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

<span class="kw">let </span><font color="Maroon" title="c1">w</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="modelc_2.html#K25" title="MODELC_2:func.25">Inf_seq</a> <a href="modelc_2.html#K43" title="MODELC_2:func.43">AtomicFamily</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">v</font> being   <a href="modelc_3.html#V5" title="MODELC_3:attr.5">neg-inner-most</a>  <a href="modelc_2.html#NM1" title="MODELC_2:NM.1">LTL-formula</a><br/>  for <font color="Olive" title="b2">U</font> being    <a href="orders_1.html#M1" title="ORDERS_1:mode.1">Choice_Function</a> of  <a href="orders_1.html#K1" title="ORDERS_1:func.1">BOOL</a> <span class="p1">(<span class="default"><a href="modelc_3.html#K1" title="MODELC_3:func.1">Subformulae</a> <font color="Olive" title="b1">v</font></span>)</span><br/>  for <font color="Olive" title="b3">s</font> being   <a href="modelc_3.html#V1" title="MODELC_3:attr.1">strict</a>   <a href="modelc_3.html#V3" title="MODELC_3:attr.3">elementary</a>   <a href="modelc_3.html#L1" title="MODELC_3:struct.1">LTLnode</a> of <font color="Olive" title="b1">v</font>  st <font color="Maroon" title="c1">w</font> <a href="modelc_2.html#R8" title="MODELC_2:pred.8">|=</a>  <a href="modelc_3.html#K14" title="MODELC_3:func.14">*</a> <span class="p1">(<span class="default"><a href="modelc_3.html#K12" title="MODELC_3:func.12">'X'</a> <font color="Olive" title="b3">s</font></span>)</span> holds <br/>(  <a href="modelc_3.html#K36" title="MODELC_3:func.36">chosen_next</a> (<font color="Maroon" title="c1">w</font>,<font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">U</font>,<font color="Olive" title="b3">s</font>) <a href="modelc_3.html#R6" title="MODELC_3:pred.6">is_next_of</a> <font color="Olive" title="b3">s</font> &amp; <font color="Maroon" title="c1">w</font> <a href="modelc_2.html#R8" title="MODELC_2:pred.8">|=</a>  <a href="modelc_3.html#K14" title="MODELC_3:func.14">*</a> <span class="p1">(<span class="default"><a href="modelc_3.html#K36" title="MODELC_3:func.36">chosen_next</a> (<font color="Maroon" title="c1">w</font>,<font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">U</font>,<font color="Olive" title="b3">s</font>)</span>)</span> )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">v</font> be   <a href="modelc_3.html#V5" title="MODELC_3:attr.5">neg-inner-most</a>  <a href="modelc_2.html#NM1" title="MODELC_2:NM.1">LTL-formula</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">U</font> being    <a href="orders_1.html#M1" title="ORDERS_1:mode.1">Choice_Function</a> of  <a href="orders_1.html#K1" title="ORDERS_1:func.1">BOOL</a> <span class="p1">(<span class="default"><a href="modelc_3.html#K1" title="MODELC_3:func.1">Subformulae</a> <font color="Maroon" title="c2">v</font></span>)</span><br/>  for <font color="Olive" title="b2">s</font> being   <a href="modelc_3.html#V1" title="MODELC_3:attr.1">strict</a>   <a href="modelc_3.html#V3" title="MODELC_3:attr.3">elementary</a>   <a href="modelc_3.html#L1" title="MODELC_3:struct.1">LTLnode</a> of <font color="Maroon" title="c2">v</font>  st <font color="Maroon" title="c1">w</font> <a href="modelc_2.html#R8" title="MODELC_2:pred.8">|=</a>  <a href="modelc_3.html#K14" title="MODELC_3:func.14">*</a> <span class="p1">(<span class="default"><a href="modelc_3.html#K12" title="MODELC_3:func.12">'X'</a> <font color="Olive" title="b2">s</font></span>)</span> holds <br/>(  <a href="modelc_3.html#K36" title="MODELC_3:func.36">chosen_next</a> (<font color="Maroon" title="c1">w</font>,<font color="Maroon" title="c2">v</font>,<font color="Olive" title="b1">U</font>,<font color="Olive" title="b2">s</font>) <a href="modelc_3.html#R6" title="MODELC_3:pred.6">is_next_of</a> <font color="Olive" title="b2">s</font> &amp; <font color="Maroon" title="c1">w</font> <a href="modelc_2.html#R8" title="MODELC_2:pred.8">|=</a>  <a href="modelc_3.html#K14" title="MODELC_3:func.14">*</a> <span class="p1">(<span class="default"><a href="modelc_3.html#K36" title="MODELC_3:func.36">chosen_next</a> (<font color="Maroon" title="c1">w</font>,<font color="Maroon" title="c2">v</font>,<font color="Olive" title="b1">U</font>,<font color="Olive" title="b2">s</font>)</span>)</span> )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">U</font> be    <a href="orders_1.html#M1" title="ORDERS_1:mode.1">Choice_Function</a> of  <a href="orders_1.html#K1" title="ORDERS_1:func.1">BOOL</a> <span class="p1">(<span class="default"><a href="modelc_3.html#K1" title="MODELC_3:func.1">Subformulae</a> <font color="Maroon" title="c2">v</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">s</font> being   <a href="modelc_3.html#V1" title="MODELC_3:attr.1">strict</a>   <a href="modelc_3.html#V3" title="MODELC_3:attr.3">elementary</a>   <a href="modelc_3.html#L1" title="MODELC_3:struct.1">LTLnode</a> of <font color="Maroon" title="c2">v</font>  st <font color="Maroon" title="c1">w</font> <a href="modelc_2.html#R8" title="MODELC_2:pred.8">|=</a>  <a href="modelc_3.html#K14" title="MODELC_3:func.14">*</a> <span class="p1">(<span class="default"><a href="modelc_3.html#K12" title="MODELC_3:func.12">'X'</a> <font color="Olive" title="b1">s</font></span>)</span> holds <br/>(  <a href="modelc_3.html#K36" title="MODELC_3:func.36">chosen_next</a> (<font color="Maroon" title="c1">w</font>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c3">U</font>,<font color="Olive" title="b1">s</font>) <a href="modelc_3.html#R6" title="MODELC_3:pred.6">is_next_of</a> <font color="Olive" title="b1">s</font> &amp; <font color="Maroon" title="c1">w</font> <a href="modelc_2.html#R8" title="MODELC_2:pred.8">|=</a>  <a href="modelc_3.html#K14" title="MODELC_3:func.14">*</a> <span class="p1">(<span class="default"><a href="modelc_3.html#K36" title="MODELC_3:func.36">chosen_next</a> (<font color="Maroon" title="c1">w</font>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c3">U</font>,<font color="Olive" title="b1">s</font>)</span>)</span> )</span><br/><span class="kw">let </span><font color="Maroon" title="c4">s</font> be   <a href="modelc_3.html#V1" title="MODELC_3:attr.1">strict</a>   <a href="modelc_3.html#V3" title="MODELC_3:attr.3">elementary</a>   <a href="modelc_3.html#L1" title="MODELC_3:struct.1">LTLnode</a> of <font color="Maroon" title="c2">v</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">w</font> <a href="modelc_2.html#R8" title="MODELC_2:pred.8">|=</a>  <a href="modelc_3.html#K14" title="MODELC_3:func.14">*</a> <span class="p1">(<span class="default"><a href="modelc_3.html#K12" title="MODELC_3:func.12">'X'</a> <font color="Maroon" title="c4">s</font></span>)</span> implies (  <a href="modelc_3.html#K36" title="MODELC_3:func.36">chosen_next</a> (<font color="Maroon" title="c1">w</font>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c3">U</font>,<font color="Maroon" title="c4">s</font>) <a href="modelc_3.html#R6" title="MODELC_3:pred.6">is_next_of</a> <font color="Maroon" title="c4">s</font> &amp; <font color="Maroon" title="c1">w</font> <a href="modelc_2.html#R8" title="MODELC_2:pred.8">|=</a>  <a href="modelc_3.html#K14" title="MODELC_3:func.14">*</a> <span class="p1">(<span class="default"><a href="modelc_3.html#K36" title="MODELC_3:func.36">chosen_next</a> (<font color="Maroon" title="c1">w</font>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c3">U</font>,<font color="Maroon" title="c4">s</font>)</span>)</span> ) )</span><br/>







