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<span class="kw">let </span><font color="Maroon" title="c1">tm1</font>, <font color="Maroon" title="c2">tm2</font> be    <a href="turing_1.html#L1" title="TURING_1:struct.1">TuringStr</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">s1</font> being   <a href="turing_1.html#NM4" title="TURING_1:NM.4">All-State</a> of <font color="Maroon" title="c1">tm1</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#NK1" title="NUMBERS:NK.1">NAT</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">tm1</font><br/>  for <font color="Olive" title="b4">s2</font> being   <a href="turing_1.html#NM4" title="TURING_1:NM.4">All-State</a> of <font color="Maroon" title="c2">tm2</font><br/>  for <font color="Olive" title="b5">s3</font> being   <a href="turing_1.html#NM4" title="TURING_1:NM.4">All-State</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">tm1</font> <a href="turing_1.html#K31" title="TURING_1:func.31">';'</a> <font color="Maroon" title="c2">tm2</font></span>)</span>  st <font color="Olive" title="b1">s1</font> is  <a href="turing_1.html#V2" title="TURING_1:attr.2">Accept-Halt</a>  &amp; <font color="Olive" title="b1">s1</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"> the <a href="turing_1.html#U4" title="TURING_1:sel.4">InitS</a> of <font color="Maroon" title="c1">tm1</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; <font color="Olive" title="b4">s2</font> is  <a href="turing_1.html#V2" title="TURING_1:attr.2">Accept-Halt</a>  &amp; <font color="Olive" title="b4">s2</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"> the <a href="turing_1.html#U4" title="TURING_1:sel.4">InitS</a> of <font color="Maroon" title="c2">tm2</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Result</a> <font color="Olive" title="b1">s1</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2_3</a></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Result</a> <font color="Olive" title="b1">s1</font></span>)</span> <a href="mcart_1.html#K3" title="MCART_1:func.3">`3_3</a></span>)</span></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span> &amp; <font color="Olive" title="b5">s3</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"> the <a href="turing_1.html#U4" title="TURING_1:sel.4">InitS</a> of <span class="p2">(<span class="default"><font color="Maroon" title="c1">tm1</font> <a href="turing_1.html#K31" title="TURING_1:func.31">';'</a> <font color="Maroon" title="c2">tm2</font></span>)</span>,<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> holds <br/>( <font color="Olive" title="b5">s3</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#K10" title="TURING_1:func.10">Result</a> <font color="Olive" title="b5">s3</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2_3</a>  <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">Result</a> <font color="Olive" title="b4">s2</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2_3</a>  &amp; <span class="p1">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Result</a> <font color="Olive" title="b5">s3</font></span>)</span> <a href="mcart_1.html#K3" title="MCART_1:func.3">`3_3</a>  <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">Result</a> <font color="Olive" title="b4">s2</font></span>)</span> <a href="mcart_1.html#K3" title="MCART_1:func.3">`3_3</a>  )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">s1</font> be   <a href="turing_1.html#NM4" title="TURING_1:NM.4">All-State</a> of <font color="Maroon" title="c1">tm1</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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#NK1" title="NUMBERS:NK.1">NAT</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">tm1</font><br/>  for <font color="Olive" title="b3">s2</font> being   <a href="turing_1.html#NM4" title="TURING_1:NM.4">All-State</a> of <font color="Maroon" title="c2">tm2</font><br/>  for <font color="Olive" title="b4">s3</font> being   <a href="turing_1.html#NM4" title="TURING_1:NM.4">All-State</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">tm1</font> <a href="turing_1.html#K31" title="TURING_1:func.31">';'</a> <font color="Maroon" title="c2">tm2</font></span>)</span>  st <font color="Maroon" title="c3">s1</font> is  <a href="turing_1.html#V2" title="TURING_1:attr.2">Accept-Halt</a>  &amp; <font color="Maroon" title="c3">s1</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"> the <a href="turing_1.html#U4" title="TURING_1:sel.4">InitS</a> of <font color="Maroon" title="c1">tm1</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; <font color="Olive" title="b3">s2</font> is  <a href="turing_1.html#V2" title="TURING_1:attr.2">Accept-Halt</a>  &amp; <font color="Olive" title="b3">s2</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"> the <a href="turing_1.html#U4" title="TURING_1:sel.4">InitS</a> of <font color="Maroon" title="c2">tm2</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Result</a> <font color="Maroon" title="c3">s1</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2_3</a></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Result</a> <font color="Maroon" title="c3">s1</font></span>)</span> <a href="mcart_1.html#K3" title="MCART_1:func.3">`3_3</a></span>)</span></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span> &amp; <font color="Olive" title="b4">s3</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"> the <a href="turing_1.html#U4" title="TURING_1:sel.4">InitS</a> of <span class="p2">(<span class="default"><font color="Maroon" title="c1">tm1</font> <a href="turing_1.html#K31" title="TURING_1:func.31">';'</a> <font color="Maroon" title="c2">tm2</font></span>)</span>,<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> holds <br/>( <font color="Olive" title="b4">s3</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#K10" title="TURING_1:func.10">Result</a> <font color="Olive" title="b4">s3</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2_3</a>  <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">Result</a> <font color="Olive" title="b3">s2</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2_3</a>  &amp; <span class="p1">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Result</a> <font color="Olive" title="b4">s3</font></span>)</span> <a href="mcart_1.html#K3" title="MCART_1:func.3">`3_3</a>  <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">Result</a> <font color="Olive" title="b3">s2</font></span>)</span> <a href="mcart_1.html#K3" title="MCART_1:func.3">`3_3</a>  )</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#NK1" title="NUMBERS:NK.1">NAT</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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">tm1</font><br/>  for <font color="Olive" title="b2">s2</font> being   <a href="turing_1.html#NM4" title="TURING_1:NM.4">All-State</a> of <font color="Maroon" title="c2">tm2</font><br/>  for <font color="Olive" title="b3">s3</font> being   <a href="turing_1.html#NM4" title="TURING_1:NM.4">All-State</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">tm1</font> <a href="turing_1.html#K31" title="TURING_1:func.31">';'</a> <font color="Maroon" title="c2">tm2</font></span>)</span>  st <font color="Maroon" title="c3">s1</font> is  <a href="turing_1.html#V2" title="TURING_1:attr.2">Accept-Halt</a>  &amp; <font color="Maroon" title="c3">s1</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"> the <a href="turing_1.html#U4" title="TURING_1:sel.4">InitS</a> of <font color="Maroon" title="c1">tm1</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; <font color="Olive" title="b2">s2</font> is  <a href="turing_1.html#V2" title="TURING_1:attr.2">Accept-Halt</a>  &amp; <font color="Olive" title="b2">s2</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"> the <a href="turing_1.html#U4" title="TURING_1:sel.4">InitS</a> of <font color="Maroon" title="c2">tm2</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Result</a> <font color="Maroon" title="c3">s1</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2_3</a></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Result</a> <font color="Maroon" title="c3">s1</font></span>)</span> <a href="mcart_1.html#K3" title="MCART_1:func.3">`3_3</a></span>)</span></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span> &amp; <font color="Olive" title="b3">s3</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"> the <a href="turing_1.html#U4" title="TURING_1:sel.4">InitS</a> of <span class="p2">(<span class="default"><font color="Maroon" title="c1">tm1</font> <a href="turing_1.html#K31" title="TURING_1:func.31">';'</a> <font color="Maroon" title="c2">tm2</font></span>)</span>,<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> holds <br/>( <font color="Olive" title="b3">s3</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#K10" title="TURING_1:func.10">Result</a> <font color="Olive" title="b3">s3</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2_3</a>  <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">Result</a> <font color="Olive" title="b2">s2</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2_3</a>  &amp; <span class="p1">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Result</a> <font color="Olive" title="b3">s3</font></span>)</span> <a href="mcart_1.html#K3" title="MCART_1:func.3">`3_3</a>  <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">Result</a> <font color="Olive" title="b2">s2</font></span>)</span> <a href="mcart_1.html#K3" title="MCART_1:func.3">`3_3</a>  )</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">tm1</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">s2</font> being   <a href="turing_1.html#NM4" title="TURING_1:NM.4">All-State</a> of <font color="Maroon" title="c2">tm2</font><br/>  for <font color="Olive" title="b2">s3</font> being   <a href="turing_1.html#NM4" title="TURING_1:NM.4">All-State</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">tm1</font> <a href="turing_1.html#K31" title="TURING_1:func.31">';'</a> <font color="Maroon" title="c2">tm2</font></span>)</span>  st <font color="Maroon" title="c3">s1</font> is  <a href="turing_1.html#V2" title="TURING_1:attr.2">Accept-Halt</a>  &amp; <font color="Maroon" title="c3">s1</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"> the <a href="turing_1.html#U4" title="TURING_1:sel.4">InitS</a> of <font color="Maroon" title="c1">tm1</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; <font color="Olive" title="b1">s2</font> is  <a href="turing_1.html#V2" title="TURING_1:attr.2">Accept-Halt</a>  &amp; <font color="Olive" title="b1">s2</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"> the <a href="turing_1.html#U4" title="TURING_1:sel.4">InitS</a> of <font color="Maroon" title="c2">tm2</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Result</a> <font color="Maroon" title="c3">s1</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2_3</a></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Result</a> <font color="Maroon" title="c3">s1</font></span>)</span> <a href="mcart_1.html#K3" title="MCART_1:func.3">`3_3</a></span>)</span></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span> &amp; <font color="Olive" title="b2">s3</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"> the <a href="turing_1.html#U4" title="TURING_1:sel.4">InitS</a> of <span class="p2">(<span class="default"><font color="Maroon" title="c1">tm1</font> <a href="turing_1.html#K31" title="TURING_1:func.31">';'</a> <font color="Maroon" title="c2">tm2</font></span>)</span>,<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> holds <br/>( <font color="Olive" title="b2">s3</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#K10" title="TURING_1:func.10">Result</a> <font color="Olive" title="b2">s3</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2_3</a>  <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">Result</a> <font color="Olive" title="b1">s2</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2_3</a>  &amp; <span class="p1">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Result</a> <font color="Olive" title="b2">s3</font></span>)</span> <a href="mcart_1.html#K3" title="MCART_1:func.3">`3_3</a>  <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">Result</a> <font color="Olive" title="b1">s2</font></span>)</span> <a href="mcart_1.html#K3" title="MCART_1:func.3">`3_3</a>  )</span><br/><span class="kw">let </span><font color="Maroon" title="c6">s2</font> be   <a href="turing_1.html#NM4" title="TURING_1:NM.4">All-State</a> of <font color="Maroon" title="c2">tm2</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">s3</font> being   <a href="turing_1.html#NM4" title="TURING_1:NM.4">All-State</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">tm1</font> <a href="turing_1.html#K31" title="TURING_1:func.31">';'</a> <font color="Maroon" title="c2">tm2</font></span>)</span>  st <font color="Maroon" title="c3">s1</font> is  <a href="turing_1.html#V2" title="TURING_1:attr.2">Accept-Halt</a>  &amp; <font color="Maroon" title="c3">s1</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"> the <a href="turing_1.html#U4" title="TURING_1:sel.4">InitS</a> of <font color="Maroon" title="c1">tm1</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; <font color="Maroon" title="c6">s2</font> is  <a href="turing_1.html#V2" title="TURING_1:attr.2">Accept-Halt</a>  &amp; <font color="Maroon" title="c6">s2</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"> the <a href="turing_1.html#U4" title="TURING_1:sel.4">InitS</a> of <font color="Maroon" title="c2">tm2</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Result</a> <font color="Maroon" title="c3">s1</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2_3</a></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Result</a> <font color="Maroon" title="c3">s1</font></span>)</span> <a href="mcart_1.html#K3" title="MCART_1:func.3">`3_3</a></span>)</span></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span> &amp; <font color="Olive" title="b1">s3</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"> the <a href="turing_1.html#U4" title="TURING_1:sel.4">InitS</a> of <span class="p2">(<span class="default"><font color="Maroon" title="c1">tm1</font> <a href="turing_1.html#K31" title="TURING_1:func.31">';'</a> <font color="Maroon" title="c2">tm2</font></span>)</span>,<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> holds <br/>( <font color="Olive" title="b1">s3</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#K10" title="TURING_1:func.10">Result</a> <font color="Olive" title="b1">s3</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2_3</a>  <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">Result</a> <font color="Maroon" title="c6">s2</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2_3</a>  &amp; <span class="p1">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Result</a> <font color="Olive" title="b1">s3</font></span>)</span> <a href="mcart_1.html#K3" title="MCART_1:func.3">`3_3</a>  <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">Result</a> <font color="Maroon" title="c6">s2</font></span>)</span> <a href="mcart_1.html#K3" title="MCART_1:func.3">`3_3</a>  )</span><br/><span class="kw">let </span><font color="Maroon" title="c7">s3</font> be   <a href="turing_1.html#NM4" title="TURING_1:NM.4">All-State</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">tm1</font> <a href="turing_1.html#K31" title="TURING_1:func.31">';'</a> <font color="Maroon" title="c2">tm2</font></span>)</span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">s1</font> is  <a href="turing_1.html#V2" title="TURING_1:attr.2">Accept-Halt</a>  &amp; <font color="Maroon" title="c3">s1</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"> the <a href="turing_1.html#U4" title="TURING_1:sel.4">InitS</a> of <font color="Maroon" title="c1">tm1</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; <font color="Maroon" title="c6">s2</font> is  <a href="turing_1.html#V2" title="TURING_1:attr.2">Accept-Halt</a>  &amp; <font color="Maroon" title="c6">s2</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"> the <a href="turing_1.html#U4" title="TURING_1:sel.4">InitS</a> of <font color="Maroon" title="c2">tm2</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Result</a> <font color="Maroon" title="c3">s1</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2_3</a></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Result</a> <font color="Maroon" title="c3">s1</font></span>)</span> <a href="mcart_1.html#K3" title="MCART_1:func.3">`3_3</a></span>)</span></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span> &amp; <font color="Maroon" title="c7">s3</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"> the <a href="turing_1.html#U4" title="TURING_1:sel.4">InitS</a> of <span class="p2">(<span class="default"><font color="Maroon" title="c1">tm1</font> <a href="turing_1.html#K31" title="TURING_1:func.31">';'</a> <font color="Maroon" title="c2">tm2</font></span>)</span>,<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> implies ( <font color="Maroon" title="c7">s3</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#K10" title="TURING_1:func.10">Result</a> <font color="Maroon" title="c7">s3</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2_3</a>  <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">Result</a> <font color="Maroon" title="c6">s2</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2_3</a>  &amp; <span class="p1">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Result</a> <font color="Maroon" title="c7">s3</font></span>)</span> <a href="mcart_1.html#K3" title="MCART_1:func.3">`3_3</a>  <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">Result</a> <font color="Maroon" title="c6">s2</font></span>)</span> <a href="mcart_1.html#K3" title="MCART_1:func.3">`3_3</a>  ) )</span><br/>











