<?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#K27" title="MODELC_2:func.27">Inf_seq</a> <a href="modelc_2.html#K46" title="MODELC_2:func.46">AtomicFamily</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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>  st <font color="Maroon" title="c1">w</font> <a href="modelc_2.html#R7" title="MODELC_2:pred.7">|=</a> <font color="Olive" title="b1">v</font> holds <br/>for <font color="Olive" title="b3">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds <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="modelc_3.html#K37" title="MODELC_3:func.37">chosen_run</a> <font color="Maroon" title="c1">w</font>,<font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">U</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<font color="Olive" title="b1">v</font> <a href="modelc_3.html#R6" title="MODELC_3:pred.6">is_next_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"><a href="modelc_3.html#K37" title="MODELC_3:func.37">chosen_run</a> <font color="Maroon" title="c1">w</font>,<font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">U</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">n</font></span>)</span>,<font color="Olive" title="b1">v</font> &amp;  <a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <font color="Maroon" title="c1">w</font>,<font color="Olive" title="b3">n</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> <span class="p2">(<span class="default"><a href="modelc_3.html#K10" title="MODELC_3:func.10">CastNode</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="modelc_3.html#K37" title="MODELC_3:func.37">chosen_run</a> <font color="Maroon" title="c1">w</font>,<font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">U</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">n</font></span>)</span>,<font color="Olive" title="b1">v</font></span>)</span></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="Red">::  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>  st <font color="Maroon" title="c1">w</font> <a href="modelc_2.html#R7" title="MODELC_2:pred.7">|=</a> <font color="Maroon" title="c2">v</font> holds <br/>for <font color="Olive" title="b2">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds <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="modelc_3.html#K37" title="MODELC_3:func.37">chosen_run</a> <font color="Maroon" title="c1">w</font>,<font color="Maroon" title="c2">v</font>,<font color="Olive" title="b1">U</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b2">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c2">v</font> <a href="modelc_3.html#R6" title="MODELC_3:pred.6">is_next_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"><a href="modelc_3.html#K37" title="MODELC_3:func.37">chosen_run</a> <font color="Maroon" title="c1">w</font>,<font color="Maroon" title="c2">v</font>,<font color="Olive" title="b1">U</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">n</font></span>)</span>,<font color="Maroon" title="c2">v</font> &amp;  <a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <font color="Maroon" title="c1">w</font>,<font color="Olive" title="b2">n</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> <span class="p2">(<span class="default"><a href="modelc_3.html#K10" title="MODELC_3:func.10">CastNode</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="modelc_3.html#K37" title="MODELC_3:func.37">chosen_run</a> <font color="Maroon" title="c1">w</font>,<font color="Maroon" title="c2">v</font>,<font color="Olive" title="b1">U</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">n</font></span>)</span>,<font color="Maroon" title="c2">v</font></span>)</span></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="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">w</font> <a href="modelc_2.html#R7" title="MODELC_2:pred.7">|=</a> <font color="Maroon" title="c2">v</font> implies for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds <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="modelc_3.html#K37" title="MODELC_3:func.37">chosen_run</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="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c2">v</font> <a href="modelc_3.html#R6" title="MODELC_3:pred.6">is_next_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"><a href="modelc_3.html#K37" title="MODELC_3:func.37">chosen_run</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="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span>,<font color="Maroon" title="c2">v</font> &amp;  <a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <font color="Maroon" title="c1">w</font>,<font color="Olive" title="b1">n</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> <span class="p2">(<span class="default"><a href="modelc_3.html#K10" title="MODELC_3:func.10">CastNode</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="modelc_3.html#K37" title="MODELC_3:func.37">chosen_run</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="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span>,<font color="Maroon" title="c2">v</font></span>)</span></span>)</span> ) )</span><br/>





<span class="kw">set </span><font color="Maroon" title="c4">s</font> =  <a href="modelc_3.html#K11" title="MODELC_3:func.11">init</a> <font color="Maroon" title="c2">v</font>;<br/>
<span class="kw">deffunc </span><font color="Maroon">H<sub>1</sub></font>(  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</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"><a href="modelc_3.html#K37" title="MODELC_3:func.37">chosen_run</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="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1</span>)</span>,<font color="Maroon" title="c2">v</font>;<br/>
<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <span class="kw">means </span>( <font color="Maroon">H<sub>1</sub></font>($1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1) <a href="modelc_3.html#R6" title="MODELC_3:pred.6">is_next_of</a> <font color="Maroon">H<sub>1</sub></font>($1) &amp;  <a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <font color="Maroon" title="c1">w</font>,$1 <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">H<sub>1</sub></font>($1)</span>)</span> );<br/>