<span class="kw">set </span><font color="Maroon" title="c5">LN</font> =  <a href="modelc_3.html#K22" title="MODELC_3:func.22">LTLNodes</a> <font color="Maroon" title="c2">v</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c6">N</font> =  <a href="modelc_3.html#K12" title="MODELC_3:func.12">'X'</a> <font color="Maroon" title="c4">s</font>;<br/>
<span class="kw">assume </span><a NAME="E1:150"/><span class="lab"><font color="Green" title="E120">A1</font></span>: 
<font color="Maroon" title="c1">w</font> <a href="modelc_2.html#R8" title="MODELC_2:pred.8">|=</a>  <a href="modelc_3.html#K14" title="MODELC_3:func.14">*</a> <span class="p1">(<span class="default"><a href="modelc_3.html#K12" title="MODELC_3:func.12">'X'</a> <font color="Maroon" title="c4">s</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="modelc_3.html#K36" title="MODELC_3:func.36">chosen_next</a> (<font color="Maroon" title="c1">w</font>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c3">U</font>,<font color="Maroon" title="c4">s</font>) <a href="modelc_3.html#R6" title="MODELC_3:pred.6">is_next_of</a> <font color="Maroon" title="c4">s</font> &amp; <font color="Maroon" title="c1">w</font> <a href="modelc_2.html#R8" title="MODELC_2:pred.8">|=</a>  <a href="modelc_3.html#K14" title="MODELC_3:func.14">*</a> <span class="p1">(<span class="default"><a href="modelc_3.html#K36" title="MODELC_3:func.36">chosen_next</a> (<font color="Maroon" title="c1">w</font>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c3">U</font>,<font color="Maroon" title="c4">s</font>)</span>)</span> )</span><br/>

<span class="kw">set </span><font color="Maroon" title="c7">n</font> =  <a href="modelc_3.html#K35" title="MODELC_3:func.35">chosen_succ_end_num</a> (<font color="Maroon" title="c1">w</font>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c3">U</font>,<span class="p1">(<span class="default"><a href="modelc_3.html#K12" title="MODELC_3:func.12">'X'</a> <font color="Maroon" title="c4">s</font></span>)</span>);<br/>
<span class="kw">set </span><font color="Maroon" title="c8">f</font> =  <a href="modelc_3.html#K26" title="MODELC_3:func.26">choice_succ_func</a> (<font color="Maroon" title="c1">w</font>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c3">U</font>);<br/>
<span class="kw">set </span><font color="Maroon" title="c9">nextnode</font> =  <a href="modelc_3.html#K10" title="MODELC_3:func.10">CastNode</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="modelc_3.html#K26" title="MODELC_3:func.26">choice_succ_func</a> (<font color="Maroon" title="c1">w</font>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c3">U</font>)</span>)</span> <a href="modelc_1.html#K30" title="MODELC_1:func.30">|**</a> <span class="p3">(<span class="default"><a href="modelc_3.html#K35" title="MODELC_3:func.35">chosen_succ_end_num</a> (<font color="Maroon" title="c1">w</font>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c3">U</font>,<span class="p4">(<span class="default"><a href="modelc_3.html#K12" title="MODELC_3:func.12">'X'</a> <font color="Maroon" title="c4">s</font></span>)</span>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="modelc_3.html#K12" title="MODELC_3:func.12">'X'</a> <font color="Maroon" title="c4">s</font></span>)</span></span>)</span>,<font color="Maroon" title="c2">v</font>);<br/>
<a NAME="E2:150"/><span class="lab"><font color="Green" title="E121">A2</font></span>: 
 <a href="modelc_3.html#K12" title="MODELC_3:func.12">'X'</a> <font color="Maroon" title="c4">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="modelc_3.html#K22" title="MODELC_3:func.22">LTLNodes</a> <font color="Maroon" title="c2">v</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="modelc_3.html#D30" target="_self" title="MODELC_3:def.30">Def30</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="150_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="150_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:150_1_1"/>
(  not  <a href="modelc_3.html#K12" title="MODELC_3:func.12">'X'</a> <font color="Maroon" title="c4">s</font> is  <a href="modelc_3.html#V3" title="MODELC_3:attr.3">elementary</a>  or  <a href="modelc_3.html#K12" title="MODELC_3:func.12">'X'</a> <font color="Maroon" title="c4">s</font> is  <a href="modelc_3.html#V3" title="MODELC_3:attr.3">elementary</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="150_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:150_1_1_1"/><span class="lab"><font color="Green" title="E122">A3</font></span>: 
 not  <a href="modelc_3.html#K12" title="MODELC_3:func.12">'X'</a> <font color="Maroon" title="c4">s</font> is  <a href="modelc_3.html#V3" title="MODELC_3:attr.3">elementary</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="modelc_3.html#K36" title="MODELC_3:func.36">chosen_next</a> (<font color="Maroon" title="c1">w</font>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c3">U</font>,<font color="Maroon" title="c4">s</font>) <a href="modelc_3.html#R6" title="MODELC_3:pred.6">is_next_of</a> <font color="Maroon" title="c4">s</font> &amp; <font color="Maroon" title="c1">w</font> <a href="modelc_2.html#R8" title="MODELC_2:pred.8">|=</a>  <a href="modelc_3.html#K14" title="MODELC_3:func.14">*</a> <span class="p1">(<span class="default"><a href="modelc_3.html#K36" title="MODELC_3:func.36">chosen_next</a> (<font color="Maroon" title="c1">w</font>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c3">U</font>,<font color="Maroon" title="c4">s</font>)</span>)</span> )</span><br/><div class="add"><span class="kw">deffunc </span><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ) <span class="kw">-&gt; </span>  <a href="modelc_3.html#V1" title="MODELC_3:attr.1">strict</a>   <a href="modelc_3.html#L1" title="MODELC_3:struct.1">LTLnode</a> of <font color="Maroon" title="c2">v</font> =  <a href="modelc_3.html#K10" title="MODELC_3:func.10">CastNode</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="modelc_3.html#K26" title="MODELC_3:func.26">choice_succ_func</a> (<font color="Maroon" title="c1">w</font>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c3">U</font>)</span>)</span> <a href="modelc_1.html#K30" title="MODELC_1:func.30">|**</a> <span class="p3">(<span class="default"><a href="modelc_2.html#K1" title="MODELC_2:func.1">CastNat</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="modelc_2.html#K1" title="MODELC_2:func.1">CastNat</a> $1</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="modelc_3.html#K12" title="MODELC_3:func.12">'X'</a> <font color="Maroon" title="c4">s</font></span>)</span></span>)</span>,<font color="Maroon" title="c2">v</font>);<br/><span class="kw">set </span><font color="Maroon" title="c10">n1</font> = <span class="p1">(<span class="default"><a href="modelc_3.html#K35" title="MODELC_3:func.35">chosen_succ_end_num</a> (<font color="Maroon" title="c1">w</font>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c3">U</font>,<span class="p2">(<span class="default"><a href="modelc_3.html#K12" title="MODELC_3:func.12">'X'</a> <font color="Maroon" title="c4">s</font></span>)</span>)</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1;<br/><a NAME="E2:150_1_1_1"/>
