>,<font color="Maroon" title="c14">T</font>)</span>)</span>,<font color="Maroon" title="c13">n3</font>,<font color="Maroon" title="c1">F</font>)</span>)</span></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span></span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="mcart_1.html#T64" title="MCART_1:th.64">MCART_1:64</a></span>
<br/>.= 
<span class="p1"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default">4,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E50:61"><span class="lab"><font color="Green" title="E73">A37</font></span></a>, <a class="ref" href="turing_1.html#T30" target="_self" title="TURING_1:th.30">Th34</a>, <a class="ref" href="mcart_1.html#T64" title="MCART_1:th.64">MCART_1:64</a></span>
;<br/><a NAME="E3:61_20"/><span class="kw">then </span><span class="lab"><font color="Green" title="E82">A58</font></span>: 
 <a href="turing_1.html#K6" title="TURING_1:func.6">offset</a> <span class="p1">(<span class="default"><a href="turing_1.html#K8" title="TURING_1:func.8">TRAN</a> <span class="p2"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c21">p3</font>,<font color="Maroon" title="c12">hh</font>,<span class="p3">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p4">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p5">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<font color="Maroon" title="c3">t</font>,<font color="Maroon" title="c7">h1</font>,<font color="Maroon" title="c14">T</font>)</span>)</span>,<font color="Maroon" title="c11">nk</font>,<font color="Maroon" title="c14">T</font>)</span>)</span>,<font color="Maroon" title="c13">n3</font>,<font color="Maroon" title="c1">F</font>)</span>)</span></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="mcart_1.html#T64" title="MCART_1:th.64">MCART_1:64</a></span>;<br/> <a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p1">(<span class="default"><span class="p2"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c21">p3</font>,<font color="Maroon" title="c12">hh</font>,<span class="p3">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p4">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p5">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<font color="Maroon" title="c3">t</font>,<font color="Maroon" title="c7">h1</font>,<font color="Maroon" title="c14">T</font>)</span>)</span>,<font color="Maroon" title="c11">nk</font>,<font color="Maroon" title="c14">T</font>)</span>)</span>,<font color="Maroon" title="c13">n3</font>,<font color="Maroon" title="c1">F</font>)</span>)</span></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span> <a href="mcart_1.html#K7" title="MCART_1:func.7">`3</a></span>)</span>,<span class="p1">(<span class="default"><a href="turing_1.html#K7" title="TURING_1:func.7">Head</a> <span class="p2"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c21">p3</font>,<font color="Maroon" title="c12">hh</font>,<span class="p3">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p4">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p5">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<font color="Maroon" title="c3">t</font>,<font color="Maroon" title="c7">h1</font>,<font color="Maroon" title="c14">T</font>)</span>)</span>,<font color="Maroon" title="c11">nk</font>,<font color="Maroon" title="c14">T</font>)</span>)</span>,<font color="Maroon" title="c13">n3</font>,<font color="Maroon" title="c1">F</font>)</span>)</span></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K8" title="TURING_1:func.8">TRAN</a> <span class="p3"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c21">p3</font>,<font color="Maroon" title="c12">hh</font>,<span class="p4">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p5">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p0">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<font color="Maroon" title="c3">t</font>,<font color="Maroon" title="c7">h1</font>,<font color="Maroon" title="c14">T</font>)</span>)</span>,<font color="Maroon" title="c11">nk</font>,<font color="Maroon" title="c14">T</font>)</span>)</span>,<font color="Maroon" title="c13">n3</font>,<font color="Maroon" title="c1">F</font>)</span>)</span></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span></span>)</span> <a href="mcart_1.html#K6" title="MCART_1:func.6">`2</a></span>)</span>) = 
 <a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p1">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p2">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p3">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<font color="Maroon" title="c3">t</font>,<font color="Maroon" title="c7">h1</font>,<font color="Maroon" title="c14">T</font>)</span>)</span>,<font color="Maroon" title="c11">nk</font>,<font color="Maroon" title="c14">T</font>)</span>)</span>,<font color="Maroon" title="c13">n3</font>,<font color="Maroon" title="c1">F</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="turing_1.html#K7" title="TURING_1:func.7">Head</a> <span class="p2"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c21">p3</font>,<font color="Maroon" title="c12">hh</font>,<span class="p3">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p4">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p5">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<font color="Maroon" title="c3">t</font>,<font color="Maroon" title="c7">h1</font>,<font color="Maroon" title="c14">T</font>)</span>)</span>,<font color="Maroon" title="c11">nk</font>,<font