<span class="kw">set </span><font color="Maroon" title="c8">p0</font> =  the <a href="turing_1.html#U4" title="TURING_1:sel.4">InitS</a> of <font color="Maroon" title="c1">tm1</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c9">q0</font> =  the <a href="turing_1.html#U4" title="TURING_1:sel.4">InitS</a> of <font color="Maroon" title="c2">tm2</font>;<br/>
<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:94"/><span class="lab"><font color="Green" title="E65">A1</font></span>: 
<font color="Maroon" title="c3">s1</font> is  <a href="turing_1.html#V2" title="TURING_1:attr.2">Accept-Halt</a> 
 <span class="kw">and </span><br/><a NAME="E2:94"/><span class="lab"><font color="Green" title="E66">A2</font></span>: 
<font color="Maroon" title="c3">s1</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"> the <a href="turing_1.html#U4" title="TURING_1:sel.4">InitS</a> of <font color="Maroon" title="c1">tm1</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="E3:94"/><span class="lab"><font color="Green" title="E67">A3</font></span>: 
<font color="Maroon" title="c6">s2</font> is  <a href="turing_1.html#V2" title="TURING_1:attr.2">Accept-Halt</a> 
 <span class="kw">and </span><br/><a NAME="E4:94"/><span class="lab"><font color="Green" title="E68">A4</font></span>: 
<font color="Maroon" title="c6">s2</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"> the <a href="turing_1.html#U4" title="TURING_1:sel.4">InitS</a> of <font color="Maroon" title="c2">tm2</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Result</a> <font color="Maroon" title="c3">s1</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2_3</a></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Result</a> <font color="Maroon" title="c3">s1</font></span>)</span> <a href="mcart_1.html#K3" title="MCART_1:func.3">`3_3</a></span>)</span></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span>
 <span class="kw">and </span><br/><a NAME="E5:94"/><span class="lab"><font color="Green" title="E69">A5</font></span>: 
<font color="Maroon" title="c7">s3</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"> the <a href="turing_1.html#U4" title="TURING_1:sel.4">InitS</a> of <span class="p2">(<span class="default"><font color="Maroon" title="c1">tm1</font> <a href="turing_1.html#K31" title="TURING_1:func.31">';'</a> <font color="Maroon" title="c2">tm2</font></span>)</span>,<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>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c7">s3</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#K10" title="TURING_1:func.10">Result</a> <font color="Maroon" title="c7">s3</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2_3</a>  <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">Result</a> <font color="Maroon" title="c6">s2</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2_3</a>  &amp; <span class="p1">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Result</a> <font color="Maroon" title="c7">s3</font></span>)</span> <a href="mcart_1.html#K3" title="MCART_1:func.3">`3_3</a>  <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">Result</a> <font color="Maroon" title="c6">s2</font></span>)</span> <a href="mcart_1.html#K3" title="MCART_1:func.3">`3_3</a>  )</span><br/>

<span class="kw">set </span><font color="Maroon" title="c10">pF</font> =  the <a href="turing_1.html#U5" title="TURING_1:sel.5">AcceptS</a> of <font color="Maroon" title="c1">tm1</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c11">qF</font> =  the <a href="turing_1.html#U5" title="TURING_1:sel.5">AcceptS</a> of <font color="Maroon" title="c2">tm2</font>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c12">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><span class="kw"> such that </span><br/><a NAME="E7:94"/><span class="lab"><font color="Green" title="E70">A6</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <font color="Maroon" title="c3">s1</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c12">k</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1_3</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="turing_1.html#U5" title="TURING_1:sel.5">AcceptS</a> of <font color="Maroon" title="c1">tm1</font>
 <span class="kw">and </span><br/><a NAME="E8:94"/><span class="lab"><font color="Green" title="E71">A7</font></span>: 
 <a href="turing_1.html#K10" title="TURING_1:func.10">Result</a> <font color="Maroon" title="c3">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <font color="Maroon" title="c3">s1</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c12">k</font>
 <span class="kw">and </span><br/><a NAME="E9:94"/><span class="lab"><font color="Green" title="E72">A8</font></span>: 
 for <font color="Olive" title="b1">i</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">k</font> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <font color="Maroon" title="c3">s1</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1_3</a>  <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">tm1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:94"><span class="lab"><font color="Green" title="E65">A1</font></span></a>, <a class="ref" href="turing_1.html#T13" target="_self" onmouseover="rs('turing_1/T13')" onmouseout="rh()">Th13</a></span>;<br/>
<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">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="c12">k</font> implies ( <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <font color="Maroon" title="c3">s1</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> $1</span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1_3</a></span>)</span>, the <a href="turing_1.html#U4" title="TURING_1:sel.4">InitS</a> of <font color="Maroon" title="c2">tm2</font></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">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <font color="Maroon" title="c7">s3</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> $1</span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1_3</a>  &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <font color="Maroon" title="c3">s1</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> $1</span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2_3</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <font color="Maroon" title="c7">s3</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> $1</span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2_3</a>  &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <font color="Maroon" title="c3">s1</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> $1</span>)</span> <a href="mcart_1.html#K3" title="MCART_1:func.3">`3_3</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <font color="Maroon" title="c7">s3</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> $1</span>)</span> <a href="mcart_1.html#K3" title="MCART_1:func.3">`3_3</a>  ) );<br/>
<a NAME="E10:94"/><span class="lab"><font color="Green" title="E73">A9</font></span>: 
 for <font color="Olive" title="b1">i</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">i</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="94_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c13">i</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="c13">i</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c13">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1] )</span><br/>