 ex <font color="Olive" title="b1">L</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a> st <br/>(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="modelc_3.html#K35" title="MODELC_3:func.35">chosen_succ_end_num</a> (<font color="Maroon" title="c1">w</font>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c3">U</font>,<span class="p2">(<span class="default"><a href="modelc_3.html#K12" title="MODELC_3:func.12">'X'</a> <font color="Maroon" title="c4">s</font></span>)</span>)</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; (  for <font color="Olive" title="b2">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b2">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Olive" title="b1">L</font> holds <br/><font color="Olive" title="b1">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b2">k</font>) ) )
 <span class="kw">from</span> <span class="lab"><a class="ref" href="finseq_1.html#S2" title="FINSEQ_1:sch.2">FINSEQ_1:sch 2</a>();<br/></span><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c11">L</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a><span class="kw"> such that </span><br/><a NAME="E4:150_1_1_1"/><span class="lab"><font color="Green" title="E123">A4</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="modelc_3.html#K35" title="MODELC_3:func.35">chosen_succ_end_num</a> (<font color="Maroon" title="c1">w</font>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c3">U</font>,<span class="p2">(<span class="default"><a href="modelc_3.html#K12" title="MODELC_3:func.12">'X'</a> <font color="Maroon" title="c4">s</font></span>)</span>)</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <span class="kw">and </span><br/><a NAME="E5:150_1_1_1"/><span class="lab"><font color="Green" title="E124">A5</font></span>: 
 for <font color="Olive" title="b1">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c11">L</font> holds <br/><font color="Maroon" title="c11">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">k</font>)
 ;<br/><a NAME="E6:150_1_1_1"/><span class="lab"><font color="Green" title="E125">A6</font></span>: 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="modelc_3.html#K35" title="MODELC_3:func.35">chosen_succ_end_num</a> (<font color="Maroon" title="c1">w</font>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c3">U</font>,<span class="p3">(<span class="default"><a href="modelc_3.html#K12" title="MODELC_3:func.12">'X'</a> <font color="Maroon" title="c4">s</font></span>)</span>)</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c11">L</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:150_1_1_1"><span class="lab"><font color="Green" title="E123">A4</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/><a NAME="E7:150_1_1_1"/><span class="lab"><font color="Green" title="E126">A7</font></span>: 
 for <font color="Olive" title="b1">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k</font> &amp; <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="modelc_3.html#K35" title="MODELC_3:func.35">chosen_succ_end_num</a> (<font color="Maroon" title="c1">w</font>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c3">U</font>,<span class="p2">(<span class="default"><a href="modelc_3.html#K12" title="MODELC_3:func.12">'X'</a> <font color="Maroon" title="c4">s</font></span>)</span>)</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 holds <br/><font color="Maroon" title="c11">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="modelc_3.html#K10" title="MODELC_3:func.10">CastNode</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="modelc_3.html#K26" title="MODELC_3:func.26">choice_succ_func</a> (<font color="Maroon" title="c1">w</font>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c3">U</font>)</span>)</span> <a href="modelc_1.html#K30" title="MODELC_1:func.30">|**</a> <span class="p3">(<span class="default"><a href="modelc_2.html#K1" title="MODELC_2:func.1">CastNat</a> <span class="p4">(<span class="default"><font color="Olive" title="b1">k</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="modelc_3.html#K12" title="MODELC_3:func.12">'X'</a> <font color="Maroon" title="c4">s</font></span>)</span></span>)</span>,<font color="Maroon" title="c2">v</font>)
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="150_1_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c12">k</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">k</font> &amp; <font color="Maroon" title="c12">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="modelc_3.html#K35" title="MODELC_3:func.35">chosen_succ_end_num</a> (<font color="Maroon" title="c1">w</font>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c3">U</font>,<span class="p2">(<span class="default"><a href="modelc_3.html#K12" title="MODELC_3:func.12">'X'</a> <font color="Maroon" title="c4">s</font></span>)</span>)</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 implies <font color="Maroon" title="c11">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="modelc_3.html#K10" title="MODELC_3:func.10">CastNode</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="modelc_3.html#K26" title="MODELC_3:func.26">choice_succ_func</a> (<font color="Maroon" title="c1">w</font>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c3">U</font>)</span>)</span> <a href="modelc_1.html#K30" title="MODELC_1:func.30">|**</a> <span class="p3">(<span class="default"><a href="modelc_2.html#K1" title="MODELC_2:func.1">CastNat</a> <span class="p4">(<span class="default"><font color="Maroon" title="c12">k</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="modelc_3.html#K12" title="MODELC_3:func.12">'X'</a> <font color="Maroon" title="c4">s</font></span>)</span></span>)</span>,<font color="Maroon" title="c2">v</font>) )</span><br/>

<span class="kw">assume </span><a NAME="E1:150_1_1_1_1"/>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">k</font> &amp; <font color="Maroon" title="c12">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="modelc_3.html#K35" title="MODELC_3:func.35">chosen_succ_end_num</a> (<font color="Maroon" title="c1">w</font>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c3">U</font>,<span class="p2">(<span class="default"><a href="modelc_3.html#K12" title="MODELC_3:func.12">'X'</a> <font color="Maroon" title="c4">s</font></span>)</span>)</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c11">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="modelc_3.html#K10" title="MODELC_3:func.10">CastNode</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="modelc_3.html#K26" title="MODELC_3:func.26">choice_succ_func</a> (<font color="Maroon" title="c1">w</font>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c3">U</font>)</span>)</span> <a href="modelc_1.html#K30" title="MODELC_1:func.30">|**</a> <span class="p3">(<span class="default"><a href="modelc_2.html#K1" title="MODELC_2:func.1">CastNat</a> <span class="p4">(<span class="default"><font color="Maroon" title="c12">k</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="modelc_3.html#K12" title="MODELC_3:func.12">'X'</a> <font color="Maroon" title="c4">s</font></span>)</span></span>)</span>,<font color="Maroon" title="c2">v</font>)</span><br/>