color="Maroon" title="c14">T</font>)</span>)</span>,<font color="Maroon" title="c13">n3</font>,<font color="Maroon" title="c1">F</font>)</span>)</span></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K8" title="TURING_1:func.8">TRAN</a> <span class="p3"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c21">p3</font>,<font color="Maroon" title="c12">hh</font>,<span class="p4">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p5">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p0">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<font color="Maroon" title="c3">t</font>,<font color="Maroon" title="c7">h1</font>,<font color="Maroon" title="c14">T</font>)</span>)</span>,<font color="Maroon" title="c11">nk</font>,<font color="Maroon" title="c14">T</font>)</span>)</span>,<font color="Maroon" title="c13">n3</font>,<font color="Maroon" title="c1">F</font>)</span>)</span></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span></span>)</span> <a href="mcart_1.html#K6" title="MCART_1:func.6">`2</a></span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="mcart_1.html#T64" title="MCART_1:th.64">MCART_1:64</a></span>
<br/>.= 
 <a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p1">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p2">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p3">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<font color="Maroon" title="c3">t</font>,<font color="Maroon" title="c7">h1</font>,<font color="Maroon" title="c14">T</font>)</span>)</span>,<font color="Maroon" title="c11">nk</font>,<font color="Maroon" title="c14">T</font>)</span>)</span>,<font color="Maroon" title="c13">n3</font>,<font color="Maroon" title="c1">F</font>)</span>)</span>,<font color="Maroon" title="c4">h</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K8" title="TURING_1:func.8">TRAN</a> <span class="p3"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c21">p3</font>,<font color="Maroon" title="c12">hh</font>,<span class="p4">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p5">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p0">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<font color="Maroon" title="c3">t</font>,<font color="Maroon" title="c7">h1</font>,<font color="Maroon" title="c14">T</font>)</span>)</span>,<font color="Maroon" title="c11">nk</font>,<font color="Maroon" title="c14">T</font>)</span>)</span>,<font color="Maroon" title="c13">n3</font>,<font color="Maroon" title="c1">F</font>)</span>)</span></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span></span>)</span> <a href="mcart_1.html#K6" title="MCART_1:func.6">`2</a></span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="mcart_1.html#T64" title="MCART_1:th.64">MCART_1:64</a></span>
<br/>.= 
 <a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p1">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p2">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p3">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<font color="Maroon" title="c3">t</font>,<font color="Maroon" title="c7">h1</font>,<font color="Maroon" title="c14">T</font>)</span>)</span>,<font color="Maroon" title="c11">nk</font>,<font color="Maroon" title="c14">T</font>)</span>)</span>,<font color="Maroon" title="c13">n3</font>,<font color="Maroon" title="c1">F</font>)</span>)</span>,<font color="Maroon" title="c4">h</font>,<font color="Maroon" title="c1">F</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:61_20"><span class="lab"><font color="Green" title="E81">A57</font></span></a>, <a class="ref" href="mcart_1.html#T64" title="MCART_1:th.64">MCART_1:64</a></span>
<br/>.= 
 <a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p1">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p2">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<font color="Maroon" title="c3">t</font>,<font color="Maroon" title="c7">h1</font>,<font color="Maroon" title="c14">T</font>)</span>)</span>,<font color="Maroon" title="c11">nk</font>,<font color="Maroon" title="c14">T</font>)</span>)</span>,<font color="Maroon" title="c13">n3</font>,<font color="Maroon" title="c1">F</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E50:61"><span class="lab"><font color="Green" title="E73">A37</font></span></a>, <a class="ref" href="turing_1.html#T24" target="_self" title="TURING_1:th.24">Th28</a></span>
;<br/><span class="kw">hence </span> <a href="turing_1.html#K9" title="TURING_1:func.9">Following</a> <span class="p1"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c21">p3</font>,<font color="Maroon" title="c12">hh</font>,<span class="p2">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p3">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p4">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<font color="Maroon" title="c3">t</font>,<font color="Maroon" title="c7">h1</font>,<font color="Maroon" title="c14">T</font>)</span>)</span>,<font color="Maroon" title="c11">nk</font>,<font color="Maroon" title="c14">T</font>)</span>)</span>,<font color="Maroon" title="c13">n3</font>,<font color="Maroon" title="c1">F</font>)</span>)</span></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span> = 
