diff -r 5499ba68188c -r f65444d29e74 thys2/Journal/SizeBound.tex
--- a/thys2/Journal/SizeBound.tex	Mon Nov 01 10:52:44 2021 +0000
+++ b/thys2/Journal/SizeBound.tex	Tue Nov 02 13:57:59 2021 +0000
@@ -1,7263 +0,0 @@
-%
-\begin{isabellebody}%
-\setisabellecontext{SizeBound}%
-%
-\isadelimtheory
-\isanewline
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{theory}\isamarkupfalse%
-\ SizeBound\isanewline
-\ \ \isakeyword{imports}\ {\isachardoublequoteopen}Lexer{\isachardoublequoteclose}\ \isanewline
-\isakeyword{begin}%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsection{Bit-Encodings%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{datatype}\isamarkupfalse%
-\ bit\ {\isacharequal}{\kern0pt}\ Z\ {\isacharbar}{\kern0pt}\ S\isanewline
-\isanewline
-\isacommand{fun}\isamarkupfalse%
-\ code\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}val\ {\isasymRightarrow}\ bit\ list{\isachardoublequoteclose}\isanewline
-\isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}code\ Void\ {\isacharequal}{\kern0pt}\ {\isacharbrackleft}{\kern0pt}{\isacharbrackright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-{\isacharbar}{\kern0pt}\ {\isachardoublequoteopen}code\ {\isacharparenleft}{\kern0pt}Char\ c{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharbrackleft}{\kern0pt}{\isacharbrackright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-{\isacharbar}{\kern0pt}\ {\isachardoublequoteopen}code\ {\isacharparenleft}{\kern0pt}Left\ v{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ Z\ {\isacharhash}{\kern0pt}\ {\isacharparenleft}{\kern0pt}code\ v{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-{\isacharbar}{\kern0pt}\ {\isachardoublequoteopen}code\ {\isacharparenleft}{\kern0pt}Right\ v{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ S\ {\isacharhash}{\kern0pt}\ {\isacharparenleft}{\kern0pt}code\ v{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-{\isacharbar}{\kern0pt}\ {\isachardoublequoteopen}code\ {\isacharparenleft}{\kern0pt}Seq\ v{\isadigit{1}}\ v{\isadigit{2}}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}code\ v{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isacharat}{\kern0pt}\ {\isacharparenleft}{\kern0pt}code\ v{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-{\isacharbar}{\kern0pt}\ {\isachardoublequoteopen}code\ {\isacharparenleft}{\kern0pt}Stars\ {\isacharbrackleft}{\kern0pt}{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharbrackleft}{\kern0pt}S{\isacharbrackright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-{\isacharbar}{\kern0pt}\ {\isachardoublequoteopen}code\ {\isacharparenleft}{\kern0pt}Stars\ {\isacharparenleft}{\kern0pt}v\ {\isacharhash}{\kern0pt}\ vs{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ \ {\isacharparenleft}{\kern0pt}Z\ {\isacharhash}{\kern0pt}\ code\ v{\isacharparenright}{\kern0pt}\ {\isacharat}{\kern0pt}\ code\ {\isacharparenleft}{\kern0pt}Stars\ vs{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isanewline
-\isacommand{fun}\isamarkupfalse%
-\ \isanewline
-\ \ Stars{\isacharunderscore}{\kern0pt}add\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}val\ {\isasymRightarrow}\ val\ {\isasymRightarrow}\ val{\isachardoublequoteclose}\isanewline
-\isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}Stars{\isacharunderscore}{\kern0pt}add\ v\ {\isacharparenleft}{\kern0pt}Stars\ vs{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ Stars\ {\isacharparenleft}{\kern0pt}v\ {\isacharhash}{\kern0pt}\ vs{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{function}\isamarkupfalse%
-\isanewline
