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<span class="kw">let </span><font color="Maroon" title="c1">tm</font> be    <a href="turing_1.html#L1" title="TURING_1:struct.1">TuringStr</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">s</font> being   <a href="turing_1.html#NM4" title="TURING_1:NM.4">All-State</a> of <font color="Maroon" title="c1">tm</font><br/>  for <font color="Olive" title="b2">p</font> being   <a href="turing_1.html#NM1" title="TURING_1:NM.1">State</a> of <font color="Maroon" title="c1">tm</font><br/>  for <font color="Olive" title="b3">h</font> being    <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> <br/>  for <font color="Olive" title="b4">t</font> being   <a href="turing_1.html#NM2" title="TURING_1:NM.2">Tape</a> of <font color="Maroon" title="c1">tm</font><br/>  for <font color="Olive" title="b5">m</font>, <font color="Olive" title="b6">d</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="Olive" title="b6">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">h</font> &amp; 1 is   <a href="turing_1.html#NM3" title="TURING_1:NM.3">Symbol</a> of <font color="Maroon" title="c1">tm</font> &amp; <font color="Olive" title="b1">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"><font color="Olive" title="b2">p</font>,<font color="Olive" title="b3">h</font>,<font color="Olive" title="b4">t</font></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span> &amp;  the <a href="turing_1.html#U3" title="TURING_1:sel.3">Tran</a> of <font color="Maroon" title="c1">tm</font> <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="Olive" title="b2">p</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="Olive" title="b2">p</font>,1,1</span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span> &amp; <font color="Olive" title="b2">p</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 <font color="Maroon" title="c1">tm</font> &amp; (  for <font color="Olive" title="b7">i</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>  st <font color="Olive" title="b6">d</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b7">i</font> &amp; <font color="Olive" title="b7">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b6">d</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b5">m</font> holds <br/><font color="Olive" title="b4">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b7">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) holds <br/><span class="p1">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <font color="Olive" title="b1">s</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b5">m</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="Olive" title="b2">p</font>,<span class="p2">(<span class="default"><font color="Olive" title="b6">d</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b5">m</font></span>)</span>,<font color="Olive" title="b4">t</font></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span></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 <font color="Maroon" title="c1">tm</font>; <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">p</font> being   <a href="turing_1.html#NM1" title="TURING_1:NM.1">State</a> of <font color="Maroon" title="c1">tm</font><br/>  for <font color="Olive" title="b2">h</font> being    <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> <br/>  for <font color="Olive" title="b3">t</font> being   <a href="turing_1.html#NM2" title="TURING_1:NM.2">Tape</a> of <font color="Maroon" title="c1">tm</font><br/>  for <font color="Olive" title="b4">m</font>, <font color="Olive" title="b5">d</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="Olive" title="b5">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">h</font> &amp; 1 is   <a href="turing_1.html#NM3" title="TURING_1:NM.3">Symbol</a> of <font color="Maroon" title="c1">tm</font> &amp; <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"><font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">h</font>,<font color="Olive" title="b3">t</font></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span> &amp;  the <a href="turing_1.html#U3" title="TURING_1:sel.3">Tran</a> of <font color="Maroon" title="c1">tm</font> <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="Olive" title="b1">p</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="Olive" title="b1">p</font>,1,1</span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span> &amp; <font color="Olive" title="b1">p</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 <font color="Maroon" title="c1">tm</font> &amp; (  for <font color="Olive" title="b6">i</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>  st <font color="Olive" title="b5">d</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b6">i</font> &amp; <font color="Olive" title="b6">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b5">d</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b4">m</font> holds <br/><font color="Olive" title="b3">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b6">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) holds <br/><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> <font color="Olive" title="b4">m</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="Olive" title="b1">p</font>,<span class="p2">(<span class="default"><font color="Olive" title="b5">d</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b4">m</font></span>)</span>,<font color="Olive" title="b3">t</font></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span></span><br/><span class="kw">let </span><font color="Maroon" title="c3">p</font> be   <a href="turing_1.html#NM1" title="TURING_1:NM.1">State</a> of <font color="Maroon" title="c1">tm</font>; <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">h</font> being    <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> <br/>  for <font color="Olive" title="b2">t</font> being   <a href="turing_1.html#NM2" title="TURING_1:NM.2">Tape</a> of <font color="Maroon" title="c1">tm</font><br/>  for <font color="Olive" title="b3">m</font>, <font color="Olive" title="b4">d</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="Olive" title="b4">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">h</font> &amp; 1 is   <a href="turing_1.html#NM3" title="TURING_1:NM.3">Symbol</a> of <font color="Maroon" title="c1">tm</font> &amp; <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"><font color="Maroon" title="c3">p</font>,<font color="Olive" title="b1">h</font>,<font color="Olive" title="b2">t</font></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span> &amp;  the <a href="turing_1.html#U3" title="TURING_1:sel.3">Tran</a> of <font color="Maroon" title="c1">tm</font> <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="c3">p</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="c3">p</font>,1,1</span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span> &amp; <font