<span class="p1"><a href="mcart_1.html#K3" title="MCART_1:func.3">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="turing_1.html#K8" title="TURING_1:func.8">TRAN</a> <span class="p4"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c21">p3</font>,<font color="Maroon" title="c12">hh</font>,<span class="p5">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p0">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p1">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<font color="Maroon" title="c3">t</font>,<font color="Maroon" title="c7">h1</font>,<font color="Maroon" title="c14">T</font>)</span>)</span>,<font color="Maroon" title="c11">nk</font>,<font color="Maroon" title="c14">T</font>)</span>)</span>,<font color="Maroon" title="c13">n3</font>,<font color="Maroon" title="c1">F</font>)</span>)</span></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span></span>)</span> <a href="mcart_1.html#K5" title="MCART_1:func.5">`1</a></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="turing_1.html#K7" title="TURING_1:func.7">Head</a> <span class="p4"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c21">p3</font>,<font color="Maroon" title="c12">hh</font>,<span class="p5">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p0">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p1">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<font color="Maroon" title="c3">t</font>,<font color="Maroon" title="c7">h1</font>,<font color="Maroon" title="c14">T</font>)</span>)</span>,<font color="Maroon" title="c11">nk</font>,<font color="Maroon" title="c14">T</font>)</span>)</span>,<font color="Maroon" title="c13">n3</font>,<font color="Maroon" title="c1">F</font>)</span>)</span></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p3">(<span class="default"><a href="turing_1.html#K6" title="TURING_1:func.6">offset</a> <span class="p4">(<span class="default"><a href="turing_1.html#K8" title="TURING_1:func.8">TRAN</a> <span class="p5"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c21">p3</font>,<font color="Maroon" title="c12">hh</font>,<span class="p0">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p1">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p2">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<font color="Maroon" title="c3">t</font>,<font color="Maroon" title="c7">h1</font>,<font color="Maroon" title="c14">T</font>)</span>)</span>,<font color="Maroon" title="c11">nk</font>,<font color="Maroon" title="c14">T</font>)</span>)</span>,<font color="Maroon" title="c13">n3</font>,<font color="Maroon" title="c1">F</font>)</span>)</span></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p3">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p4">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<font color="Maroon" title="c3">t</font>,<font color="Maroon" title="c7">h1</font>,<font color="Maroon" title="c14">T</font>)</span>)</span>,<font color="Maroon" title="c11">nk</font>,<font color="Maroon" title="c14">T</font>)</span>)</span>,<font color="Maroon" title="c13">n3</font>,<font color="Maroon" title="c1">F</font>)</span>)</span></span><a href="mcart_1.html#K3" title="MCART_1:func.3">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="turing_1.html#E54"><span class="lab"><font color="Green" title="E42">Lm7</font></span></a></span>
<br/>.= 
<span class="p1"><a href="mcart_1.html#K3" title="MCART_1:func.3">[</a><span class="default">4,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="turing_1.html#K7" title="TURING_1:func.7">Head</a> <span class="p4"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c21">p3</font>,<font color="Maroon" title="c12">hh</font>,<span class="p5">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p0">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p1">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<font color="Maroon" title="c3">t</font>,<font color="Maroon" title="c7">h1</font>,<font color="Maroon" title="c14">T</font>)</span>)</span>,<font color="Maroon" title="c11">nk</font>,<font color="Maroon" title="c14">T</font>)</span>)</span>,<font color="Maroon" title="c13">n3</font>,<font color="Maroon" title="c1">F</font>)</span>)</span></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p3">(<span class="default"><a href="turing_1.html#K6" title="TURING_1:func.6">offset</a> <span class="p4">(<span class="default"><a href="turing_1.html#K8" title="TURING_1:func.8">TRAN</a> <span class="p5"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c21">p3</font>,<font color="Maroon" title="c12">hh</font>,<span class="p0">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p1">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p2">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<font color="Maroon" title="c3">t</font>,<font color="Maroon" title="c7">h1</font>,<font color="Maroon" title="c14">T</font>)</span>)</span>,<font color="Maroon" title="c11">nk</font>,<font color="Maroon" title="c14">T</font>)</span>)</span>,<font color="Maroon" title="c13">n3</font>,<font color="Maroon" title="c1">F</font>)</span>)</span></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p3">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p4">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<font color="Maroon" title="c3">t</font>,<font color="Maroon" title="c7">h1</font>,<font color="Maroon" title="c14">T</font>)</span>)</span>,<font color="Maroon" title="c11">nk</font>,<font color="Maroon" title="c14">T</font>)</span>)</span>,<font color="Maroon" title="c13">n3</font>,<font color="Maroon" title="c1">F</font>)</span>)</span></span><a href="mcart_1.html#K3" title="MCART_1:func.3">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:61_20"><span class="lab"><font color="Green" title="E81">A57</font></span></a>, <a class="ref" href="mcart_1.html#T64" title="MCART_1:th.64">MCART_1:64</a></span>