<span class="kw">assume </span><a NAME="E1:94_1"/><span class="lab"><font color="Green" title="E74">A10</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c13">i</font>]
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="c13">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]</span><br/>

<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="94_1_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c13">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">k</font> implies ( <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <font color="Maroon" title="c3">s1</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c13">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1_3</a></span>)</span>, the <a href="turing_1.html#U4" title="TURING_1:sel.4">InitS</a> of <font color="Maroon" title="c2">tm2</font></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">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <font color="Maroon" title="c7">s3</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1_3</a>  &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <font color="Maroon" title="c3">s1</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2_3</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <font color="Maroon" title="c7">s3</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2_3</a>  &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <font color="Maroon" title="c3">s1</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="mcart_1.html#K3" title="MCART_1:func.3">`3_3</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <font color="Maroon" title="c7">s3</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="mcart_1.html#K3" title="MCART_1:func.3">`3_3</a>  ) )</span></a><div class="add"><span class="kw">set </span><font color="Maroon" title="c14">s1i1</font> = <span class="p1">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <font color="Maroon" title="c3">s1</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>;<br/><span class="kw">set </span><font color="Maroon" title="c15">s1i</font> = <span class="p1">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <font color="Maroon" title="c3">s1</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c13">i</font>;<br/><span class="kw">set </span><font color="Maroon" title="c16">s3i1</font> = <span class="p1">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <font color="Maroon" title="c7">s3</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>;<br/><span class="kw">set </span><font color="Maroon" title="c17">s3i</font> = <span class="p1">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <font color="Maroon" title="c7">s3</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c13">i</font>;<br/><a NAME="E1:94_1_1"/><span class="lab"><font color="Green" title="E75">A11</font></span>: 
<font color="Maroon" title="c13">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c13">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T29" onmouseover="rs('xreal_1/T29')" onmouseout="rh()">XREAL_1:29</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c18">f</font> =  <a href="turing_1.html#K7" title="TURING_1:func.7">TRAN</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <font color="Maroon" title="c7">s3</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c13">i</font></span>)</span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c19">h</font> =  <a href="turing_1.html#K6" title="TURING_1:func.6">Head</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <font color="Maroon" title="c3">s1</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c13">i</font></span>)</span> 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>  ;<br/><span class="kw">reconsider </span><font color="Maroon" title="c20">ss1</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <font color="Maroon" title="c3">s1</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c13">i</font></span>)</span> <a href="mcart_1.html#K3" title="MCART_1:func.3">`3_3</a>  as   <a href="turing_1.html#NM2" title="TURING_1:NM.2">Tape</a> of <font color="Maroon" title="c1">tm1</font> ;<br/><span class="kw">reconsider </span><font color="Maroon" title="c21">y</font> = <font color="Maroon" title="c20">ss1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c19">h</font> as   <a href="turing_1.html#NM3" title="TURING_1:NM.3">Symbol</a> of <font color="Maroon" title="c1">tm1</font> ;<br/><span class="kw">reconsider </span><font color="Maroon" title="c22">ss3</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <font color="Maroon" title="c7">s3</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c13">i</font></span>)</span> <a href="mcart_1.html#K3" title="MCART_1:func.3">`3_3</a>  as   <a href="turing_1.html#NM2" title="TURING_1:NM.2">Tape</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">tm1</font> <a href="turing_1.html#K31" title="TURING_1:func.31">';'</a> <font color="Maroon" title="c2">tm2</font></span>)</span> ;<br/><span class="kw">set </span><font color="Maroon" title="c23">p</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <font color="Maroon" title="c3">s1</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c13">i</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1_3</a> ;<br/><span class="kw">set </span><font color="Maroon" title="c24">g</font> =  <a href="turing_1.html#K7" title="TURING_1:func.7">TRAN</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <font color="Maroon" title="c3">s1</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c13">i</font></span>)</span>;<br/><span class="kw">assume </span><a NAME="E6:94_1_1"/><span class="lab"><font color="Green" title="E76">A12</font></span>: 
<font color="Maroon" title="c13">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">k</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <font color="Maroon" title="c3">s1</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c13">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1_3</a></span>)</span>, the <a href="turing_1.html#U4" title="TURING_1:sel.4">InitS</a> of <font color="Maroon" title="c2">tm2</font></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">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <font color="Maroon" title="c7">s3</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1_3</a>  &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <font color="Maroon" title="c3">s1</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2_3</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <font color="Maroon" title="c7">s3</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2_3</a>  &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <font color="Maroon" title="c3">s1</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="mcart_1.html#K3" title="MCART_1:func.3">`3_3</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <font color="Maroon" title="c7">s3</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="mcart_1.html#K3" title="MCART_1:func.3">`3_3</a>  )</span><br/><a NAME="E7:94_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E77">A13</font></span>: 
<font color="Maroon" title="c13">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:94_1_1"><span class="lab"><font color="Green" title="E75">A11</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/><a NAME="E8:94_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E78">A14</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <font color="Maroon" title="c3">s1</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c13">i</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1_3</a>  <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">tm1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:94"><span class="lab"><font color="Green" title="E72">A8</font></span></a></span>;<br/><a NAME="E9:94_1_1"/><span class="lab"><font color="Green" title="E79">A15</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <font color="Maroon" title="c7">s3</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c13">i</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1_3</a>  <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 <span class="p1">(<span class="default"><font color="Maroon" title="c1">tm1</font> <a href="turing_1.html#K31" title="TURING_1:func.31">';'</a> <font color="Maroon" title="c2">tm2</font></span>)</span>
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="94_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:94_1_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <font color="Maroon" title="c7">s3</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c13">i</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1_3</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="turing_1.html#U5" title="TURING_1:sel.5">AcceptS</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">tm1</font> <a href="turing_1.html#K31" title="TURING_1:func.31">';'</a> <font color="Maroon" title="c2">tm2</font></span>)</span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:94_1_1_1"/><span class="kw">then </span>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <font color="Maroon" title="c3">s1</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c13">i</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1_3</a></span>)</span>, the <a href="turing_1.html#U4" title="TURING_1:sel.4">InitS</a> of <font color="Maroon" title="c2">tm2</font></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#K1" title="DOMAIN_1:func.1">[</a><span class="default"> the <a href="turing_1.html#U5" title="TURING_1:sel.5">AcceptS</a> of <font color="Maroon" title="c1">tm1</font>, the <a href="turing_1.html#U5" title="TURING_1:sel.5">AcceptS</a> of <font color="Maroon" title="c2">tm2</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:94_1"><span class="lab"><font color="Green" title="E74">A10</font></span></a>, <a class="txt" href="#E6:94_1_1"><span class="lab"><font color="Green" title="E76">A12</font></span></a>, <a class="txt" href="#E1:94_1_1"><span class="lab"><font color="Green" title="E75">A11</font></span></a>, <a class="ref" href="turing_1.html#D31" target="_self" onmouseover="rs('turing_1/D31')" onmouseout="rh()">Def31</a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:94_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:94_1_1"><span class="lab"><font color="Green" title="E78">A14</font></span></a>, <a class="ref" href="xtuple_0.html#T1" onmouseover="rs('xtuple_0/T1')" onmouseout="rh()">XTUPLE_0:1</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="lab"><font color="Green" title="E80">A16</font></span>:  <a href="turing_1.html#K7" title="TURING_1:func.7">TRAN</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <font color="Maroon" title="c7">s3</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c13">i</font></span>)</span> = 
 the <a href="turing_1.html#U3" title="TURING_1:sel.3">Tran</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">tm1</font> <a href="turing_1.html#K31" title="TURING_1:func.31">';'</a> <font color="Maroon" title="c2">tm2</font></span>)</span> <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"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <font color="Maroon" title="c3">s1</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c13">i</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1_3</a></span>)</span>, the <a href="turing_1.html#U4" title="TURING_1:sel.4">InitS</a> of <font color="Maroon" title="c2">tm2</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<font color="Maroon" title="c21">y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:94_1"><span class="lab"><font color="Green" title="E74">A10</font></span></a>, <a class="txt" href="#E6:94_1_1"><span class="lab"><font color="Green" title="E76">A12</font></span></a>, <a class="txt" href="#E1:94_1_1"><span class="lab"><font color="Green" title="E75">A11</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>
<br/>.= 
<span class="p1"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="turing_1.html#K7" title="TURING_1:func.7">TRAN</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <font color="Maroon" title="c3">s1</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c13">i</font></span>)</span></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1_3</a></span>)</span>, the <a href="turing_1.html#U4" title="TURING_1:sel.4">InitS</a> of <font color="Maroon" title="c2">tm2</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="turing_1.html#K7" title="TURING_1:func.7">TRAN</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <font color="Maroon" title="c3">s1</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c13">i</font></span>)</span></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2_3</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">TRAN</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <font color="Maroon" title="c3">s1</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c13">i</font></span>)</span></span>)</span> <a href="mcart_1.html#K3" title="MCART_1:func.3">`3_3</a></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="#E9:94"><span class="lab"><font color="Green" title="E72">A8</font></span></a>, <a class="txt" href="#E7:94_1_1"><span class="lab"><font color="Green" title="E77">A13</font></span></a>, <a class="ref" href="turing_1.html#T43" target="_self" onmouseover="rs('turing_1/T43')" onmouseout="rh()">Th43</a></span>
;<br/><a NAME="E11:94_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E81">A17</font></span>: 
<span class="p1">(<span class="default"><a href="turing_1.html#K7" title="TURING_1:func.7">TRAN</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <font color="Maroon" title="c3">s1</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c13">i</font></span>)</span></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2_3</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="turing_1.html#K7" title="TURING_1:func.7">TRAN</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <font color="Maroon" title="c7">s3</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c13">i</font></span>)</span></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2_3</a> 
 ;<br/><span class="lab"><font color="Green" title="E82">A18</font></span>: <span class="p1">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <font color="Maroon" title="c7">s3</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> = 
 <a href="turing_1.html#K8" title="TURING_1:func.8">Following</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <font color="Maroon" title="c7">s3</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c13">i</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="turing_1.html#D7" target="_self" onmouseover="rs('turing_1/D7')" onmouseout="rh()">Def7</a></span>
<br/>.= 
<span class="p1"><a href="xtuple_0.html#K3" title="XTUPLE_0:func.3">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="turing_1.html#K7" title="TURING_1:func.7">TRAN</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <font color="Maroon" title="c7">s3</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c13">i</font></span>)</span></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1_3</a></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="turing_1.html#K6" title="TURING_1:func.6">Head</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <font color="Maroon" title="c7">s3</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c13">i</font></span>)</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#K5" title="TURING_1:func.5">offset</a> <span class="p4">(<span class="default"><a href="turing_1.html#K7" title="TURING_1:func.7">TRAN</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <font color="Maroon" title="c7">s3</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c13">i</font></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="turing_1.html#K4" title="TURING_1:func.4">Tape-Chg</a> (<font color="Maroon" title="c22">ss3</font>,<span class="p3">(<span class="default"><a href="turing_1.html#K6" title="TURING_1:func.6">Head</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <font color="Maroon" title="c7">s3</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c13">i</font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="turing_1.html#K7" title="TURING_1:func.7">TRAN</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <font color="Maroon" title="c7">s3</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c13">i</font></span>)</span></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2_3</a></span>)</span>)</span>)</span></span><a href="xtuple_0.html#K3" title="XTUPLE_0:func.3">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:94_1_1"><span class="lab"><font color="Green" title="E79">A15</font></span></a>, <a class="ref" href="turing_1.html#D6" target="_self" onmouseover="rs('turing_1/D6')" onmouseout="rh()">Def6</a></span>
;<br/><span class="lab"><font color="Green" title="E83">A19</font></span>: <span class="p1">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <font color="Maroon" title="c3">s1</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> = 
 <a href="turing_1.html#K8" title="TURING_1:func.8">Following</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <font color="Maroon" title="c3">s1</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c13">i</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="turing_1.html#D7" target="_self" onmouseover="rs('turing_1/D7')" onmouseout="rh()">Def7</a></span>
<br/>.= 
<span class="p1"><a href="xtuple_0.html#K3" title="XTUPLE_0:func.3">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="turing_1.html#K7" title="TURING_1:func.7">TRAN</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <font color="Maroon" title="c3">s1</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c13">i</font></span>)</span></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1_3</a></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c19">h</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p3">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">offset</a> <span class="p4">(<span class="default"><a href="turing_1.html#K7" title="TURING_1:func.7">TRAN</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <font color="Maroon" title="c3">s1</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c13">i</font></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="turing_1.html#K4" title="TURING_1:func.4">Tape-Chg</a> (<font color="Maroon" title="c20">ss1</font>,<font color="Maroon" title="c19">h</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="turing_1.html#K7" title="TURING_1:func.7">TRAN</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <font color="Maroon" title="c3">s1</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c13">i</font></span>)</span></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2_3</a></span>)</span>)</span>)</span></span><a href="xtuple_0.html#K3" title="XTUPLE_0:func.3">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:94_1_1"><span class="lab"><font color="Green" title="E78">A14</font></span></a>, <a class="ref" href="turing_1.html#D6" target="_self" onmouseover="rs('turing_1/D6')" onmouseout="rh()">Def6</a></span>
;<br/><span class="kw">hence </span><span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <font color="Maroon" title="c3">s1</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c13">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1_3</a></span>)</span>, the <a href="turing_1.html#U4" title="TURING_1:sel.4">InitS</a> of <font color="Maroon" title="c2">tm2</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> = 
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="turing_1.html#K7" title="TURING_1:func.7">TRAN</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <font color="Maroon" title="c3">s1</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c13">i</font></span>)</span></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1_3</a></span>)</span>, the <a href="turing_1.html#U4" title="TURING_1:sel.4">InitS</a> of <font color="Maroon" title="c2">tm2</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>