-\ \ decode{\isacharprime}{\kern0pt}\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}bit\ list\ {\isasymRightarrow}\ rexp\ {\isasymRightarrow}\ {\isacharparenleft}{\kern0pt}val\ {\isacharasterisk}{\kern0pt}\ bit\ list{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}decode{\isacharprime}{\kern0pt}\ ds\ ZERO\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}Void{\isacharcomma}{\kern0pt}\ {\isacharbrackleft}{\kern0pt}{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-{\isacharbar}{\kern0pt}\ {\isachardoublequoteopen}decode{\isacharprime}{\kern0pt}\ ds\ ONE\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}Void{\isacharcomma}{\kern0pt}\ ds{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-{\isacharbar}{\kern0pt}\ {\isachardoublequoteopen}decode{\isacharprime}{\kern0pt}\ ds\ {\isacharparenleft}{\kern0pt}CH\ d{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}Char\ d{\isacharcomma}{\kern0pt}\ ds{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-{\isacharbar}{\kern0pt}\ {\isachardoublequoteopen}decode{\isacharprime}{\kern0pt}\ {\isacharbrackleft}{\kern0pt}{\isacharbrackright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}ALT\ r{\isadigit{1}}\ r{\isadigit{2}}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}Void{\isacharcomma}{\kern0pt}\ {\isacharbrackleft}{\kern0pt}{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-{\isacharbar}{\kern0pt}\ {\isachardoublequoteopen}decode{\isacharprime}{\kern0pt}\ {\isacharparenleft}{\kern0pt}Z\ {\isacharhash}{\kern0pt}\ ds{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}ALT\ r{\isadigit{1}}\ r{\isadigit{2}}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}let\ {\isacharparenleft}{\kern0pt}v{\isacharcomma}{\kern0pt}\ ds{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ decode{\isacharprime}{\kern0pt}\ ds\ r{\isadigit{1}}\ in\ {\isacharparenleft}{\kern0pt}Left\ v{\isacharcomma}{\kern0pt}\ ds{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-{\isacharbar}{\kern0pt}\ {\isachardoublequoteopen}decode{\isacharprime}{\kern0pt}\ {\isacharparenleft}{\kern0pt}S\ {\isacharhash}{\kern0pt}\ ds{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}ALT\ r{\isadigit{1}}\ r{\isadigit{2}}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}let\ {\isacharparenleft}{\kern0pt}v{\isacharcomma}{\kern0pt}\ ds{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ decode{\isacharprime}{\kern0pt}\ ds\ r{\isadigit{2}}\ in\ {\isacharparenleft}{\kern0pt}Right\ v{\isacharcomma}{\kern0pt}\ ds{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-{\isacharbar}{\kern0pt}\ {\isachardoublequoteopen}decode{\isacharprime}{\kern0pt}\ ds\ {\isacharparenleft}{\kern0pt}SEQ\ r{\isadigit{1}}\ r{\isadigit{2}}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}let\ {\isacharparenleft}{\kern0pt}v{\isadigit{1}}{\isacharcomma}{\kern0pt}\ ds{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ decode{\isacharprime}{\kern0pt}\ ds\ r{\isadigit{1}}\ in\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ let\ {\isacharparenleft}{\kern0pt}v{\isadigit{2}}{\isacharcomma}{\kern0pt}\ ds{\isacharprime}{\kern0pt}{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ decode{\isacharprime}{\kern0pt}\ ds{\isacharprime}{\kern0pt}\ r{\isadigit{2}}\ in\ {\isacharparenleft}{\kern0pt}Seq\ v{\isadigit{1}}\ v{\isadigit{2}}{\isacharcomma}{\kern0pt}\ ds{\isacharprime}{\kern0pt}{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-{\isacharbar}{\kern0pt}\ {\isachardoublequoteopen}decode{\isacharprime}{\kern0pt}\ {\isacharbrackleft}{\kern0pt}{\isacharbrackright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}STAR\ r{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}Void{\isacharcomma}{\kern0pt}\ {\isacharbrackleft}{\kern0pt}{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-{\isacharbar}{\kern0pt}\ {\isachardoublequoteopen}decode{\isacharprime}{\kern0pt}\ {\isacharparenleft}{\kern0pt}S\ {\isacharhash}{\kern0pt}\ ds{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}STAR\ r{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}Stars\ {\isacharbrackleft}{\kern0pt}{\isacharbrackright}{\kern0pt}{\isacharcomma}{\kern0pt}\ ds{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-{\isacharbar}{\kern0pt}\ {\isachardoublequoteopen}decode{\isacharprime}{\kern0pt}\ {\isacharparenleft}{\kern0pt}Z\ {\isacharhash}{\kern0pt}\ ds{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}STAR\ r{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}let\ {\isacharparenleft}{\kern0pt}v{\isacharcomma}{\kern0pt}\ ds{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ decode{\isacharprime}{\kern0pt}\ ds\ r\ in\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ let\ {\isacharparenleft}{\kern0pt}vs{\isacharcomma}{\kern0pt}\ ds{\isacharprime}{\kern0pt}{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ decode{\isacharprime}{\kern0pt}\ ds{\isacharprime}{\kern0pt}\ {\isacharparenleft}{\kern0pt}STAR\ r{\isacharparenright}{\kern0pt}\ \isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ in\ {\isacharparenleft}{\kern0pt}Stars{\isacharunderscore}{\kern0pt}add\ v\ vs{\isacharcomma}{\kern0pt}\ ds{\isacharprime}{\kern0pt}{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ pat{\isacharunderscore}{\kern0pt}completeness\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ decode{\isacharprime}{\kern0pt}{\isacharunderscore}{\kern0pt}smaller{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}decode{\isacharprime}{\kern0pt}{\isacharunderscore}{\kern0pt}dom\ {\isacharparenleft}{\kern0pt}ds{\isacharcomma}{\kern0pt}\ r{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}length\ {\isacharparenleft}{\kern0pt}snd\ {\isacharparenleft}{\kern0pt}decode{\isacharprime}{\kern0pt}\ ds\ r{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymle}\ length\ ds{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ assms\isanewline
-\isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}induct\ ds\ r{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}auto\ simp\ add{\isacharcolon}{\kern0pt}\ decode{\isacharprime}{\kern0pt}{\isachardot}{\kern0pt}psimps\ split{\isacharcolon}{\kern0pt}\ prod{\isachardot}{\kern0pt}split{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{using}\isamarkupfalse%
-\ dual{\isacharunderscore}{\kern0pt}order{\isachardot}{\kern0pt}trans\ \isacommand{apply}\isamarkupfalse%
-\ blast\isanewline
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}meson\ dual{\isacharunderscore}{\kern0pt}order{\isachardot}{\kern0pt}trans\ le{\isacharunderscore}{\kern0pt}SucI{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{termination}\isamarkupfalse%
-\ {\isachardoublequoteopen}decode{\isacharprime}{\kern0pt}{\isachardoublequoteclose}\ \ \isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}relation\ {\isachardoublequoteopen}inv{\isacharunderscore}{\kern0pt}image\ {\isacharparenleft}{\kern0pt}measure{\isacharparenleft}{\kern0pt}{\isacharpercent}{\kern0pt}cs{\isachardot}{\kern0pt}\ size\ cs{\isacharparenright}{\kern0pt}\ {\isacharless}{\kern0pt}{\isacharasterisk}{\kern0pt}lex{\isacharasterisk}{\kern0pt}{\isachargreater}{\kern0pt}\ measure{\isacharparenleft}{\kern0pt}{\isacharpercent}{\kern0pt}s{\isachardot}{\kern0pt}\ size\ s{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharpercent}{\kern0pt}{\isacharparenleft}{\kern0pt}ds{\isacharcomma}{\kern0pt}r{\isacharparenright}{\kern0pt}{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}r{\isacharcomma}{\kern0pt}ds{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}{\isacharparenright}{\kern0pt}\ \isanewline
-\isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}auto\ dest{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ decode{\isacharprime}{\kern0pt}{\isacharunderscore}{\kern0pt}smaller{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ less{\isacharunderscore}{\kern0pt}Suc{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}le\ snd{\isacharunderscore}{\kern0pt}conv{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\isanewline
-\ \ decode\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}bit\ list\ {\isasymRightarrow}\ rexp\ {\isasymRightarrow}\ val\ option{\isachardoublequoteclose}\isanewline