<a NAME="E2:150_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c12">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="modelc_3.html#K35" title="MODELC_3:func.35">chosen_succ_end_num</a> (<font color="Maroon" title="c1">w</font>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c3">U</font>,<span class="p3">(<span class="default"><a href="modelc_3.html#K12" title="MODELC_3:func.12">'X'</a> <font color="Maroon" title="c4">s</font></span>)</span>)</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/>
<a NAME="E3:150_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c11">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="modelc_3.html#K10" title="MODELC_3:func.10">CastNode</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="modelc_3.html#K26" title="MODELC_3:func.26">choice_succ_func</a> (<font color="Maroon" title="c1">w</font>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c3">U</font>)</span>)</span> <a href="modelc_1.html#K30" title="MODELC_1:func.30">|**</a> <span class="p3">(<span class="default"><a href="modelc_2.html#K1" title="MODELC_2:func.1">CastNat</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="modelc_2.html#K1" title="MODELC_2:func.1">CastNat</a> <font color="Maroon" title="c12">k</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="modelc_3.html#K12" title="MODELC_3:func.12">'X'</a> <font color="Maroon" title="c4">s</font></span>)</span></span>)</span>,<font color="Maroon" title="c2">v</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:150_1_1_1"><span class="lab"><font color="Green" title="E124">A5</font></span></a>, <a class="txt" href="#E6:150_1_1_1"><span class="lab"><font color="Green" title="E125">A6</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:150_1_1_1_1"/>
<font color="Maroon" title="c11">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="modelc_3.html#K10" title="MODELC_3:func.10">CastNode</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="modelc_3.html#K26" title="MODELC_3:func.26">choice_succ_func</a> (<font color="Maroon" title="c1">w</font>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c3">U</font>)</span>)</span> <a href="modelc_1.html#K30" title="MODELC_1:func.30">|**</a> <span class="p3">(<span class="default"><a href="modelc_2.html#K1" title="MODELC_2:func.1">CastNat</a> <span class="p4">(<span class="default"><font color="Maroon" title="c12">k</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="modelc_3.html#K12" title="MODELC_3:func.12">'X'</a> <font color="Maroon" title="c4">s</font></span>)</span></span>)</span>,<font color="Maroon" title="c2">v</font>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="modelc_2.html#D1" title="MODELC_2:def.1">MODELC_2:def 1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E8:150_1_1_1"/>
 for <font color="Olive" title="b1">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k</font> &amp; <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">L</font> holds <br/> ex <font color="Olive" title="b2">N1</font>, <font color="Olive" title="b3">M1</font> being   <a href="modelc_3.html#V1" title="MODELC_3:attr.1">strict</a>   <a href="modelc_3.html#L1" title="MODELC_3:struct.1">LTLnode</a> of <font color="Maroon" title="c2">v</font> st <br/>( <font color="Olive" title="b2">N1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font> &amp; <font color="Olive" title="b3">M1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> &amp; <font color="Olive" title="b3">M1</font> <a href="modelc_3.html#R4" title="MODELC_3:pred.4">is_succ_of</a> <font color="Olive" title="b2">N1</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="150_1_1_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c12">k</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">k</font> &amp; <font color="Maroon" title="c12">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">L</font> implies  ex <font color="Olive" title="b1">N1</font>, <font color="Olive" title="b2">M1</font> being   <a href="modelc_3.html#V1" title="MODELC_3:attr.1">strict</a>   <a href="modelc_3.html#L1" title="MODELC_3:struct.1">LTLnode</a> of <font color="Maroon" title="c2">v</font> st <br/>( <font color="Olive" title="b1">N1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">k</font> &amp; <font color="Olive" title="b2">M1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> &amp; <font color="Olive" title="b2">M1</font> <a href="modelc_3.html#R4" title="MODELC_3:pred.4">is_succ_of</a> <font color="Olive" title="b1">N1</font> ) )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:150_1_1_1_2"/><span class="lab"><font color="Green" title="E127">A8</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">k</font>
 <span class="kw">and </span><br/><a NAME="E2:150_1_1_1_2"/><span class="lab"><font color="Green" title="E128">A9</font></span>: 
<font color="Maroon" title="c12">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">L</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">N1</font>, <font color="Olive" title="b2">M1</font> being   <a href="modelc_3.html#V1" title="MODELC_3:attr.1">strict</a>   <a href="modelc_3.html#L1" title="MODELC_3:struct.1">LTLnode</a> of <font color="Maroon" title="c2">v</font> st <br/>( <font color="Olive" title="b1">N1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">k</font> &amp; <font color="Olive" title="b2">M1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> &amp; <font color="Olive" title="b2">M1</font> <a href="modelc_3.html#R4" title="MODELC_3:pred.4">is_succ_of</a> <font color="Olive" title="b1">N1</font> )</span><br/>

<span class="kw">set </span><font color="Maroon" title="c13">k1</font> = <font color="Maroon" title="c12">k</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c14">k1</font> = <font color="Maroon" title="c12">k</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 as   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:150_1_1_1_2"><span class="lab"><font color="Green" title="E127">A8</font></span></a>, <a class="ref" href="nat_1.html#T21" title="NAT_1:th.21">NAT_1:21</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c15">M1</font> =  <a href="modelc_3.html#K10" title="MODELC_3:func.10">CastNode</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="modelc_3.html#K26" title="MODELC_3:func.26">choice_succ_func</a> (<font color="Maroon" title="c1">w</font>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c3">U</font>)</span>)</span> <a href="modelc_1.html#K30" title="MODELC_1:func.30">|**</a> <span class="p3">(<span class="default"><font color="Maroon" title="c14">k1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="modelc_3.html#K12" title="MODELC_3:func.12">'X'</a> <font color="Maroon" title="c4">s</font></span>)</span></span>)</span>,<font color="Maroon" title="c2">v</font>);<br/>
<span class="kw">set </span><font color="Maroon" title="c16">kp</font> = <font color="Maroon" title="c12">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1;<br/>