<br/>.= 
<span class="p1"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default">4,<span class="p2">(<span class="default"><font color="Maroon" title="c4">h</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span>,<span class="p2">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p3">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p4">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<font color="Maroon" title="c3">t</font>,<font color="Maroon" title="c7">h1</font>,<font color="Maroon" title="c14">T</font>)</span>)</span>,<font color="Maroon" title="c11">nk</font>,<font color="Maroon" title="c14">T</font>)</span>)</span>,<font color="Maroon" title="c13">n3</font>,<font color="Maroon" title="c1">F</font>)</span>)</span></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:61_20"><span class="lab"><font color="Green" title="E82">A58</font></span></a>, <a class="ref" href="mcart_1.html#T64" title="MCART_1:th.64">MCART_1:64</a></span>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="p1">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <font color="Maroon" title="c2">s</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <font color="Maroon" title="c2">s</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p3">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="turing_1.html#T10" target="_self" title="TURING_1:th.10">Th13</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <span class="p2">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Following</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <font color="Maroon" title="c2">s</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> 1</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="turing_1.html#D7" target="_self" title="TURING_1:def.7">Def8</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <span class="p2">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Following</a> <span class="p3"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c8">p1</font>,<font color="Maroon" title="c7">h1</font>,<font color="Maroon" title="c3">t</font></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span></span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:61"><span class="lab"><font color="Green" title="E58">A18</font></span></a>, <a class="ref" href="turing_1.html#T9" target="_self" title="TURING_1:th.9">Th12</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <span class="p4"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c19">p1</font>,<font color="Maroon" title="c10">i2</font>,<span class="p5">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<font color="Maroon" title="c3">t</font>,<font color="Maroon" title="c7">h1</font>,<font color="Maroon" title="c14">T</font>)</span>)</span></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:61"><span class="lab"><font color="Green" title="E61">A21</font></span></a>, <a class="ref" href="turing_1.html#T10" target="_self" title="TURING_1:th.10">Th13</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <span class="p2">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Following</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <span class="p5"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c19">p1</font>,<font color="Maroon" title="c10">i2</font>,<span class="p0">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<font color="Maroon" title="c3">t</font>,<font color="Maroon" title="c7">h1</font>,<font color="Maroon" title="c14">T</font>)</span>)</span></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="turing_1.html#D7" target="_self" title="TURING_1:def.7">Def8</a></span>
;<br/>
<a NAME="E62:61"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <font color="Maroon" title="c2">s</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <span class="p2"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c21">p3</font>,<font color="Maroon" title="c11">nk</font>,<span class="p3">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p4">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p5">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<font color="Maroon" title="c3">t</font>,<font color="Maroon" title="c7">h1</font>,<font color="Maroon" title="c14">T</font>)</span>)</span>,<font color="Maroon" title="c11">nk</font>,<font color="Maroon" title="c14">T</font>)</span>)</span>,<font color="Maroon" title="c13">n3</font>,<font color="Maroon" title="c1">F</font>)</span>)</span></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E36:61"><span class="lab"><font color="Green" title="E62">A22</font></span></a>, <a class="txt" href="#E55:61"><span class="lab"><font color="Green" title="E76">A43</font></span></a>, <a class="txt" href="#E53:61"><span class="lab"><font color="Green" title="E75">A42</font></span></a>, <a class="ref" href="turing_1.html#D7" target="_self" title="TURING_1:def.7">Def8</a></span>;<br/>
<a NAME="E63:61"/><span class="kw">then </span><span class="lab"><font color="Green" title="E81">A59</font></span>: 