<br/>.= 
<span class="p1">(<span class="default"><a href="turing_1.html#K7" title="TURING_1:func.7">TRAN</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <font color="Maroon" title="c7">s3</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c13">i</font></span>)</span></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1_3</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:94_1_1"><span class="lab"><font color="Green" title="E80">A16</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <font color="Maroon" title="c7">s3</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1_3</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:94_1_1"><span class="lab"><font color="Green" title="E82">A18</font></span></a></span>
;<br/> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <font color="Maroon" title="c3">s1</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2_3</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <font color="Maroon" title="c7">s3</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2_3</a>  &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <font color="Maroon" title="c3">s1</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="mcart_1.html#K3" title="MCART_1:func.3">`3_3</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <font color="Maroon" title="c7">s3</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="mcart_1.html#K3" title="MCART_1:func.3">`3_3</a>  )</span><br/><a NAME="E15:94_1_1"/>
 <a href="turing_1.html#K5" title="TURING_1:func.5">offset</a> <span class="p1">(<span class="default"><a href="turing_1.html#K7" title="TURING_1:func.7">TRAN</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <font color="Maroon" title="c3">s1</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c13">i</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="turing_1.html#K5" title="TURING_1:func.5">offset</a> <span class="p1">(<span class="default"><a href="turing_1.html#K7" title="TURING_1:func.7">TRAN</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <font color="Maroon" title="c7">s3</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c13">i</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:94_1_1"><span class="lab"><font color="Green" title="E80">A16</font></span></a></span>;<br/><span class="kw">hence </span><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <font color="Maroon" title="c3">s1</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2_3</a>  = 
<span class="p1">(<span class="default"><a href="turing_1.html#K6" title="TURING_1:func.6">Head</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <font color="Maroon" title="c7">s3</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c13">i</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">offset</a> <span class="p2">(<span class="default"><a href="turing_1.html#K7" title="TURING_1:func.7">TRAN</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <font color="Maroon" title="c7">s3</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c13">i</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:94_1"><span class="lab"><font color="Green" title="E74">A10</font></span></a>, <a class="txt" href="#E6:94_1_1"><span class="lab"><font color="Green" title="E76">A12</font></span></a>, <a class="txt" href="#E1:94_1_1"><span class="lab"><font color="Green" title="E75">A11</font></span></a>, <a class="txt" href="#E13:94_1_1"><span class="lab"><font color="Green" title="E83">A19</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <font color="Maroon" title="c7">s3</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2_3</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:94_1_1"><span class="lab"><font color="Green" title="E82">A18</font></span></a></span>
;<br/> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <font color="Maroon" title="c3">s1</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="mcart_1.html#K3" title="MCART_1:func.3">`3_3</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <font color="Maroon" title="c7">s3</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="mcart_1.html#K3" title="MCART_1:func.3">`3_3</a> </span><br/><span class="kw">thus </span><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <font color="Maroon" title="c3">s1</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="mcart_1.html#K3" title="MCART_1:func.3">`3_3</a>  = 
<font color="Maroon" title="c22">ss3</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">h</font> <a href="turing_1.html#K2" title="TURING_1:func.2">.--&gt;</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="turing_1.html#K7" title="TURING_1:func.7">TRAN</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <font color="Maroon" title="c3">s1</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c13">i</font></span>)</span></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2_3</a></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:94_1"><span class="lab"><font color="Green" title="E74">A10</font></span></a>, <a class="txt" href="#E6:94_1_1"><span class="lab"><font color="Green" title="E76">A12</font></span></a>, <a class="txt" href="#E1:94_1_1"><span class="lab"><font color="Green" title="E75">A11</font></span></a>, <a class="txt" href="#E13:94_1_1"><span class="lab"><font color="Green" title="E83">A19</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <font color="Maroon" title="c7">s3</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="mcart_1.html#K3" title="MCART_1:func.3">`3_3</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:94_1"><span class="lab"><font color="Green" title="E74">A10</font></span></a>, <a class="txt" href="#E6:94_1_1"><span class="lab"><font color="Green" title="E76">A12</font></span></a>, <a class="txt" href="#E1:94_1_1"><span class="lab"><font color="Green" title="E75">A11</font></span></a>, <a class="txt" href="#E11:94_1_1"><span class="lab"><font color="Green" title="E81">A17</font></span></a>, <a class="txt" href="#E12:94_1_1"><span class="lab"><font color="Green" title="E82">A18</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="E3:94_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c13">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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">set </span><font color="Maroon" title="c13">s1k</font> = <span class="p1">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <font color="Maroon" title="c3">s1</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c12">k</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c14">s3k</font> = <span class="p1">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <font color="Maroon" title="c7">s3</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c12">k</font>;<br/>
<a NAME="E11:94"/><span class="lab"><font color="Green" title="E74">A20</font></span>: 
<font color="Maroon" title="c7">s3</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"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"> the <a href="turing_1.html#U4" title="TURING_1:sel.4">InitS</a> of <font color="Maroon" title="c1">tm1</font>, the <a href="turing_1.html#U4" title="TURING_1:sel.4">InitS</a> of <font color="Maroon" title="c2">tm2</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<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">by</span> <span class="lab"><a class="txt" href="#E5:94"><span class="lab"><font color="Green" title="E69">A5</font></span></a>, <a class="ref" href="turing_1.html#D31" target="_self" onmouseover="rs('turing_1/D31')" onmouseout="rh()">Def31</a></span>;<br/>
<a NAME="E12:94"/><span class="lab"><font color="Green" title="E75">A21</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> ]
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="94_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:94_2"/>
 <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">k</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <font color="Maroon" title="c3">s1</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1_3</a></span>)</span>, the <a href="turing_1.html#U4" title="TURING_1:sel.4">InitS</a> of <font color="Maroon" title="c2">tm2</font></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">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <font color="Maroon" title="c7">s3</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1_3</a>  &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <font color="Maroon" title="c3">s1</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2_3</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <font color="Maroon" title="c7">s3</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2_3</a>  &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <font color="Maroon" title="c3">s1</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span> <a href="mcart_1.html#K3" title="MCART_1:func.3">`3_3</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <font color="Maroon" title="c7">s3</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span> <a href="mcart_1.html#K3" title="MCART_1:func.3">`3_3</a>  )</span><br/>