<a NAME="E4:150_1_1_1_2"/>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c12">k</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> <span class="p1">(<span class="default"><a href="modelc_3.html#K35" title="MODELC_3:func.35">chosen_succ_end_num</a> (<font color="Maroon" title="c1">w</font>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c3">U</font>,<span class="p2">(<span class="default"><a href="modelc_3.html#K12" title="MODELC_3:func.12">'X'</a> <font color="Maroon" title="c4">s</font></span>)</span>)</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:150_1_1_1"><span class="lab"><font color="Green" title="E123">A4</font></span></a>, <a class="txt" href="#E1:150_1_1_1_2"><span class="lab"><font color="Green" title="E127">A8</font></span></a>, <a class="txt" href="#E2:150_1_1_1_2"><span class="lab"><font color="Green" title="E128">A9</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E129">A10</font></span>: <font color="Maroon" title="c11">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> = 
 <a href="modelc_3.html#K10" title="MODELC_3:func.10">CastNode</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="modelc_3.html#K26" title="MODELC_3:func.26">choice_succ_func</a> (<font color="Maroon" title="c1">w</font>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c3">U</font>)</span>)</span> <a href="modelc_1.html#K30" title="MODELC_1:func.30">|**</a> <span class="p3">(<span class="default"><a href="modelc_2.html#K1" title="MODELC_2:func.1">CastNat</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c12">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="modelc_3.html#K12" title="MODELC_3:func.12">'X'</a> <font color="Maroon" title="c4">s</font></span>)</span></span>)</span>,<font color="Maroon" title="c2">v</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:150_1_1_1"><span class="lab"><font color="Green" title="E126">A7</font></span></a></span>
<br/>.= 
 <a href="modelc_3.html#K10" title="MODELC_3:func.10">CastNode</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="modelc_3.html#K26" title="MODELC_3:func.26">choice_succ_func</a> (<font color="Maroon" title="c1">w</font>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c3">U</font>)</span>)</span> <a href="modelc_1.html#K30" title="MODELC_1:func.30">|**</a> <span class="p3">(<span class="default"><font color="Maroon" title="c14">k1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="modelc_3.html#K12" title="MODELC_3:func.12">'X'</a> <font color="Maroon" title="c4">s</font></span>)</span></span>)</span>,<font color="Maroon" title="c2">v</font>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="modelc_2.html#D1" title="MODELC_2:def.1">MODELC_2:def 1</a></span>
;<br/>
<span class="kw">set </span><font color="Maroon" title="c17">N1</font> =  <a href="modelc_3.html#K10" title="MODELC_3:func.10">CastNode</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="modelc_3.html#K26" title="MODELC_3:func.26">choice_succ_func</a> (<font color="Maroon" title="c1">w</font>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c3">U</font>)</span>)</span> <a href="modelc_1.html#K30" title="MODELC_1:func.30">|**</a> <font color="Maroon" title="c14">k1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="modelc_3.html#K12" title="MODELC_3:func.12">'X'</a> <font color="Maroon" title="c4">s</font></span>)</span></span>)</span>,<font color="Maroon" title="c2">v</font>);<br/>
<a NAME="E6:150_1_1_1_2"/>
<font color="Maroon" title="c12">k</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="modelc_3.html#K35" title="MODELC_3:func.35">chosen_succ_end_num</a> (<font color="Maroon" title="c1">w</font>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c3">U</font>,<span class="p3">(<span class="default"><a href="modelc_3.html#K12" title="MODELC_3:func.12">'X'</a> <font color="Maroon" title="c4">s</font></span>)</span>)</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:150_1_1_1"><span class="lab"><font color="Green" title="E123">A4</font></span></a>, <a class="txt" href="#E2:150_1_1_1_2"><span class="lab"><font color="Green" title="E128">A9</font></span></a>, <a class="ref" href="xreal_1.html#T14" title="XREAL_1:th.14">XREAL_1:14</a></span>;<br/>
<a NAME="E7:150_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E130">A11</font></span>: 
 <a href="modelc_3.html#K10" title="MODELC_3:func.10">CastNode</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="modelc_3.html#K26" title="MODELC_3:func.26">choice_succ_func</a> (<font color="Maroon" title="c1">w</font>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c3">U</font>)</span>)</span> <a href="modelc_1.html#K30" title="MODELC_1:func.30">|**</a> <span class="p3">(<span class="default"><font color="Maroon" title="c14">k1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="modelc_3.html#K12" title="MODELC_3:func.12">'X'</a> <font color="Maroon" title="c4">s</font></span>)</span></span>)</span>,<font color="Maroon" title="c2">v</font>) <a href="modelc_3.html#R4" title="MODELC_3:pred.4">is_succ_of</a>  <a href="modelc_3.html#K10" title="MODELC_3:func.10">CastNode</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="modelc_3.html#K26" title="MODELC_3:func.26">choice_succ_func</a> (<font color="Maroon" title="c1">w</font>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c3">U</font>)</span>)</span> <a href="modelc_1.html#K30" title="MODELC_1:func.30">|**</a> <font color="Maroon" title="c14">k1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="modelc_3.html#K12" title="MODELC_3:func.12">'X'</a> <font color="Maroon" title="c4">s</font></span>)</span></span>)</span>,<font color="Maroon" title="c2">v</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:150"><span class="lab"><font color="Green" title="E120">A1</font></span></a>, <a class="txt" href="#E1:150_1_1_1"><span class="lab"><font color="Green" title="E122">A3</font></span></a>, <a class="ref" href="modelc_3.html#D48" target="_self" title="MODELC_3:def.48">Def48</a></span>;<br/>
<font color="Maroon" title="c11">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">k</font> = 
 <a href="modelc_3.html#K10" title="MODELC_3:func.10">CastNode</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="modelc_3.html#K26" title="MODELC_3:func.26">choice_succ_func</a> (<font color="Maroon" title="c1">w</font>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c3">U</font>)</span>)</span> <a href="modelc_1.html#K30" title="MODELC_1:func.30">|**</a> <span class="p3">(<span class="default"><a href="modelc_2.html#K1" title="MODELC_2:func.1">CastNat</a> <font color="Maroon" title="c14">k1</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="modelc_3.html#K12" title="MODELC_3:func.12">'X'</a> <font color="Maroon" title="c4">s</font></span>)</span></span>)</span>,<font color="Maroon" title="c2">v</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:150_1_1_1"><span class="lab"><font color="Green" title="E123">A4</font></span></a>, <a class="txt" href="#E7:150_1_1_1"><span class="lab"><font color="Green" title="E126">A7</font></span></a>, <a class="txt" href="#E1:150_1_1_1_2"><span class="lab"><font color="Green" title="E127">A8</font></span></a>, <a class="txt" href="#E2:150_1_1_1_2"><span class="lab"><font color="Green" title="E128">A9</font></span></a></span>
<br/>.= 
 <a href="modelc_3.html#K10" title="MODELC_3:func.10">CastNode</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="modelc_3.html#K26" title="MODELC_3:func.26">choice_succ_func</a> (<font color="Maroon" title="c1">w</font>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c3">U</font>)</span>)</span> <a href="modelc_1.html#K30" title="MODELC_1:func.30">|**</a> <font color="Maroon" title="c14">k1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="modelc_3.html#K12" title="MODELC_3:func.12">'X'</a> <font color="Maroon" title="c4">s</font></span>)</span></span>)</span>,<font color="Maroon" title="c2">v</font>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="modelc_2.html#D1" title="MODELC_2:def.1">MODELC_2:def 1</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E9:150_1_1_1_2"/>