<span class="kw">assume </span><a NAME="E1:156"/>
<font color="Maroon" title="c1">w</font> <a href="modelc_2.html#R7" title="MODELC_2:pred.7">|=</a> <font color="Maroon" title="c2">v</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds <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="modelc_3.html#K37" title="MODELC_3:func.37">chosen_run</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="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c2">v</font> <a href="modelc_3.html#R6" title="MODELC_3:pred.6">is_next_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"><a href="modelc_3.html#K37" title="MODELC_3:func.37">chosen_run</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="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span>,<font color="Maroon" title="c2">v</font> &amp;  <a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <font color="Maroon" title="c1">w</font>,<font color="Olive" title="b1">n</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> <span class="p2">(<span class="default"><a href="modelc_3.html#K10" title="MODELC_3:func.10">CastNode</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="modelc_3.html#K37" title="MODELC_3:func.37">chosen_run</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="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span>,<font color="Maroon" title="c2">v</font></span>)</span></span>)</span> )</span><br/>

<a NAME="E2:156"/><span class="kw">then </span><span class="lab"><font color="Green" title="E125">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> <span class="p2">(<span class="default"><a href="modelc_3.html#K11" title="MODELC_3:func.11">init</a> <font color="Maroon" title="c2">v</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="modelc_3.html#T72" target="_self" title="MODELC_3:th.72">Th72</a></span>;<br/>
<span class="lab"><font color="Green" title="E126">A2</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"><a href="modelc_3.html#K37" title="MODELC_3:func.37">chosen_run</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="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>,<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"><a href="modelc_3.html#K11" title="MODELC_3:func.11">init</a> <font color="Maroon" title="c2">v</font></span>)</span>,<font color="Maroon" title="c2">v</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="modelc_3.html#D50" target="_self" title="MODELC_3:def.50">Def50</a></span>
<br/>.= 
 <a href="modelc_3.html#K11" title="MODELC_3:func.11">init</a> <font color="Maroon" title="c2">v</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/>
<a NAME="E4:156"/><span class="lab"><font color="Green" title="E127">A3</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="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="156_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">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="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">n</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1] )</span><br/>

<span class="kw">set </span><font color="Maroon" title="c6">s1</font> = <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">n</font>);<br/>
<a NAME="E1:156_2"/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">n</font>) is   <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>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="156_2_1"><span class="kw">proof </span></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="156_2_1_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="156_2_1_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:156_2_1_1_1"/>
( <font color="Maroon" title="c5">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">n</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="156_2_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:156_2_1_1_1_1"/>
<font color="Maroon" title="c5">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">n</font>) is   <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></span><br/><div class="add"><span class="kw">then </span><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">n</font>) = 
 <a href="modelc_3.html#K10" title="MODELC_3:func.10">CastNode</a> <span class="p1">(<span class="default"><a href="modelc_3.html#K11" title="MODELC_3:func.11">init</a> <font color="Maroon" title="c2">v</font></span>)</span>,<font color="Maroon" title="c2">v</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="modelc_3.html#D50" target="_self" title="MODELC_3:def.50">Def50</a></span>
<br/>.= 
 <a href="modelc_3.html#K11" title="MODELC_3:func.11">init</a> <font color="Maroon" title="c2">v</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="E3:156_2_1_1_1_1"/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">n</font>) is   <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="Red">::  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="156_2_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:156_2_1_1_1_2"/><span class="lab"><font color="Green" title="E128">A4</font></span>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">n</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">n</font>) is   <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></span><br/><div class="add"><span class="kw">set </span><font color="Maroon" title="c7">m</font> = <font color="Maroon" title="c5">n</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1;<br/><span class="kw">reconsider </span><font color="Maroon" title="c8">m</font> = <font color="Maroon" title="c5">n</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="#E1:156_2_1_1_1_2"><span class="lab"><font color="Green" title="E128">A4</font></span></a>, <a class="ref" href="nat_1.html#T20" title="NAT_1:th.20">NAT_1:20</a></span>;<br/><a NAME="E3:156_2_1_1_1_2"/>
<font color="Maroon" title="c5">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 ;<br/><span class="kw">then </span><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">n</font>) = 