<span class="lab"><font color="Green" title="E76">A22</font></span>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <font color="Maroon" title="c7">s3</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1_3</a>  = 
<font color="Maroon" title="c7">s3</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1_3</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="turing_1.html#D7" target="_self" onmouseover="rs('turing_1/D7')" onmouseout="rh()">Def7</a></span>
<br/>.= 
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"> the <a href="turing_1.html#U4" title="TURING_1:sel.4">InitS</a> of <font color="Maroon" title="c1">tm1</font>, the <a href="turing_1.html#U4" title="TURING_1:sel.4">InitS</a> of <font color="Maroon" title="c2">tm2</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:94"><span class="lab"><font color="Green" title="E74">A20</font></span></a></span>
;<br/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <font color="Maroon" title="c3">s1</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1_3</a>  = 
<font color="Maroon" title="c3">s1</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1_3</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="turing_1.html#D7" target="_self" onmouseover="rs('turing_1/D7')" onmouseout="rh()">Def7</a></span>
<br/>.= 
 the <a href="turing_1.html#U4" title="TURING_1:sel.4">InitS</a> of <font color="Maroon" title="c1">tm1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:94"><span class="lab"><font color="Green" title="E66">A2</font></span></a></span>
;<br/>
<span class="kw">hence </span><a NAME="E4:94_2"/>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <font color="Maroon" title="c3">s1</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1_3</a></span>)</span>, the <a href="turing_1.html#U4" title="TURING_1:sel.4">InitS</a> of <font color="Maroon" title="c2">tm2</font></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">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <font color="Maroon" title="c7">s3</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1_3</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:94_2"><span class="lab"><font color="Green" title="E76">A22</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <font color="Maroon" title="c3">s1</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2_3</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <font color="Maroon" title="c7">s3</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2_3</a>  &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <font color="Maroon" title="c3">s1</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span> <a href="mcart_1.html#K3" title="MCART_1:func.3">`3_3</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <font color="Maroon" title="c7">s3</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span> <a href="mcart_1.html#K3" title="MCART_1:func.3">`3_3</a>  )</span><br/>

<span class="kw">thus </span><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <font color="Maroon" title="c3">s1</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2_3</a>  = 
<font color="Maroon" title="c3">s1</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2_3</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="turing_1.html#D7" target="_self" onmouseover="rs('turing_1/D7')" onmouseout="rh()">Def7</a></span>
<br/>.= 
<font color="Maroon" title="c4">h</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:94"><span class="lab"><font color="Green" title="E66">A2</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c7">s3</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2_3</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:94"><span class="lab"><font color="Green" title="E69">A5</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <font color="Maroon" title="c7">s3</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2_3</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="turing_1.html#D7" target="_self" onmouseover="rs('turing_1/D7')" onmouseout="rh()">Def7</a></span>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <font color="Maroon" title="c3">s1</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span> <a href="mcart_1.html#K3" title="MCART_1:func.3">`3_3</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <font color="Maroon" title="c7">s3</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span> <a href="mcart_1.html#K3" title="MCART_1:func.3">`3_3</a> </span><br/>

<span class="kw">thus </span><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <font color="Maroon" title="c3">s1</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span> <a href="mcart_1.html#K3" title="MCART_1:func.3">`3_3</a>  = 
<font color="Maroon" title="c3">s1</font> <a href="mcart_1.html#K3" title="MCART_1:func.3">`3_3</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="turing_1.html#D7" target="_self" onmouseover="rs('turing_1/D7')" onmouseout="rh()">Def7</a></span>
<br/>.= 
<font color="Maroon" title="c5">t</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:94"><span class="lab"><font color="Green" title="E66">A2</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c7">s3</font> <a href="mcart_1.html#K3" title="MCART_1:func.3">`3_3</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:94"><span class="lab"><font color="Green" title="E69">A5</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <font color="Maroon" title="c7">s3</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span> <a href="mcart_1.html#K3" title="MCART_1:func.3">`3_3</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="turing_1.html#D7" target="_self" onmouseover="rs('turing_1/D7')" onmouseout="rh()">Def7</a></span>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="E13:94"/><span class="lab"><font color="Green" title="E76">A23</font></span>: 
 for <font color="Olive" title="b1">i</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">i</font>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S2" onmouseover="rs('nat_1/S2')" onmouseout="rh()">NAT_1:sch 2</a>(<a class="txt" href="#E12:94"><span class="lab"><font color="Green" title="E75">A21</font></span></a>, <a class="txt" href="#E10:94"><span class="lab"><font color="Green" title="E73">A9</font></span></a>);<br/></span>
<a NAME="E14:94"/><span class="kw">then </span><span class="lab"><font color="Green" title="E77">A24</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <font color="Maroon" title="c3">s1</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c12">k</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2_3</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <font color="Maroon" title="c7">s3</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c12">k</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2_3</a> 
 
;<br/>
<span class="kw">consider </span><font color="Maroon" title="c15">m</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><span class="kw"> such that </span><br/><a NAME="E16:94"/><span class="lab"><font color="Green" title="E78">A25</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <font color="Maroon" title="c6">s2</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c15">m</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1_3</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="turing_1.html#U5" title="TURING_1:sel.5">AcceptS</a> of <font color="Maroon" title="c2">tm2</font>
 <span class="kw">and </span><br/><a NAME="E17:94"/><span class="lab"><font color="Green" title="E79">A26</font></span>: 
 <a href="turing_1.html#K10" title="TURING_1:func.10">Result</a> <font color="Maroon" title="c6">s2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <font color="Maroon" title="c6">s2</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c15">m</font>
 <span class="kw">and </span><br/><a NAME="E18:94"/><span class="lab"><font color="Green" title="E80">A27</font></span>: 
 for <font color="Olive" title="b1">i</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c15">m</font> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <font color="Maroon" title="c6">s2</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1_3</a>  <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="c2">tm2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:94"><span class="lab"><font color="Green" title="E67">A3</font></span></a>, <a class="ref" href="turing_1.html#T13" target="_self" onmouseover="rs('turing_1/T13')" onmouseout="rh()">Th13</a></span>;<br/>
<span class="kw">defpred </span><font color="Maroon">S<sub>2</sub></font>[  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">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="c15">m</font> implies ( <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"> the <a href="turing_1.html#U5" title="TURING_1:sel.5">AcceptS</a> of <font color="Maroon" title="c1">tm1</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <font color="Maroon" title="c6">s2</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> $1</span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1_3</a></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">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <font color="Maroon" title="c7">s3</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c12">k</font></span>)</span></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> $1</span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1_3</a>  &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <font color="Maroon" title="c6">s2</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> $1</span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2_3</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <font color="Maroon" title="c7">s3</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c12">k</font></span>)</span></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> $1</span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2_3</a>  &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <font color="Maroon" title="c6">s2</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> $1</span>)</span> <a href="mcart_1.html#K3" title="MCART_1:func.3">`3_3</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <font color="Maroon" title="c7">s3</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c12">k</font></span>)</span></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> $1</span>)</span> <a href="mcart_1.html#K3" title="MCART_1:func.3">`3_3</a>  ) );<br/>
<a NAME="E19:94"/><span class="lab"><font color="Green" title="E81">A28</font></span>: 
 for <font color="Olive" title="b1">i</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">i</font>] holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="94_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c16">i</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c16">i</font>] implies <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c16">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1] )</span><br/>

<span class="kw">assume </span><a NAME="E1:94_3"/><span class="lab"><font color="Green" title="E82">A29</font></span>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c16">i</font>]
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c16">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]</span><br/>