 ex <font color="Olive" title="b1">N1</font>, <font color="Olive" title="b2">M1</font> being   <a href="modelc_3.html#V1" title="MODELC_3:attr.1">strict</a>   <a href="modelc_3.html#L1" title="MODELC_3:struct.1">LTLnode</a> of <font color="Maroon" title="c2">v</font> st <br/>( <font color="Olive" title="b1">N1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">k</font> &amp; <font color="Olive" title="b2">M1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> &amp; <font color="Olive" title="b2">M1</font> <a href="modelc_3.html#R4" title="MODELC_3:pred.4">is_succ_of</a> <font color="Olive" title="b1">N1</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:150_1_1_1_2"><span class="lab"><font color="Green" title="E130">A11</font></span></a>, <a class="txt" href="#E5:150_1_1_1_2"><span class="lab"><font color="Green" title="E129">A10</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E9:150_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E127">A12</font></span>: 
<font color="Maroon" title="c11">L</font> <a href="modelc_3.html#R5" title="MODELC_3:pred.5">is_Finseq_for</a> <font color="Maroon" title="c2">v</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="modelc_3.html#D19" target="_self" title="MODELC_3:def.19">Def19</a></span>;<br/><a NAME="E10:150_1_1_1"/><span class="lab"><font color="Green" title="E128">A13</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="modelc_3.html#K35" title="MODELC_3:func.35">chosen_succ_end_num</a> (<font color="Maroon" title="c1">w</font>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c3">U</font>,<span class="p2">(<span class="default"><a href="modelc_3.html#K12" title="MODELC_3:func.12">'X'</a> <font color="Maroon" title="c4">s</font></span>)</span>)</span>)</span> <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="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E129">A14</font></span>: <font color="Maroon" title="c11">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">L</font></span>)</span> = 
 <a href="modelc_3.html#K10" title="MODELC_3:func.10">CastNode</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="modelc_3.html#K26" title="MODELC_3:func.26">choice_succ_func</a> (<font color="Maroon" title="c1">w</font>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c3">U</font>)</span>)</span> <a href="modelc_1.html#K30" title="MODELC_1:func.30">|**</a> <span class="p3">(<span class="default"><a href="modelc_2.html#K1" title="MODELC_2:func.1">CastNat</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="modelc_3.html#K35" title="MODELC_3:func.35">chosen_succ_end_num</a> (<font color="Maroon" title="c1">w</font>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c3">U</font>,<span class="p1">(<span class="default"><a href="modelc_3.html#K12" title="MODELC_3:func.12">'X'</a> <font color="Maroon" title="c4">s</font></span>)</span>)</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="modelc_3.html#K12" title="MODELC_3:func.12">'X'</a> <font color="Maroon" title="c4">s</font></span>)</span></span>)</span>,<font color="Maroon" title="c2">v</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:150_1_1_1"><span class="lab"><font color="Green" title="E123">A4</font></span></a>, <a class="txt" href="#E7:150_1_1_1"><span class="lab"><font color="Green" title="E126">A7</font></span></a></span>
<br/>.= 
 <a href="modelc_3.html#K10" title="MODELC_3:func.10">CastNode</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="modelc_3.html#K26" title="MODELC_3:func.26">choice_succ_func</a> (<font color="Maroon" title="c1">w</font>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c3">U</font>)</span>)</span> <a href="modelc_1.html#K30" title="MODELC_1:func.30">|**</a> <span class="p3">(<span class="default"><a href="modelc_3.html#K35" title="MODELC_3:func.35">chosen_succ_end_num</a> (<font color="Maroon" title="c1">w</font>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c3">U</font>,<span class="p4">(<span class="default"><a href="modelc_3.html#K12" title="MODELC_3:func.12">'X'</a> <font color="Maroon" title="c4">s</font></span>)</span>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="modelc_3.html#K12" title="MODELC_3:func.12">'X'</a> <font color="Maroon" title="c4">s</font></span>)</span></span>)</span>,<font color="Maroon" title="c2">v</font>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="modelc_2.html#D1" title="MODELC_2:def.1">MODELC_2:def 1</a></span>
;<br/><a NAME="E12:150_1_1_1"/><span class="lab"><font color="Green" title="E130">A15</font></span>: 
 <a href="modelc_3.html#K10" title="MODELC_3:func.10">CastNode</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="modelc_3.html#K26" title="MODELC_3:func.26">choice_succ_func</a> (<font color="Maroon" title="c1">w</font>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c3">U</font>)</span>)</span> <a href="modelc_1.html#K30" title="MODELC_1:func.30">|**</a> <span class="p3">(<span class="default"><a href="modelc_3.html#K35" title="MODELC_3:func.35">chosen_succ_end_num</a> (<font color="Maroon" title="c1">w</font>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c3">U</font>,<span class="p4">(<span class="default"><a href="modelc_3.html#K12" title="MODELC_3:func.12">'X'</a> <font color="Maroon" title="c4">s</font></span>)</span>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="modelc_3.html#K12" title="MODELC_3:func.12">'X'</a> <font color="Maroon" title="c4">s</font></span>)</span></span>)</span>,<font color="Maroon" title="c2">v</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="modelc_3.html#K36" title="MODELC_3:func.36">chosen_next</a> (<font color="Maroon" title="c1">w</font>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c3">U</font>,<font color="Maroon" title="c4">s</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:150"><span class="lab"><font color="Green" title="E120">A1</font></span></a>, <a class="txt" href="#E1:150_1_1_1"><span class="lab"><font color="Green" title="E122">A3</font></span></a>, <a class="ref" href="modelc_3.html#D49" target="_self" title="MODELC_3:def.49">Def49</a></span>;<br/><font color="Maroon" title="c11">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 = 
 <a href="modelc_3.html#K10" title="MODELC_3:func.10">CastNode</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="modelc_3.html#K26" title="MODELC_3:func.26">choice_succ_func</a> (<font color="Maroon" title="c1">w</font>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c3">U</font>)</span>)</span> <a href="modelc_1.html#K30" title="MODELC_1:func.30">|**</a> <span class="p3">(<span class="default"><a href="modelc_2.html#K1" title="MODELC_2:func.1">CastNat</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="modelc_3.html#K12" title="MODELC_3:func.12">'X'</a> <font color="Maroon" title="c4">s</font></span>)</span></span>)</span>,<font color="Maroon" title="c2">v</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:150_1_1_1"><span class="lab"><font color="Green" title="E128">A13</font></span></a>, <a class="txt" href="#E7:150_1_1_1"><span class="lab"><font color="Green" title="E126">A7</font></span></a></span>