 <a href="modelc_3.html#K10" title="MODELC_3:func.10">CastNode</a> <span class="p1">(<span class="default"><a href="modelc_3.html#K36" title="MODELC_3:func.36">chosen_next</a> <span class="p2">(<span class="default"><a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <font color="Maroon" title="c1">w</font>,<font color="Maroon" title="c8">m</font></span>)</span>,<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#K10" title="MODELC_3:func.10">CastNode</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="modelc_3.html#K37" title="MODELC_3:func.37">chosen_run</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="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">m</font></span>)</span>,<font color="Maroon" title="c2">v</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_3.html#D50" target="_self" title="MODELC_3:def.50">Def50</a></span>
<br/>.= 
 <a href="modelc_3.html#K36" title="MODELC_3:func.36">chosen_next</a> <span class="p1">(<span class="default"><a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <font color="Maroon" title="c1">w</font>,<font color="Maroon" title="c8">m</font></span>)</span>,<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#K10" title="MODELC_3:func.10">CastNode</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="modelc_3.html#K37" title="MODELC_3:func.37">chosen_run</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="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">m</font></span>)</span>,<font color="Maroon" title="c2">v</font></span>)</span>
<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="E5:156_2_1_1_1_2"/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">n</font>) is   <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="Red">::  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="E2:156_2_1"/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">n</font>) is   <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="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c7">s1</font> = <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">n</font>) as   <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> ;<br/>
<span class="kw">set </span><font color="Maroon" title="c8">n1</font> = <font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1;<br/>
<span class="kw">set </span><font color="Maroon" title="c9">w1</font> =  <a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <font color="Maroon" title="c1">w</font>,<font color="Maroon" title="c5">n</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c10">w2</font> =  <a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <font color="Maroon" title="c1">w</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c11">s2</font> = <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1);<br/>
<span class="kw">set </span><font color="Maroon" title="c12">s3</font> = <font color="Maroon">H<sub>1</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1);<br/>
<span class="lab"><font color="Green" title="E128">A5</font></span>: <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1) = 
 <a href="modelc_3.html#K10" title="MODELC_3:func.10">CastNode</a> <span class="p1">(<span class="default"><a href="modelc_3.html#K36" title="MODELC_3:func.36">chosen_next</a> <span class="p2">(<span class="default"><a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <font color="Maroon" title="c1">w</font>,<font color="Maroon" title="c5">n</font></span>)</span>,<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#K10" title="MODELC_3:func.10">CastNode</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="modelc_3.html#K37" title="MODELC_3:func.37">chosen_run</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="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">n</font></span>)</span>,<font color="Maroon" title="c2">v</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_3.html#D50" target="_self" title="MODELC_3:def.50">Def50</a></span>
<br/>.= 
 <a href="modelc_3.html#K36" title="MODELC_3:func.36">chosen_next</a> <span class="p1">(<span class="default"><a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <font color="Maroon" title="c1">w</font>,<font color="Maroon" title="c5">n</font></span>)</span>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c3">U</font>,<font color="Maroon" title="c7">s1</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">then </span><span class="kw">reconsider </span><font color="Maroon" title="c13">s2</font> = <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1) as   <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> ;<br/>
<span class="lab"><font color="Green" title="E129">A6</font></span>: <font color="Maroon">H<sub>1</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1) = 
 <a href="modelc_3.html#K10" title="MODELC_3:func.10">CastNode</a> <span class="p1">(<span class="default"><a href="modelc_3.html#K36" title="MODELC_3:func.36">chosen_next</a> <span class="p2">(<span class="default"><a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <font color="Maroon" title="c1">w</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<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#K10" title="MODELC_3:func.10">CastNode</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="modelc_3.html#K37" title="MODELC_3:func.37">chosen_run</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="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c2">v</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_3.html#D50" target="_self" title="MODELC_3:def.50">Def50</a></span>
<br/>.= 
 <a href="modelc_3.html#K36" title="MODELC_3:func.36">chosen_next</a> <span class="p1">(<span class="default"><a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <font color="Maroon" title="c1">w</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c3">U</font>,<font color="Maroon" title="c13">s2</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">assume </span><a NAME="E6:156_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">n</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]</span><br/>