<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="94_3_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c16">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">m</font> implies ( <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"> the <a href="turing_1.html#U5" title="TURING_1:sel.5">AcceptS</a> of <font color="Maroon" title="c1">tm1</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <font color="Maroon" title="c6">s2</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c16">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1_3</a></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">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <font color="Maroon" title="c7">s3</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c12">k</font></span>)</span></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c16">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1_3</a>  &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <font color="Maroon" title="c6">s2</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c16">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2_3</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <font color="Maroon" title="c7">s3</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c12">k</font></span>)</span></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c16">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2_3</a>  &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <font color="Maroon" title="c6">s2</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c16">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="mcart_1.html#K3" title="MCART_1:func.3">`3_3</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <font color="Maroon" title="c7">s3</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c12">k</font></span>)</span></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c16">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="mcart_1.html#K3" title="MCART_1:func.3">`3_3</a>  ) )</span></a><div class="add"><span class="kw">set </span><font color="Maroon" title="c17">s2i1</font> = <span class="p1">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <font color="Maroon" title="c6">s2</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>;<br/><span class="kw">set </span><font color="Maroon" title="c18">s2i</font> = <span class="p1">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <font color="Maroon" title="c6">s2</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c16">i</font>;<br/><span class="kw">set </span><font color="Maroon" title="c19">ski1</font> = <span class="p1">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <font color="Maroon" title="c7">s3</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c12">k</font></span>)</span></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>;<br/><span class="kw">set </span><font color="Maroon" title="c20">ski</font> = <span class="p1">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <font color="Maroon" title="c7">s3</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c12">k</font></span>)</span></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c16">i</font>;<br/><a NAME="E1:94_3_1"/><span class="lab"><font color="Green" title="E83">A30</font></span>: 
<font color="Maroon" title="c16">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c16">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T29" onmouseover="rs('xreal_1/T29')" onmouseout="rh()">XREAL_1:29</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c21">ssk</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <font color="Maroon" title="c7">s3</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c12">k</font></span>)</span></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c16">i</font></span>)</span> <a href="mcart_1.html#K3" title="MCART_1:func.3">`3_3</a>  as   <a href="turing_1.html#NM2" title="TURING_1:NM.2">Tape</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">tm1</font> <a href="turing_1.html#K31" title="TURING_1:func.31">';'</a> <font color="Maroon" title="c2">tm2</font></span>)</span> ;<br/><span class="kw">set </span><font color="Maroon" title="c22">f</font> =  <a href="turing_1.html#K7" title="TURING_1:func.7">TRAN</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <font color="Maroon" title="c7">s3</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c12">k</font></span>)</span></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c16">i</font></span>)</span>;<br/><span class="kw">set </span><font color="Maroon" title="c23">q</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <font color="Maroon" title="c6">s2</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c16">i</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1_3</a> ;<br/><span class="kw">set </span><font color="Maroon" title="c24">g</font> =  <a href="turing_1.html#K7" title="TURING_1:func.7">TRAN</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <font color="Maroon" title="c6">s2</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c16">i</font></span>)</span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c25">h</font> =  <a href="turing_1.html#K6" title="TURING_1:func.6">Head</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <font color="Maroon" title="c6">s2</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c16">i</font></span>)</span> 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>  ;<br/><span class="kw">reconsider </span><font color="Maroon" title="c26">ss2</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <font color="Maroon" title="c6">s2</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c16">i</font></span>)</span> <a href="mcart_1.html#K3" title="MCART_1:func.3">`3_3</a>  as   <a href="turing_1.html#NM2" title="TURING_1:NM.2">Tape</a> of <font color="Maroon" title="c2">tm2</font> ;<br/><span class="kw">reconsider </span><font color="Maroon" title="c27">y</font> = <font color="Maroon" title="c26">ss2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c25">h</font> as   <a href="turing_1.html#NM3" title="TURING_1:NM.3">Symbol</a> of <font color="Maroon" title="c2">tm2</font> ;<br/><span class="kw">assume </span><a NAME="E6:94_3_1"/><span class="lab"><font color="Green" title="E84">A31</font></span>: 
<font color="Maroon" title="c16">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">m</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"> the <a href="turing_1.html#U5" title="TURING_1:sel.5">AcceptS</a> of <font color="Maroon" title="c1">tm1</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <font color="Maroon" title="c6">s2</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c16">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1_3</a></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">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <font color="Maroon" title="c7">s3</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c12">k</font></span>)</span></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c16">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1_3</a>  &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <font color="Maroon" title="c6">s2</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c16">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2_3</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <font color="Maroon" title="c7">s3</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c12">k</font></span>)</span></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c16">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2_3</a>  &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <font color="Maroon" title="c6">s2</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c16">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="mcart_1.html#K3" title="MCART_1:func.3">`3_3</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <font color="Maroon" title="c7">s3</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c12">k</font></span>)</span></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c16">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="mcart_1.html#K3" title="MCART_1:func.3">`3_3</a>  )</span><br/><span class="kw">then </span><span class="lab"><font color="Green" title="E85">A32</font></span>:  <a href="turing_1.html#K7" title="TURING_1:func.7">TRAN</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <font color="Maroon" title="c7">s3</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c12">k</font></span>)</span></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c16">i</font></span>)</span> = 
 the <a href="turing_1.html#U3" title="TURING_1:sel.3">Tran</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">tm1</font> <a href="turing_1.html#K31" title="TURING_1:func.31">';'</a> <font color="Maroon" title="c2">tm2</font></span>)</span> <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"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"> the <a href="turing_1.html#U5" title="TURING_1:sel.5">AcceptS</a> of <font color="Maroon" title="c1">tm1</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <font color="Maroon" title="c6">s2</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c16">i</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1_3</a></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<font color="Maroon" title="c27">y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:94_3"><span class="lab"><font color="Green" title="E82">A29</font></span></a>, <a class="txt" href="#E1:94_3_1"><span class="lab"><font color="Green" title="E83">A30</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>
<br/>.= 
<span class="p1"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"> the <a href="turing_1.html#U5" title="TURING_1:sel.5">AcceptS</a> of <font color="Maroon" title="c1">tm1</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="turing_1.html#K7" title="TURING_1:func.7">TRAN</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <font color="Maroon" title="c6">s2</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c16">i</font></span>)</span></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1_3</a></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="turing_1.html#K7" title="TURING_1:func.7">TRAN</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <font color="Maroon" title="c6">s2</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c16">i</font></span>)</span></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2_3</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">TRAN</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <font color="Maroon" title="c6">s2</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c16">i</font></span>)</span></span>)</span> <a href="mcart_1.html#K3" title="MCART_1:func.3">`3_3</a></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#T44" target="_self" onmouseover="rs('turing_1/T44')" onmouseout="rh()">Th44</a></span>
;<br/><a NAME="E8:94_3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E86">A33</font></span>: 
<span class="p1">(<span class="default"><a href="turing_1.html#K7" title="TURING_1:func.7">TRAN</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <font color="Maroon" title="c6">s2</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c16">i</font></span>)</span></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2_3</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="turing_1.html#K7" title="TURING_1:func.7">TRAN</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <font color="Maroon" title="c7">s3</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c12">k</font></span>)</span></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c16">i</font></span>)</span></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2_3</a> 
 ;<br/><a NAME="E9:94_3_1"/>
<font color="Maroon" title="c16">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c15">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:94_3_1"><span class="lab"><font color="Green" title="E84">A31</font></span></a>, <a class="txt" href="#E1:94_3_1"><span class="lab"><font color="Green" title="E83">A30</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/><a NAME="E10:94_3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E87">A34</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <font color="Maroon" title="c6">s2</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c16">i</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1_3</a>  <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="c2">tm2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:94"><span class="lab"><font color="Green" title="E80">A27</font></span></a></span>;<br/><a NAME="E11:94_3_1"/><span class="lab"><font color="Green" title="E88">A35</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <font color="Maroon" title="c7">s3</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c12">k</font></span>)</span></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c16">i</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1_3</a>  <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 <span class="p1">(<span class="default"><font color="Maroon" title="c1">tm1</font> <a href="turing_1.html#K31" title="TURING_1:func.31">';'</a> <font color="Maroon" title="c2">tm2</font></span>)</span>
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="94_3_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:94_3_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <font color="Maroon" title="c7">s3</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c12">k</font></span>)</span></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c16">i</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1_3</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="turing_1.html#U5" title="TURING_1:sel.5">AcceptS</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">tm1</font> <a href="turing_1.html#K31" title="TURING_1:func.31">';'</a> <font color="Maroon" title="c2">tm2</font></span>)</span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:94_3_1_2"/><span class="kw">then </span>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"> the <a href="turing_1.html#U5" title="TURING_1:sel.5">AcceptS</a> of <font color="Maroon" title="c1">tm1</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <font color="Maroon" title="c6">s2</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c16">i</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1_3</a></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#K1" title="DOMAIN_1:func.1">[</a><span class="default"> the <a href="turing_1.html#U5" title="TURING_1:sel.5">AcceptS</a> of <font color="Maroon" title="c1">tm1</font>, the <a href="turing_1.html#U5" title="TURING_1:sel.5">AcceptS</a> of <font color="Maroon" title="c2">tm2</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:94_3"><span class="lab"><font color="Green" title="E82">A29</font></span></a>, <a class="txt" href="#E6:94_3_1"><span class="lab"><font color="Green" title="E84">A31</font></span></a>, <a class="txt" href="#E1:94_3_1"><span class="lab"><font color="Green" title="E83">A30</font></span></a>, <a class="ref" href="turing_1.html#D31" target="_self" onmouseover="rs('turing_1/D31')" onmouseout="rh()">Def31</a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:94_3_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:94_3_1"><span class="lab"><font color="Green" title="E87">A34</font></span></a>, <a class="ref" href="xtuple_0.html#T1" onmouseover="rs('xtuple_0/T1')" onmouseout="rh()">XTUPLE_0:1</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="lab"><font color="Green" title="E89">A36</font></span>: <span class="p1">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <font color="Maroon" title="c7">s3</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c12">k</font></span>)</span></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> = 
 <a href="turing_1.html#K8" title="TURING_1:func.8">Following</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <font color="Maroon" title="c7">s3</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c12">k</font></span>)</span></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c16">i</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="turing_1.html#D7" target="_self" onmouseover="rs('turing_1/D7')" onmouseout="rh()">Def7</a></span>
<br/>.= 
<span class="p1"><a href="xtuple_0.html#K3" title="XTUPLE_0:func.3">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="turing_1.html#K7" title="TURING_1:func.7">TRAN</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <font color="Maroon" title="c7">s3</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c12">k</font></span>)</span></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c16">i</font></span>)</span></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1_3</a></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="turing_1.html#K6" title="TURING_1:func.6">Head</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <font color="Maroon" title="c7">s3</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c12">k</font></span>)</span></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c16">i</font></span>)</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#K5" title="TURING_1:func.5">offset</a> <span class="p4">(<span class="default"><a href="turing_1.html#K7" title="TURING_1:func.7">TRAN</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <font color="Maroon" title="c7">s3</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c12">k</font></span>)</span></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c16">i</font></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="turing_1.html#K4" title="TURING_1:func.4">Tape-Chg</a> (<font color="Maroon" title="c21">ssk</font>,<span class="p3">(<span class="default"><a href="turing_1.html#K6" title="TURING_1:func.6">Head</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <font color="Maroon" title="c7">s3</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c12">k</font></span>)</span></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c16">i</font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="turing_1.html#K7" title="TURING_1:func.7">TRAN</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <font color="Maroon" title="c7">s3</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c12">k</font></span>)</span></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c16">i</font></span>)</span></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2_3</a></span>)</span>)</span>)</span></span><a href="xtuple_0.html#K3" title="XTUPLE_0:func.3">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:94_3_1"><span class="lab"><font color="Green" title="E88">A35</font></span></a>, <a class="ref" href="turing_1.html#D6" target="_self" onmouseover="rs('turing_1/D6')" onmouseout="rh()">Def6</a></span>
;<br/><span class="lab"><font color="Green" title="E90">A37</font></span>: <span class="p1">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <font color="Maroon" title="c6">s2</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> = 
 <a href="turing_1.html#K8" title="TURING_1:func.8">Following</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <font color="Maroon" title="c6">s2</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c16">i</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="turing_1.html#D7" target="_self" onmouseover="rs('turing_1/D7')" onmouseout="rh()">Def7</a></span>
<br/>.= 
<span class="p1"><a href="xtuple_0.html#K3" title="XTUPLE_0:func.3">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="turing_1.html#K7" title="TURING_1:func.7">TRAN</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <font color="Maroon" title="c6">s2</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c16">i</font></span>)</span></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1_3</a></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c25">h</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p3">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">offset</a> <span class="p4">(<span class="default"><a href="turing_1.html#K7" title="TURING_1:func.7">TRAN</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <font color="Maroon" title="c6">s2</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c16">i</font></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="turing_1.html#K4" title="TURING_1:func.4">Tape-Chg</a> (<font color="Maroon" title="c26">ss2</font>,<font color="Maroon" title="c25">h</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="turing_1.html#K7" title="TURING_1:func.7">TRAN</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <font color="Maroon" title="c6">s2</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c16">i</font></span>)</span></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2_3</a></span>)</span>)</span>)</span></span><a href="xtuple_0.html#K3" title="XTUPLE_0:func.3">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:94_3_1"><span class="lab"><font color="Green" title="E87">A34</font></span></a>, <a class="ref" href="turing_1.html#D6" target="_self" onmouseover="rs('turing_1/D6')" onmouseout="rh()">Def6</a></span>
;<br/><span class="kw">hence </span><span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"> the <a href="turing_1.html#U5" title="TURING_1:sel.5">AcceptS</a> of <font color="Maroon" title="c1">tm1</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <font color="Maroon" title="c6">s2</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c16">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1_3</a></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> = 
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"> the <a href="turing_1.html#U5" title="TURING_1:sel.5">AcceptS</a> of <font color="Maroon" title="c1">tm1</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="turing_1.html#K7" title="TURING_1:func.7">TRAN</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <font color="Maroon" title="c6">s2</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c16">i</font></span>)</span></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1_3</a></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>