color="Maroon" title="c3">p</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 <font color="Maroon" title="c1">tm</font> &amp; (  for <font color="Olive" title="b5">i</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>  st <font color="Olive" title="b4">d</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b5">i</font> &amp; <font color="Olive" title="b5">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b4">d</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b3">m</font> holds <br/><font color="Olive" title="b2">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b5">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) holds <br/><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> <font color="Olive" title="b3">m</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="c3">p</font>,<span class="p2">(<span class="default"><font color="Olive" title="b4">d</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b3">m</font></span>)</span>,<font color="Olive" title="b2">t</font></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span></span><br/><span class="kw">let </span><font color="Maroon" title="c4">h</font> be    <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> ; <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 <font color="Maroon" title="c1">tm</font><br/>  for <font color="Olive" title="b2">m</font>, <font color="Olive" title="b3">d</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="Olive" title="b3">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">h</font> &amp; 1 is   <a href="turing_1.html#NM3" title="TURING_1:NM.3">Symbol</a> of <font color="Maroon" title="c1">tm</font> &amp; <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"><font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">h</font>,<font color="Olive" title="b1">t</font></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span> &amp;  the <a href="turing_1.html#U3" title="TURING_1:sel.3">Tran</a> of <font color="Maroon" title="c1">tm</font> <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="c3">p</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="c3">p</font>,1,1</span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span> &amp; <font color="Maroon" title="c3">p</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 <font color="Maroon" title="c1">tm</font> &amp; (  for <font color="Olive" title="b4">i</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>  st <font color="Olive" title="b3">d</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b4">i</font> &amp; <font color="Olive" title="b4">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b3">d</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b2">m</font> holds <br/><font color="Olive" title="b1">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) holds <br/><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> <font color="Olive" title="b2">m</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="c3">p</font>,<span class="p2">(<span class="default"><font color="Olive" title="b3">d</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b2">m</font></span>)</span>,<font color="Olive" title="b1">t</font></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span></span><br/><span class="kw">let </span><font color="Maroon" title="c5">t</font> be   <a href="turing_1.html#NM2" title="TURING_1:NM.2">Tape</a> of <font color="Maroon" title="c1">tm</font>; <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">m</font>, <font color="Olive" title="b2">d</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="Olive" title="b2">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">h</font> &amp; 1 is   <a href="turing_1.html#NM3" title="TURING_1:NM.3">Symbol</a> of <font color="Maroon" title="c1">tm</font> &amp; <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"><font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">h</font>,<font color="Maroon" title="c5">t</font></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span> &amp;  the <a href="turing_1.html#U3" title="TURING_1:sel.3">Tran</a> of <font color="Maroon" title="c1">tm</font> <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="c3">p</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="c3">p</font>,1,1</span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span> &amp; <font color="Maroon" title="c3">p</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 <font color="Maroon" title="c1">tm</font> &amp; (  for <font color="Olive" title="b3">i</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>  st <font color="Olive" title="b2">d</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b3">i</font> &amp; <font color="Olive" title="b3">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">d</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b1">m</font> holds <br/><font color="Maroon" title="c5">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) holds <br/><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> <font color="Olive" title="b1">m</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="c3">p</font>,<span class="p2">(<span class="default"><font color="Olive" title="b2">d</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b1">m</font></span>)</span>,<font color="Maroon" title="c5">t</font></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span></span><br/><span class="kw">let </span><font color="Maroon" title="c6">m</font>, <font color="Maroon" title="c7">d</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="c7">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">h</font> &amp; 1 is   <a href="turing_1.html#NM3" title="TURING_1:NM.3">Symbol</a> of <font color="Maroon" title="c1">tm</font> &amp; <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"><font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">h</font>,<font color="Maroon" title="c5">t</font></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span> &amp;  the <a href="turing_1.html#U3" title="TURING_1:sel.3">Tran</a> of <font color="Maroon" title="c1">tm</font> <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="c3">p</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="c3">p</font>,1,1</span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span> &amp; <font color="Maroon" title="c3">p</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 <font color="Maroon" title="c1">tm</font> &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="c7">d</font> <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> <font color="Maroon" title="c7">d</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c6">m</font> holds <br/><font color="Maroon" title="c5">t</font> <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 ) implies <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> <font color="Maroon" title="c6">m</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="c3">p</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c7">d</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c6">m</font></span>)</span>,<font color="Maroon" title="c5">t</font></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span> )</span><br/>