<br/>.= 
 <a href="modelc_3.html#K10" title="MODELC_3:func.10">CastNode</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="modelc_3.html#K26" title="MODELC_3:func.26">choice_succ_func</a> (<font color="Maroon" title="c1">w</font>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c3">U</font>)</span>)</span> <a href="modelc_1.html#K30" title="MODELC_1:func.30">|**</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="modelc_3.html#K12" title="MODELC_3:func.12">'X'</a> <font color="Maroon" title="c4">s</font></span>)</span></span>)</span>,<font color="Maroon" title="c2">v</font>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="modelc_2.html#D1" title="MODELC_2:def.1">MODELC_2:def 1</a></span>
<br/>.= 
 <a href="modelc_3.html#K10" title="MODELC_3:func.10">CastNode</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <span class="p3">(<span class="default"><a href="modelc_3.html#K22" title="MODELC_3:func.22">LTLNodes</a> <font color="Maroon" title="c2">v</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="modelc_3.html#K12" title="MODELC_3:func.12">'X'</a> <font color="Maroon" title="c4">s</font></span>)</span></span>)</span>,<font color="Maroon" title="c2">v</font>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_7.html#T84" title="FUNCT_7:th.84">FUNCT_7:84</a></span>
<br/>.= 
 <a href="modelc_3.html#K10" title="MODELC_3:func.10">CastNode</a> (<span class="p1">(<span class="default"><a href="modelc_3.html#K12" title="MODELC_3:func.12">'X'</a> <font color="Maroon" title="c4">s</font></span>)</span>,<font color="Maroon" title="c2">v</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:150"><span class="lab"><font color="Green" title="E121">A2</font></span></a>, <a class="ref" href="funct_1.html#T18" title="FUNCT_1:th.18">FUNCT_1:18</a></span>
<br/>.= 
 <a href="modelc_3.html#K12" title="MODELC_3:func.12">'X'</a> <font color="Maroon" title="c4">s</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="modelc_3.html#D16" target="_self" title="MODELC_3:def.16">Def16</a></span>
;<br/><span class="kw">hence </span><a NAME="E14:150_1_1_1"/>
(  <a href="modelc_3.html#K36" title="MODELC_3:func.36">chosen_next</a> (<font color="Maroon" title="c1">w</font>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c3">U</font>,<font color="Maroon" title="c4">s</font>) <a href="modelc_3.html#R6" title="MODELC_3:pred.6">is_next_of</a> <font color="Maroon" title="c4">s</font> &amp; <font color="Maroon" title="c1">w</font> <a href="modelc_2.html#R8" title="MODELC_2:pred.8">|=</a>  <a href="modelc_3.html#K14" title="MODELC_3:func.14">*</a> <span class="p1">(<span class="default"><a href="modelc_3.html#K36" title="MODELC_3:func.36">chosen_next</a> (<font color="Maroon" title="c1">w</font>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c3">U</font>,<font color="Maroon" title="c4">s</font>)</span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:150"><span class="lab"><font color="Green" title="E120">A1</font></span></a>, <a class="txt" href="#E1:150_1_1_1"><span class="lab"><font color="Green" title="E122">A3</font></span></a>, <a class="txt" href="#E12:150_1_1_1"><span class="lab"><font color="Green" title="E130">A15</font></span></a>, <a class="txt" href="#E10:150_1_1_1"><span class="lab"><font color="Green" title="E128">A13</font></span></a>, <a class="txt" href="#E4:150_1_1_1"><span class="lab"><font color="Green" title="E123">A4</font></span></a>, <a class="txt" href="#E11:150_1_1_1"><span class="lab"><font color="Green" title="E129">A14</font></span></a>, <a class="txt" href="#E9:150_1_1_1"><span class="lab"><font color="Green" title="E127">A12</font></span></a>, <a class="ref" href="modelc_3.html#D20" target="_self" title="MODELC_3:def.20">Def20</a>, <a class="ref" href="modelc_3.html#D48" target="_self" title="MODELC_3:def.48">Def48</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="150_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:150_1_1_2"/>
 <a href="modelc_3.html#K12" title="MODELC_3:func.12">'X'</a> <font color="Maroon" title="c4">s</font> is  <a href="modelc_3.html#V3" title="MODELC_3:attr.3">elementary</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="modelc_3.html#K36" title="MODELC_3:func.36">chosen_next</a> (<font color="Maroon" title="c1">w</font>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c3">U</font>,<font color="Maroon" title="c4">s</font>) <a href="modelc_3.html#R6" title="MODELC_3:pred.6">is_next_of</a> <font color="Maroon" title="c4">s</font> &amp; <font color="Maroon" title="c1">w</font> <a href="modelc_2.html#R8" title="MODELC_2:pred.8">|=</a>  <a href="modelc_3.html#K14" title="MODELC_3:func.14">*</a> <span class="p1">(<span class="default"><a href="modelc_3.html#K36" title="MODELC_3:func.36">chosen_next</a> (<font color="Maroon" title="c1">w</font>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c3">U</font>,<font color="Maroon" title="c4">s</font>)</span>)</span> )</span><br/><div class="add"><a NAME="E2:150_1_1_2"/><span class="kw">then </span>
 the <a href="modelc_3.html#U2" title="MODELC_3:sel.2">LTLnew</a> of <span class="p1">(<span class="default"><a href="modelc_3.html#K12" title="MODELC_3:func.12">'X'</a> <font color="Maroon" title="c4">s</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="modelc_3.html#U2" title="MODELC_3:sel.2">LTLnew</a> of <span class="p1">(<span class="default"><a href="modelc_3.html#K9" title="MODELC_3:func.9">FinalNode</a> <font color="Maroon" title="c2">v</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="modelc_3.html#D11" target="_self" title="MODELC_3:def.11">Def11</a></span>;<br/><a NAME="E3:150_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E122">A16</font></span>: 
 <a href="modelc_3.html#K36" title="MODELC_3:func.36">chosen_next</a> (<font color="Maroon" title="c1">w</font>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c3">U</font>,<font color="Maroon" title="c4">s</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="modelc_3.html#K12" title="MODELC_3:func.12">'X'</a> <font color="Maroon" title="c4">s</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:150"><span class="lab"><font color="Green" title="E120">A1</font></span></a>, <a class="ref" href="modelc_3.html#D49" target="_self" title="MODELC_3:def.49">Def49</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c10">L</font> = <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="modelc_3.html#K12" title="MODELC_3:func.12">'X'</a> <font color="Maroon" title="c4">s</font></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span>;<br/><a NAME="E4:150_1_1_2"/><span class="lab"><font color="Green" title="E123">A17</font></span>: 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="modelc_3.html#K12" title="MODELC_3:func.12">'X'</a> <font color="Maroon" title="c4">s</font></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T38" title="FINSEQ_1:th.38">FINSEQ_1:38</a></span>;<br/><a NAME="E5:150_1_1_2"/><span class="lab"><font color="Green" title="E124">A18</font></span>: 