<br/>.= 
<span class="p1">(<span class="default"><a href="turing_1.html#K7" title="TURING_1:func.7">TRAN</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <font color="Maroon" title="c7">s3</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c12">k</font></span>)</span></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c16">i</font></span>)</span></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1_3</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:94_3_1"><span class="lab"><font color="Green" title="E85">A32</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <font color="Maroon" title="c7">s3</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c12">k</font></span>)</span></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c16">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1_3</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:94_3_1"><span class="lab"><font color="Green" title="E89">A36</font></span></a></span>
;<br/> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <font color="Maroon" title="c6">s2</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c16">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2_3</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <font color="Maroon" title="c7">s3</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c12">k</font></span>)</span></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c16">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2_3</a>  &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <font color="Maroon" title="c6">s2</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c16">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="mcart_1.html#K3" title="MCART_1:func.3">`3_3</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <font color="Maroon" title="c7">s3</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c12">k</font></span>)</span></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c16">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="mcart_1.html#K3" title="MCART_1:func.3">`3_3</a>  )</span><br/><a NAME="E15:94_3_1"/>
 <a href="turing_1.html#K5" title="TURING_1:func.5">offset</a> <span class="p1">(<span class="default"><a href="turing_1.html#K7" title="TURING_1:func.7">TRAN</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <font color="Maroon" title="c6">s2</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c16">i</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="turing_1.html#K5" title="TURING_1:func.5">offset</a> <span class="p1">(<span class="default"><a href="turing_1.html#K7" title="TURING_1:func.7">TRAN</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <font color="Maroon" title="c7">s3</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c12">k</font></span>)</span></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c16">i</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:94_3_1"><span class="lab"><font color="Green" title="E85">A32</font></span></a></span>;<br/><span class="kw">hence </span><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <font color="Maroon" title="c6">s2</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c16">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2_3</a>  = 
<span class="p1">(<span class="default"><a href="turing_1.html#K6" title="TURING_1:func.6">Head</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <font color="Maroon" title="c7">s3</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c12">k</font></span>)</span></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c16">i</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">offset</a> <span class="p2">(<span class="default"><a href="turing_1.html#K7" title="TURING_1:func.7">TRAN</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <font color="Maroon" title="c7">s3</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c12">k</font></span>)</span></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c16">i</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:94_3"><span class="lab"><font color="Green" title="E82">A29</font></span></a>, <a class="txt" href="#E6:94_3_1"><span class="lab"><font color="Green" title="E84">A31</font></span></a>, <a class="txt" href="#E1:94_3_1"><span class="lab"><font color="Green" title="E83">A30</font></span></a>, <a class="txt" href="#E13:94_3_1"><span class="lab"><font color="Green" title="E90">A37</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <font color="Maroon" title="c7">s3</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c12">k</font></span>)</span></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c16">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2_3</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:94_3_1"><span class="lab"><font color="Green" title="E89">A36</font></span></a></span>
;<br/> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <font color="Maroon" title="c6">s2</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c16">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="mcart_1.html#K3" title="MCART_1:func.3">`3_3</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <font color="Maroon" title="c7">s3</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c12">k</font></span>)</span></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c16">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="mcart_1.html#K3" title="MCART_1:func.3">`3_3</a> </span><br/><span class="kw">thus </span><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <font color="Maroon" title="c6">s2</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c16">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="mcart_1.html#K3" title="MCART_1:func.3">`3_3</a>  = 
<font color="Maroon" title="c21">ssk</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c25">h</font> <a href="turing_1.html#K2" title="TURING_1:func.2">.--&gt;</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="turing_1.html#K7" title="TURING_1:func.7">TRAN</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <font color="Maroon" title="c6">s2</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c16">i</font></span>)</span></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2_3</a></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:94_3"><span class="lab"><font color="Green" title="E82">A29</font></span></a>, <a class="txt" href="#E6:94_3_1"><span class="lab"><font color="Green" title="E84">A31</font></span></a>, <a class="txt" href="#E1:94_3_1"><span class="lab"><font color="Green" title="E83">A30</font></span></a>, <a class="txt" href="#E13:94_3_1"><span class="lab"><font color="Green" title="E90">A37</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <font color="Maroon" title="c7">s3</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c12">k</font></span>)</span></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c16">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="mcart_1.html#K3" title="MCART_1:func.3">`3_3</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:94_3"><span class="lab"><font color="Green" title="E82">A29</font></span></a>, <a class="txt" href="#E6:94_3_1"><span class="lab"><font color="Green" title="E84">A31</font></span></a>, <a class="txt" href="#E1:94_3_1"><span class="lab"><font color="Green" title="E83">A30</font></span></a>, <a class="txt" href="#E8:94_3_1"><span class="lab"><font color="Green" title="E86">A33</font></span></a>, <a class="txt" href="#E12:94_3_1"><span class="lab"><font color="Green" title="E89">A36</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="E3:94_3"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c16">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="E20:94"/><span class="lab"><font color="Green" title="E82">A38</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <font color="Maroon" title="c3">s1</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c12">k</font></span>)</span> <a href="mcart_1.html#K3" title="MCART_1:func.3">`3_3</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <font color="Maroon" title="c7">s3</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c12">k</font></span>)</span> <a href="mcart_1.html#K3" title="MCART_1:func.3">`3_3</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:94"><span class="lab"><font color="Green" title="E76">A23</font></span></a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c16">s2m</font> = <span class="p1">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <font color="Maroon" title="c6">s2</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c15">m</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c17">skm</font> = <span class="p1">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <font color="Maroon" title="c7">s3</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c12">k</font></span>)</span></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c15">m</font>;<br/>
<a NAME="E21:94"/><span class="lab"><font color="Green" title="E83">A39</font></span>: 
<span class="p1">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <font color="Maroon" title="c7">s3</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c15">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <font color="Maroon" title="c7">s3</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c12">k</font></span>)</span></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c15">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="turing_1.html#T10" target="_self" onmouseover="rs('turing_1/T10')" onmouseout="rh()">Th10</a></span>;<br/>
<a NAME="E22:94"/><span class="lab"><font color="Green" title="E84">A40</font></span>: 
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <font color="Maroon" title="c3">s1</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c12">k</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1_3</a></span>)</span>, the <a href="turing_1.html#U4" title="TURING_1:sel.4">InitS</a> of <font color="Maroon" title="c2">tm2</font></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">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <font color="Maroon" title="c7">s3</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c12">k</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1_3</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:94"><span class="lab"><font color="Green" title="E76">A23</font></span></a></span>;<br/>
<a NAME="E23:94"/><span class="lab"><font color="Green" title="E85">A41</font></span>: 
<font color="Maroon">S<sub>2</sub></font>[ <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> ]
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="94_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:94_4"/>
 <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">m</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"> the <a href="turing_1.html#U5" title="TURING_1:sel.5">AcceptS</a> of <font color="Maroon" title="c1">tm1</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <font color="Maroon" title="c6">s2</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1_3</a></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">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <font color="Maroon" title="c7">s3</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c12">k</font></span>)</span></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1_3</a>  &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <font color="Maroon" title="c6">s2</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2_3</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <font color="Maroon" title="c7">s3</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c12">k</font></span>)</span></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2_3</a>  &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <font color="Maroon" title="c6">s2</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span> <a href="mcart_1.html#K3" title="MCART_1:func.3">`3_3</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <font color="Maroon" title="c7">s3</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c12">k</font></span>)</span></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span> <a href="mcart_1.html#K3" title="MCART_1:func.3">`3_3</a>  )</span><br/>