<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:50"/><span class="lab"><font color="Green" title="E35">A1</font></span>: 
<font color="Maroon" title="c7">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">h</font>
 <span class="kw">and </span><br/><a NAME="E2:50"/><span class="lab"><font color="Green" title="E36">A2</font></span>: 
1 is   <a href="turing_1.html#NM3" title="TURING_1:NM.3">Symbol</a> of <font color="Maroon" title="c1">tm</font>
 <span class="kw">and </span><br/><a NAME="E3:50"/><span class="lab"><font color="Green" title="E37">A3</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"><font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">h</font>,<font color="Maroon" title="c5">t</font></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span>
 <span class="kw">and </span><br/><a NAME="E4:50"/><span class="lab"><font color="Green" title="E38">A4</font></span>: 
 the <a href="turing_1.html#U3" title="TURING_1:sel.3">Tran</a> of <font color="Maroon" title="c1">tm</font> <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="c3">p</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="c3">p</font>,1,1</span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span>
 <span class="kw">and </span><br/><a NAME="E5:50"/><span class="lab"><font color="Green" title="E39">A5</font></span>: 
<font color="Maroon" title="c3">p</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 <font color="Maroon" title="c1">tm</font>
 <span class="kw">and </span><br/><a NAME="E6:50"/><span class="lab"><font color="Green" title="E40">A6</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 <font color="Maroon" title="c7">d</font> <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> <font color="Maroon" title="c7">d</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c6">m</font> holds <br/><font color="Maroon" title="c5">t</font> <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
 ; <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> <font color="Maroon" title="c2">s</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c6">m</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="c3">p</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c7">d</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c6">m</font></span>)</span>,<font color="Maroon" title="c5">t</font></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span></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>( $1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">m</font> implies <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> $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"><font color="Maroon" title="c3">p</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c7">d</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> $1</span>)</span>,<font color="Maroon" title="c5">t</font></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span> );<br/>
<a NAME="E7:50"/><span class="lab"><font color="Green" title="E41">A7</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="50_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c8">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="c8">k</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1] )</span><br/>

<span class="kw">assume </span><a NAME="E1:50_1"/><span class="lab"><font color="Green" title="E42">A8</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">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="c8">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="50_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">reconsider </span><font color="Maroon" title="c9">T</font> = 1 as   <a href="turing_1.html#NM3" title="TURING_1:NM.3">Symbol</a> of <font color="Maroon" title="c1">tm</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:50"><span class="lab"><font color="Green" title="E36">A2</font></span></a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c10">dk</font> = <font color="Maroon" title="c7">d</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c8">k</font>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c11">ik</font> = <font color="Maroon" title="c7">d</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c8">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="c12">sk</font> = <span class="p1"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c11">ik</font>,<font color="Maroon" title="c5">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="c13">tt</font> = <span class="p1"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c11">ik</font>,<font color="Maroon" title="c5">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 <font color="Maroon" title="c1">tm</font> ;<br/><span class="kw">assume </span><a NAME="E4:50_1_1"/><span class="lab"><font color="Green" title="E43">A9</font></span>: 
<font color="Maroon" title="c8">k</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="c6">m</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> <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"><font color="Maroon" title="c8">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"><font color="Maroon" title="c3">p</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c7">d</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c5">t</font></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span></span><br/><a NAME="E5:50_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c8">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">m</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E6:50_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c7">d</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c8">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">d</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c6">m</font>
 <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="E7:50_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E44">A10</font></span>: 
<font color="Maroon" title="c5">t</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">ik</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:50"><span class="lab"><font color="Green" title="E40">A6</font></span></a>, <a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></span>;<br/><span class="lab"><font color="Green" title="E45">A11</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="c3">p</font>,<font color="Maroon" title="c11">ik</font>,<font color="Maroon" title="c5">t</font></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span> = 