 for <font color="Olive" title="b1">n</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="modelc_3.html#K12" title="MODELC_3:func.12">'X'</a> <font color="Maroon" title="c4">s</font></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> holds <br/><span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="modelc_3.html#K12" title="MODELC_3:func.12">'X'</a> <font color="Maroon" title="c4">s</font></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="modelc_3.html#K12" title="MODELC_3:func.12">'X'</a> <font color="Maroon" title="c4">s</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="150_1_1_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c11">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c11">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="modelc_3.html#K12" title="MODELC_3:func.12">'X'</a> <font color="Maroon" title="c4">s</font></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> implies <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="modelc_3.html#K12" title="MODELC_3:func.12">'X'</a> <font color="Maroon" title="c4">s</font></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="modelc_3.html#K12" title="MODELC_3:func.12">'X'</a> <font color="Maroon" title="c4">s</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:150_1_1_2_1"/>
<font color="Maroon" title="c11">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="modelc_3.html#K12" title="MODELC_3:func.12">'X'</a> <font color="Maroon" title="c4">s</font></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="modelc_3.html#K12" title="MODELC_3:func.12">'X'</a> <font color="Maroon" title="c4">s</font></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="modelc_3.html#K12" title="MODELC_3:func.12">'X'</a> <font color="Maroon" title="c4">s</font></span><br/>

<a NAME="E2:150_1_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c11">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:150_1_1_2"><span class="lab"><font color="Green" title="E123">A17</font></span></a>, <a class="ref" href="finseq_1.html#T2" title="FINSEQ_1:th.2">FINSEQ_1:2</a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:150_1_1_2_1"/>
<span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="modelc_3.html#K12" title="MODELC_3:func.12">'X'</a> <font color="Maroon" title="c4">s</font></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="modelc_3.html#K12" title="MODELC_3:func.12">'X'</a> <font color="Maroon" title="c4">s</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T40" title="FINSEQ_1:th.40">FINSEQ_1:40</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E6:150_1_1_2"/>
 for <font color="Olive" title="b1">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k</font> &amp; <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="modelc_3.html#K12" title="MODELC_3:func.12">'X'</a> <font color="Maroon" title="c4">s</font></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> holds <br/> ex <font color="Olive" title="b2">N1</font>, <font color="Olive" title="b3">M1</font> being   <a href="modelc_3.html#V1" title="MODELC_3:attr.1">strict</a>   <a href="modelc_3.html#L1" title="MODELC_3:struct.1">LTLnode</a> of <font color="Maroon" title="c2">v</font> st <br/>( <font color="Olive" title="b2">N1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="modelc_3.html#K12" title="MODELC_3:func.12">'X'</a> <font color="Maroon" title="c4">s</font></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font> &amp; <font color="Olive" title="b3">M1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="modelc_3.html#K12" title="MODELC_3:func.12">'X'</a> <font color="Maroon" title="c4">s</font></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> &amp; <font color="Olive" title="b3">M1</font> <a href="modelc_3.html#R4" title="MODELC_3:pred.4">is_succ_of</a> <font color="Olive" title="b2">N1</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T39" title="FINSEQ_1:th.39">FINSEQ_1:39</a></span>;<br/><a NAME="E7:150_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E125">A19</font></span>: 
<span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="modelc_3.html#K12" title="MODELC_3:func.12">'X'</a> <font color="Maroon" title="c4">s</font></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> <a href="modelc_3.html#R5" title="MODELC_3:pred.5">is_Finseq_for</a> <font color="Maroon" title="c2">v</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="modelc_3.html#D19" target="_self" title="MODELC_3:def.19">Def19</a></span>;<br/><a NAME="E8:150_1_1_2"/>
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/><a NAME="E9:150_1_1_2"/><span class="kw">then </span>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="modelc_3.html#K12" title="MODELC_3:func.12">'X'</a> <font color="Maroon" title="c4">s</font></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="modelc_3.html#K12" title="MODELC_3:func.12">'X'</a> <font color="Maroon" title="c4">s</font></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="modelc_3.html#K12" title="MODELC_3:func.12">'X'</a> <font color="Maroon" title="c4">s</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:150_1_1_2"><span class="lab"><font color="Green" title="E123">A17</font></span></a>, <a class="txt" href="#E5:150_1_1_2"><span class="lab"><font color="Green" title="E124">A18</font></span></a>, <a class="ref" href="finseq_1.html#T39" title="FINSEQ_1:th.39">FINSEQ_1:39</a></span>;<br/><span class="kw">hence </span><a NAME="E10:150_1_1_2"/>
(  <a href="modelc_3.html#K36" title="MODELC_3:func.36">chosen_next</a> (<font color="Maroon" title="c1">w</font>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c3">U</font>,<font color="Maroon" title="c4">s</font>) <a href="modelc_3.html#R6" title="MODELC_3:pred.6">is_next_of</a> <font color="Maroon" title="c4">s</font> &amp; <font color="Maroon" title="c1">w</font> <a href="modelc_2.html#R8" title="MODELC_2:pred.8">|=</a>  <a href="modelc_3.html#K14" title="MODELC_3:func.14">*</a> <span class="p1">(<span class="default"><a href="modelc_3.html#K36" title="MODELC_3:func.36">chosen_next</a> (<font color="Maroon" title="c1">w</font>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c3">U</font>,<font color="Maroon" title="c4">s</font>)</span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:150"><span class="lab"><font color="Green" title="E120">A1</font></span></a>, <a class="txt" href="#E3:150_1_1_2"><span class="lab"><font color="Green" title="E122">A16</font></span></a>, <a class="txt" href="#E7:150_1_1_2"><span class="lab"><font color="Green" title="E125">A19</font></span></a>, <a class="ref" href="modelc_3.html#D20" target="_self" title="MODELC_3:def.20">Def20</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E4:150"/>
(  <a href="modelc_3.html#K36" title="MODELC_3:func.36">chosen_next</a> (<font color="Maroon" title="c1">w</font>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c3">U</font>,<font color="Maroon" title="c4">s</font>) <a href="modelc_3.html#R6" title="MODELC_3:pred.6">is_next_of</a> <font color="Maroon" title="c4">s</font> &amp; <font color="Maroon" title="c1">w</font> <a href="modelc_2.html#R8" title="MODELC_2:pred.8">|=</a>  <a href="modelc_3.html#K14" title="MODELC_3:func.14">*</a> <span class="p1">(<span class="default"><a href="modelc_3.html#K36" title="MODELC_3:func.36">chosen_next</a> (<font color="Maroon" title="c1">w</font>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c3">U</font>,<font color="Maroon" title="c4">s</font>)</span>)</span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