<span class="kw">thus </span><span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"> the <a href="turing_1.html#U5" title="TURING_1:sel.5">AcceptS</a> of <font color="Maroon" title="c1">tm1</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <font color="Maroon" title="c6">s2</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1_3</a></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> = 
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"> the <a href="turing_1.html#U5" title="TURING_1:sel.5">AcceptS</a> of <font color="Maroon" title="c1">tm1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">s2</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1_3</a></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="turing_1.html#D7" target="_self" onmouseover="rs('turing_1/D7')" onmouseout="rh()">Def7</a></span>
<br/>.= 
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"> the <a href="turing_1.html#U5" title="TURING_1:sel.5">AcceptS</a> of <font color="Maroon" title="c1">tm1</font>, the <a href="turing_1.html#U4" title="TURING_1:sel.4">InitS</a> of <font color="Maroon" title="c2">tm2</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:94"><span class="lab"><font color="Green" title="E68">A4</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <font color="Maroon" title="c7">s3</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c12">k</font></span>)</span></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1_3</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:94"><span class="lab"><font color="Green" title="E70">A6</font></span></a>, <a class="txt" href="#E22:94"><span class="lab"><font color="Green" title="E84">A40</font></span></a>, <a class="ref" href="turing_1.html#D7" target="_self" onmouseover="rs('turing_1/D7')" onmouseout="rh()">Def7</a></span>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <font color="Maroon" title="c6">s2</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2_3</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <font color="Maroon" title="c7">s3</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c12">k</font></span>)</span></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2_3</a>  &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <font color="Maroon" title="c6">s2</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span> <a href="mcart_1.html#K3" title="MCART_1:func.3">`3_3</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <font color="Maroon" title="c7">s3</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c12">k</font></span>)</span></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span> <a href="mcart_1.html#K3" title="MCART_1:func.3">`3_3</a>  )</span><br/>

<span class="kw">thus </span><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <font color="Maroon" title="c6">s2</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2_3</a>  = 
<font color="Maroon" title="c6">s2</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2_3</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="turing_1.html#D7" target="_self" onmouseover="rs('turing_1/D7')" onmouseout="rh()">Def7</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <font color="Maroon" title="c7">s3</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c12">k</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2_3</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:94"><span class="lab"><font color="Green" title="E68">A4</font></span></a>, <a class="txt" href="#E8:94"><span class="lab"><font color="Green" title="E71">A7</font></span></a>, <a class="txt" href="#E14:94"><span class="lab"><font color="Green" title="E77">A24</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <font color="Maroon" title="c7">s3</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c12">k</font></span>)</span></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2_3</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="turing_1.html#D7" target="_self" onmouseover="rs('turing_1/D7')" onmouseout="rh()">Def7</a></span>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <font color="Maroon" title="c6">s2</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span> <a href="mcart_1.html#K3" title="MCART_1:func.3">`3_3</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <font color="Maroon" title="c7">s3</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c12">k</font></span>)</span></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span> <a href="mcart_1.html#K3" title="MCART_1:func.3">`3_3</a> </span><br/>

<span class="kw">thus </span><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <font color="Maroon" title="c6">s2</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span> <a href="mcart_1.html#K3" title="MCART_1:func.3">`3_3</a>  = 
<font color="Maroon" title="c6">s2</font> <a href="mcart_1.html#K3" title="MCART_1:func.3">`3_3</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="turing_1.html#D7" target="_self" onmouseover="rs('turing_1/D7')" onmouseout="rh()">Def7</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <font color="Maroon" title="c7">s3</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c12">k</font></span>)</span> <a href="mcart_1.html#K3" title="MCART_1:func.3">`3_3</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:94"><span class="lab"><font color="Green" title="E68">A4</font></span></a>, <a class="txt" href="#E8:94"><span class="lab"><font color="Green" title="E71">A7</font></span></a>, <a class="txt" href="#E20:94"><span class="lab"><font color="Green" title="E82">A38</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <font color="Maroon" title="c7">s3</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c12">k</font></span>)</span></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span> <a href="mcart_1.html#K3" title="MCART_1:func.3">`3_3</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="turing_1.html#D7" target="_self" onmouseover="rs('turing_1/D7')" onmouseout="rh()">Def7</a></span>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="E24:94"/><span class="lab"><font color="Green" title="E86">A42</font></span>: 
 for <font color="Olive" title="b1">i</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">i</font>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S2" onmouseover="rs('nat_1/S2')" onmouseout="rh()">NAT_1:sch 2</a>(<a class="txt" href="#E23:94"><span class="lab"><font color="Green" title="E85">A41</font></span></a>, <a class="txt" href="#E19:94"><span class="lab"><font color="Green" title="E81">A28</font></span></a>);<br/></span>
<a NAME="E25:94"/><span class="kw">then </span>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"> the <a href="turing_1.html#U5" title="TURING_1:sel.5">AcceptS</a> of <font color="Maroon" title="c1">tm1</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <font color="Maroon" title="c6">s2</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c15">m</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1_3</a></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">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <font color="Maroon" title="c7">s3</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c12">k</font></span>)</span></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c15">m</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1_3</a> 
 
;<br/>
<a NAME="E26:94"/><span class="kw">then </span><span class="lab"><font color="Green" title="E87">A43</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <font color="Maroon" title="c7">s3</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c15">m</font></span>)</span></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1_3</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="turing_1.html#U5" title="TURING_1:sel.5">AcceptS</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">tm1</font> <a href="turing_1.html#K31" title="TURING_1:func.31">';'</a> <font color="Maroon" title="c2">tm2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:94"><span class="lab"><font color="Green" title="E78">A25</font></span></a>, <a class="txt" href="#E21:94"><span class="lab"><font color="Green" title="E83">A39</font></span></a>, <a class="ref" href="turing_1.html#D31" target="_self" onmouseover="rs('turing_1/D31')" onmouseout="rh()">Def31</a></span>;<br/>
<span class="kw">hence </span><a NAME="E27:94"/><span class="lab"><font color="Green" title="E88">A44</font></span>: 
<font color="Maroon" title="c7">s3</font> is  <a href="turing_1.html#V2" title="TURING_1:attr.2">Accept-Halt</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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">Result</a> <font color="Maroon" title="c7">s3</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2_3</a>  <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">Result</a> <font color="Maroon" title="c6">s2</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2_3</a>  &amp; <span class="p1">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Result</a> <font color="Maroon" title="c7">s3</font></span>)</span> <a href="mcart_1.html#K3" title="MCART_1:func.3">`3_3</a>  <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">Result</a> <font color="Maroon" title="c6">s2</font></span>)</span> <a href="mcart_1.html#K3" title="MCART_1:func.3">`3_3</a>  )</span><br/>

<a NAME="E28:94"/>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <font color="Maroon" title="c6">s2</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c15">m</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2_3</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <font color="Maroon" title="c7">s3</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c12">k</font></span>)</span></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c15">m</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2_3</a>  &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <font color="Maroon" title="c6">s2</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c15">m</font></span>)</span> <a href="mcart_1.html#K3" title="MCART_1:func.3">`3_3</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <font color="Maroon" title="c7">s3</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c12">k</font></span>)</span></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c15">m</font></span>)</span> <a href="mcart_1.html#K3" title="MCART_1:func.3">`3_3</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:94"><span class="lab"><font color="Green" title="E86">A42</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E29:94"/>
( <span class="p1">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Result</a> <font color="Maroon" title="c7">s3</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2_3</a>  <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">Result</a> <font color="Maroon" title="c6">s2</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2_3</a>  &amp; <span class="p1">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Result</a> <font color="Maroon" title="c7">s3</font></span>)</span> <a href="mcart_1.html#K3" title="MCART_1:func.3">`3_3</a>  <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">Result</a> <font color="Maroon" title="c6">s2</font></span>)</span> <a href="mcart_1.html#K3" title="MCART_1:func.3">`3_3</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:94"><span class="lab"><font color="Green" title="E79">A26</font></span></a>, <a class="txt" href="#E21:94"><span class="lab"><font color="Green" title="E83">A39</font></span></a>, <a class="txt" href="#E26:94"><span class="lab"><font color="Green" title="E87">A43</font></span></a>, <a class="txt" href="#E27:94"><span class="lab"><font color="Green" title="E88">A44</font></span></a>, <a class="ref" href="turing_1.html#D9" target="_self" onmouseover="rs('turing_1/D9')" onmouseout="rh()">Def9</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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