<span class="p1">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <font color="Maroon" title="c2">s</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default">4,<font color="Maroon" title="c4">h</font>,<span class="p2">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p3">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p4">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<font color="Maroon" title="c3">t</font>,<font color="Maroon" title="c7">h1</font>,<font color="Maroon" title="c14">T</font>)</span>)</span>,<font color="Maroon" title="c11">nk</font>,<font color="Maroon" title="c14">T</font>)</span>)</span>,<font color="Maroon" title="c13">n3</font>,<font color="Maroon" title="c1">F</font>)</span>)</span></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E59:61"><span class="lab"><font color="Green" title="E79">A55</font></span></a>, <a class="txt" href="#E60:61"><span class="lab"><font color="Green" title="E80">A56</font></span></a>, <a class="ref" href="turing_1.html#D7" target="_self" title="TURING_1:def.7">Def8</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E82">A60</font></span>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <font color="Maroon" title="c2">s</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="mcart_1.html#K5" title="MCART_1:func.5">`1</a>  = 
4
<span class="kw">by</span> <span class="lab"><a class="ref" href="mcart_1.html#T64" title="MCART_1:th.64">MCART_1:64</a></span>
<br/>.= 
 the <a href="turing_1.html#U5" title="TURING_1:sel.5">AcceptS</a> of <a href="turing_1.html#K16" title="TURING_1:func.16">SuccTuring</a>
<span class="kw">by</span> <span class="lab"><a class="ref" href="turing_1.html#D17" target="_self" title="TURING_1:def.17">Def18</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E65:61"/>
<font color="Maroon" title="c2">s</font> is  <a href="turing_1.html#V2" title="TURING_1:attr.2">Accept-Halt</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="turing_1.html#D8" target="_self" title="TURING_1:def.8">Def9</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <span class="p1">(<span class="default"><a href="turing_1.html#K11" title="TURING_1:func.11">Result</a> <font color="Maroon" title="c2">s</font></span>)</span> <a href="mcart_1.html#K6" title="MCART_1:func.6">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">h</font> &amp; <span class="p1">(<span class="default"><a href="turing_1.html#K11" title="TURING_1:func.11">Result</a> <font color="Maroon" title="c2">s</font></span>)</span> <a href="mcart_1.html#K7" title="MCART_1:func.7">`3</a>  <a href="turing_1.html#R2" title="TURING_1:pred.2">storeData</a> <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c4">h</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> )</span><br/>

<a NAME="E66:61"/><span class="kw">then </span><span class="lab"><font color="Green" title="E83">A61</font></span>: 
 <a href="turing_1.html#K11" title="TURING_1:func.11">Result</a> <font color="Maroon" title="c2">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <font color="Maroon" title="c2">s</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E64:61"><span class="lab"><font color="Green" title="E82">A60</font></span></a>, <a class="ref" href="turing_1.html#D9" target="_self" title="TURING_1:def.9">Def10</a></span>;<br/>
<span class="kw">hence </span><a NAME="E67:61"/>
<span class="p1">(<span class="default"><a href="turing_1.html#K11" title="TURING_1:func.11">Result</a> <font color="Maroon" title="c2">s</font></span>)</span> <a href="mcart_1.html#K6" title="MCART_1:func.6">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">h</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E63:61"><span class="lab"><font color="Green" title="E81">A59</font></span></a>, <a class="ref" href="mcart_1.html#T64" title="MCART_1:th.64">MCART_1:64</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="turing_1.html#K11" title="TURING_1:func.11">Result</a> <font color="Maroon" title="c2">s</font></span>)</span> <a href="mcart_1.html#K7" title="MCART_1:func.7">`3</a>  <a href="turing_1.html#R2" title="TURING_1:pred.2">storeData</a> <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c4">h</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span></span><br/>

<a NAME="E68:61"/>
<span class="p1">(<span class="default"><a href="turing_1.html#K11" title="TURING_1:func.11">Result</a> <font color="Maroon" title="c2">s</font></span>)</span> <a href="mcart_1.html#K7" title="MCART_1:func.7">`3</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p1">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<span class="p2">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> (<font color="Maroon" title="c3">t</font>,<font color="Maroon" title="c7">h1</font>,<font color="Maroon" title="c14">T</font>)</span>)</span>,<font color="Maroon" title="c11">nk</font>,<font color="Maroon" title="c14">T</font>)</span>)</span>,<font color="Maroon" title="c13">n3</font>,<font color="Maroon" title="c1">F</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E63:61"><span class="lab"><font color="Green" title="E81">A59</font></span></a>, <a class="txt" href="#E66:61"><span class="lab"><font color="Green" title="E83">A61</font></span></a>, <a class="ref" href="mcart_1.html#T64" title="MCART_1:th.64">MCART_1:64</a></span>;<br/>
<span class="kw">hence </span><a NAME="E69:61"/>
<span class="p1">(<span class="default"><a href="turing_1.html#K11" title="TURING_1:func.11">Result</a> <font color="Maroon" title="c2">s</font></span>)</span> <a href="mcart_1.html#K7" title="MCART_1:func.7">`3</a>  <a href="turing_1.html#R2" title="TURING_1:pred.2">storeData</a> <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c4">h</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E51:61"><span class="lab"><font color="Green" title="E74">A41</font></span></a>, <a class="ref" href="turing_1.html#T16" target="_self" title="TURING_1:th.16">Th19</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