<a NAME="E7:156_2"/><span class="kw">then </span>
(  <a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <font color="Maroon" title="c1">w</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">n</font> <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="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <span class="p1">(<span class="default"><a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <font color="Maroon" title="c1">w</font>,<font color="Maroon" title="c5">n</font></span>)</span>,1 &amp;  <a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <font color="Maroon" title="c1">w</font>,<font color="Maroon" title="c5">n</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> <span class="p2">(<span class="default"><a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <font color="Maroon" title="c1">w</font>,<font color="Maroon" title="c5">n</font></span>)</span>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c3">U</font>,<font color="Maroon" title="c7">s1</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="modelc_3.html#T69" target="_self" title="MODELC_3:th.69">Th69</a>, <a class="ref" href="modelc_2.html#T80" title="MODELC_2:th.80">MODELC_2:80</a></span>;<br/>
<a NAME="E8:156_2"/><span class="kw">then </span>
 <a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <font color="Maroon" title="c1">w</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <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="c13">s2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:156_2"><span class="lab"><font color="Green" title="E128">A5</font></span></a>, <a class="ref" href="modelc_3.html#T70" target="_self" title="MODELC_3:th.70">Th70</a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:156_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">n</font> <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="#E5:156_2"><span class="lab"><font color="Green" title="E129">A6</font></span></a>, <a class="ref" href="modelc_3.html#T69" target="_self" title="MODELC_3:th.69">Th69</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<font color="Maroon">H<sub>1</sub></font>(<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1) = 
 <a href="modelc_3.html#K10" title="MODELC_3:func.10">CastNode</a> <span class="p1">(<span class="default"><a href="modelc_3.html#K36" title="MODELC_3:func.36">chosen_next</a> <span class="p2">(<span class="default"><a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <font color="Maroon" title="c1">w</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>,<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#K10" title="MODELC_3:func.10">CastNode</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="modelc_3.html#K37" title="MODELC_3:func.37">chosen_run</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="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>,<font color="Maroon" title="c2">v</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_3.html#D50" target="_self" title="MODELC_3:def.50">Def50</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#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>,<span class="p2">(<span class="default"><a href="modelc_3.html#K11" title="MODELC_3:func.11">init</a> <font color="Maroon" title="c2">v</font></span>)</span></span>)</span>,<font color="Maroon" title="c2">v</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:156"><span class="lab"><font color="Green" title="E126">A2</font></span></a>, <a class="ref" href="modelc_2.html#T79" title="MODELC_2:th.79">MODELC_2:79</a></span>
<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>,<span class="p1">(<span class="default"><a href="modelc_3.html#K11" title="MODELC_3:func.11">init</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#D16" target="_self" title="MODELC_3:def.16">Def16</a></span>
;<br/>
<a NAME="E6:156"/><span class="kw">then </span><span class="lab"><font color="Green" title="E128">A7</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:156"><span class="lab"><font color="Green" title="E125">A1</font></span></a>, <a class="txt" href="#E3:156"><span class="lab"><font color="Green" title="E126">A2</font></span></a>, <a class="ref" href="modelc_3.html#T69" target="_self" title="MODELC_3:th.69">Th69</a>, <a class="ref" href="modelc_2.html#T79" title="MODELC_2:th.79">MODELC_2:79</a></span>;<br/>
<a NAME="E7:156"/>
for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S2" title="NAT_1:sch.2">NAT_1:sch 2</a>(<a class="txt" href="#E6:156"><span class="lab"><font color="Green" title="E128">A7</font></span></a>, <a class="txt" href="#E4:156"><span class="lab"><font color="Green" title="E127">A3</font></span></a>);<br/></span>
<span class="kw">hence </span><a NAME="E8:156"/>
for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds <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="modelc_3.html#K37" title="MODELC_3:func.37">chosen_run</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="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c2">v</font> <a href="modelc_3.html#R6" title="MODELC_3:pred.6">is_next_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"><a href="modelc_3.html#K37" title="MODELC_3:func.37">chosen_run</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="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span>,<font color="Maroon" title="c2">v</font> &amp;  <a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <font color="Maroon" title="c1">w</font>,<font color="Olive" title="b1">n</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> <span class="p2">(<span class="default"><a href="modelc_3.html#K10" title="MODELC_3:func.10">CastNode</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="modelc_3.html#K37" title="MODELC_3:func.37">chosen_run</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="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span>,<font color="Maroon" title="c2">v</font></span>)</span></span>)</span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