 the <a href="turing_1.html#U3" title="TURING_1:sel.3">Tran</a> of <font color="Maroon" title="c1">tm</font> <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="c3">p</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c13">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="c3">p</font>,<font color="Maroon" title="c11">ik</font>,<font color="Maroon" title="c5">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/>.= 
 the <a href="turing_1.html#U3" title="TURING_1:sel.3">Tran</a> of <font color="Maroon" title="c1">tm</font> <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="c3">p</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">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="c3">p</font>,<font color="Maroon" title="c11">ik</font>,<font color="Maroon" title="c5">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/>.= 
<span class="p1"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c3">p</font>,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="#E4:50"><span class="lab"><font color="Green" title="E38">A4</font></span></a>, <a class="txt" href="#E7:50_1_1"><span class="lab"><font color="Green" title="E44">A10</font></span></a>, <a class="ref" href="mcart_1.html#T64" title="MCART_1:th.64">MCART_1:64</a></span>
;<br/><a NAME="E9:50_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E46">A12</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="c3">p</font>,<font color="Maroon" title="c11">ik</font>,<font color="Maroon" title="c5">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="lab"><font color="Green" title="E47">A13</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="c3">p</font>,<font color="Maroon" title="c11">ik</font>,<font color="Maroon" title="c5">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="c3">p</font>,<font color="Maroon" title="c11">ik</font>,<font color="Maroon" title="c5">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="c3">p</font>,<font color="Maroon" title="c11">ik</font>,<font color="Maroon" title="c5">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="c5">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="c3">p</font>,<font color="Maroon" title="c11">ik</font>,<font color="Maroon" title="c5">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="c3">p</font>,<font color="Maroon" title="c11">ik</font>,<font color="Maroon" title="c5">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="c5">t</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">d</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c8">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="c3">p</font>,<font color="Maroon" title="c11">ik</font>,<font color="Maroon" title="c5">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="c5">t</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">d</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c8">k</font></span>)</span>,<font color="Maroon" title="c9">T</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:50_1_1"><span class="lab"><font color="Green" title="E45">A11</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="c5">t</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:50_1_1"><span class="lab"><font color="Green" title="E44">A10</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">thus </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"><font color="Maroon" title="c8">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="c3">p</font>,<font color="Maroon" title="c11">ik</font>,<font color="Maroon" title="c5">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="#E1:50_1"><span class="lab"><font color="Green" title="E42">A8</font></span></a>, <a class="txt" href="#E4:50_1_1"><span class="lab"><font color="Green" title="E43">A9</font></span></a>, <a class="ref" href="turing_1.html#D7" target="_self" title="TURING_1:def.7">Def8</a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</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="c3">p</font>,<font color="Maroon" title="c11">ik</font>,<font color="Maroon" title="c5">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="c3">p</font>,<font color="Maroon" title="c11">ik</font>,<font color="Maroon" title="c5">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="c3">p</font>,<font color="Maroon" title="c11">ik</font>,<font color="Maroon" title="c5">t</font></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c5">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="#E5:50"><span class="lab"><font color="Green" title="E39">A5</font></span></a>, <a class="txt" href="#E10:50_1_1"><span class="lab"><font color="Green" title="E47">A13</font></span></a>, <a class="ref" href="turing_1.html#T25" target="_self" title="TURING_1:th.25">Th29</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="c3">p</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="c3">p</font>,<font color="Maroon" title="c11">ik</font>,<font color="Maroon" title="c5">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="c3">p</font>,<font color="Maroon" title="c11">ik</font>,<font color="Maroon" title="c5">t</font></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c5">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="#E8:50_1_1"><span class="lab"><font color="Green" title="E45">A11</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="c3">p</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c7">d</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c5">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="#E9:50_1_1"><span class="lab"><font color="Green" title="E46">A12</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="c3">p</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c7">d</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c5">t</font></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>
<span class="kw">hence </span><a NAME="E3:50_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">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="E8:50"/><span class="lab"><font color="Green" title="E42">A14</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="txt" href="#E1:50"><span class="lab"><font color="Green" title="E35">A1</font></span></a>, <a class="txt" href="#E3:50"><span class="lab"><font color="Green" title="E37">A3</font></span></a>, <a class="ref" href="turing_1.html#D7" target="_self" title="TURING_1:def.7">Def8</a></span>;<br/>
<a NAME="E9:50"/>
 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="#E8:50"><span class="lab"><font color="Green" title="E42">A14</font></span></a>, <a class="txt" href="#E7:50"><span class="lab"><font color="Green" title="E41">A7</font></span></a>);<br/></span>
<span class="kw">hence </span><a NAME="E10:50"/>
<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> <font color="Maroon" title="c6">m</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="c3">p</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c7">d</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c6">m</font></span>)</span>,<font color="Maroon" title="c5">t</font></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>
