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<span class="kw">let </span><font color="Maroon" title="c1">T</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">s</font> being   <a href="turing_1.html#NM4" title="TURING_1:NM.4">All-State</a> of <font color="Maroon" title="c1">T</font>  st <font color="Olive" title="b1">s</font> is  <a href="turing_1.html#V2" title="TURING_1:attr.2">Accept-Halt</a>  holds <br/> ex <font color="Olive" title="b2">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> st <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="Olive" title="b1">s</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b2">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">T</font> &amp;  <a href="turing_1.html#K10" title="TURING_1:func.10">Result</a> <font color="Olive" title="b1">s</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="Olive" title="b1">s</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b2">k</font> &amp; (  for <font color="Olive" title="b3">i</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b3">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">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="Olive" title="b1">s</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b3">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">T</font> ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">s</font> be   <a href="turing_1.html#NM4" title="TURING_1:NM.4">All-State</a> of <font color="Maroon" title="c1">T</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" title="c2">s</font> is  <a href="turing_1.html#V2" title="TURING_1:attr.2">Accept-Halt</a>  implies  ex <font color="Olive" title="b1">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> st <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="c2">s</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b1">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">T</font> &amp;  <a href="turing_1.html#K10" title="TURING_1:func.10">Result</a> <font color="Maroon" title="c2">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <font color="Maroon" title="c2">s</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b1">k</font> &amp; (  for <font color="Olive" title="b2">i</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b2">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">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="c2">s</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b2">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">T</font> ) ) )</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><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="c2">s</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>  <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">T</font>;<br/>
<span class="kw">assume </span><a NAME="E1:27"/><span class="lab"><font color="Green" title="E14">A1</font></span>: 
<font color="Maroon" title="c2">s</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">  ex <font color="Olive" title="b1">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> st <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="c2">s</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b1">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">T</font> &amp;  <a href="turing_1.html#K10" title="TURING_1:func.10">Result</a> <font color="Maroon" title="c2">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <font color="Maroon" title="c2">s</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b1">k</font> &amp; (  for <font color="Olive" title="b2">i</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b2">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">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="c2">s</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b2">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">T</font> ) )</span><br/>

<a NAME="E2:27"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> st <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="c2">s</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b1">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">T</font>
 
;<br/>
<a NAME="E3:27"/><span class="kw">then </span><span class="lab"><font color="Green" title="E15">A2</font></span>: 
 ex <font color="Olive" title="b1">k</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">k</font>]
 
;<br/>
<span class="kw">consider </span><font color="Maroon" title="c3">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><span class="kw"> such that </span><br/><a NAME="E5:27"/><span class="lab"><font color="Green" title="E16">A3</font></span>: 
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">k</font>] &amp; (  for <font color="Olive" title="b1">n</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">n</font>] holds <br/><font color="Maroon" title="c3">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">n</font> ) )
 <span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S5" onmouseover="rs('nat_1/S5')" onmouseout="rh()">NAT_1:sch 5</a>(<a class="txt" href="#E3:27"><span class="lab"><font color="Green" title="E15">A2</font></span></a>);<br/></span>
<span class="kw">reconsider </span><font color="Maroon" title="c4">k</font> = <font color="Maroon" title="c3">k</font> as    <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>  <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D12" onmouseover="rs('ordinal1/D12')" onmouseout="rh()">ORDINAL1:def 12</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c4">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">(<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="c2">s</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c4">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">T</font> &amp;  <a href="turing_1.html#K10" title="TURING_1:func.10">Result</a> <font color="Maroon" title="c2">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <font color="Maroon" title="c2">s</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c4">k</font> &amp; (  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="c4">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="c2">s</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">T</font> ) )</span><br/>

<span class="kw">thus </span><a NAME="E7:27"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">k</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:27"><span class="lab"><font color="Green" title="E16">A3</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"> (  <a href="turing_1.html#K10" title="TURING_1:func.10">Result</a> <font color="Maroon" title="c2">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <font color="Maroon" title="c2">s</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c4">k</font> &amp; (  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="c4">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="c2">s</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">T</font> ) )</span><br/>

<span class="kw">thus </span><a NAME="E8:27"/>
 <a href="turing_1.html#K10" title="TURING_1:func.10">Result</a> <font color="Maroon" title="c2">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <font color="Maroon" title="c2">s</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c4">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:27"><span class="lab"><font color="Green" title="E14">A1</font></span></a>, <a class="txt" href="#E5:27"><span class="lab"><font color="Green" title="E16">A3</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">  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="c4">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="c2">s</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">T</font></span><br/>

<span class="kw">thus </span><a NAME="E9:27"/>
 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="c4">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="c2">s</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">T</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:27"><span class="lab"><font color="Green" title="E16">A3</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"> verum</span><br/>


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