-\isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}decode\ ds\ r\ {\isasymequiv}\ {\isacharparenleft}{\kern0pt}let\ {\isacharparenleft}{\kern0pt}v{\isacharcomma}{\kern0pt}\ ds{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ decode{\isacharprime}{\kern0pt}\ ds\ r\ \isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ in\ {\isacharparenleft}{\kern0pt}if\ ds{\isacharprime}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharbrackleft}{\kern0pt}{\isacharbrackright}{\kern0pt}\ then\ Some\ v\ else\ None{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ decode{\isacharprime}{\kern0pt}{\isacharunderscore}{\kern0pt}code{\isacharunderscore}{\kern0pt}Stars{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}{\isasymforall}v{\isasymin}set\ vs{\isachardot}{\kern0pt}\ {\isasymTurnstile}\ v\ {\isacharcolon}{\kern0pt}\ r\ {\isasymand}\ {\isacharparenleft}{\kern0pt}{\isasymforall}x{\isachardot}{\kern0pt}\ decode{\isacharprime}{\kern0pt}\ {\isacharparenleft}{\kern0pt}code\ v\ {\isacharat}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ r\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}v{\isacharcomma}{\kern0pt}\ x{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymand}\ flat\ v\ {\isasymnoteq}\ {\isacharbrackleft}{\kern0pt}{\isacharbrackright}{\kern0pt}{\isachardoublequoteclose}\ \isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}decode{\isacharprime}{\kern0pt}\ {\isacharparenleft}{\kern0pt}code\ {\isacharparenleft}{\kern0pt}Stars\ vs{\isacharparenright}{\kern0pt}\ {\isacharat}{\kern0pt}\ ds{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}STAR\ r{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}Stars\ vs{\isacharcomma}{\kern0pt}\ ds{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ assms\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}induct\ vs{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}auto{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{done}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ decode{\isacharprime}{\kern0pt}{\isacharunderscore}{\kern0pt}code{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}{\isasymTurnstile}\ v\ {\isacharcolon}{\kern0pt}\ r{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}decode{\isacharprime}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}code\ v{\isacharparenright}{\kern0pt}\ {\isacharat}{\kern0pt}\ ds{\isacharparenright}{\kern0pt}\ r\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}v{\isacharcomma}{\kern0pt}\ ds{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ assms\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}induct\ v\ r\ arbitrary{\isacharcolon}{\kern0pt}\ ds{\isacharparenright}{\kern0pt}\ \isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}auto{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{using}\isamarkupfalse%
-\ decode{\isacharprime}{\kern0pt}{\isacharunderscore}{\kern0pt}code{\isacharunderscore}{\kern0pt}Stars\ \isacommand{by}\isamarkupfalse%
-\ blast%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ decode{\isacharunderscore}{\kern0pt}code{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}{\isasymTurnstile}\ v\ {\isacharcolon}{\kern0pt}\ r{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}decode\ {\isacharparenleft}{\kern0pt}code\ v{\isacharparenright}{\kern0pt}\ r\ {\isacharequal}{\kern0pt}\ Some\ v{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ assms\ \isacommand{unfolding}\isamarkupfalse%
-\ decode{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}smt\ append{\isacharunderscore}{\kern0pt}Nil{\isadigit{2}}\ decode{\isacharprime}{\kern0pt}{\isacharunderscore}{\kern0pt}code\ old{\isachardot}{\kern0pt}prod{\isachardot}{\kern0pt}case{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsection{Annotated Regular Expressions%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{datatype}\isamarkupfalse%
-\ arexp\ {\isacharequal}{\kern0pt}\ \isanewline
-\ \ AZERO\isanewline
-{\isacharbar}{\kern0pt}\ AONE\ {\isachardoublequoteopen}bit\ list{\isachardoublequoteclose}\isanewline
-{\isacharbar}{\kern0pt}\ ACHAR\ {\isachardoublequoteopen}bit\ list{\isachardoublequoteclose}\ char\isanewline
-{\isacharbar}{\kern0pt}\ ASEQ\ {\isachardoublequoteopen}bit\ list{\isachardoublequoteclose}\ arexp\ arexp\isanewline
-{\isacharbar}{\kern0pt}\ AALTs\ {\isachardoublequoteopen}bit\ list{\isachardoublequoteclose}\ {\isachardoublequoteopen}arexp\ list{\isachardoublequoteclose}\isanewline
-{\isacharbar}{\kern0pt}\ ASTAR\ {\isachardoublequoteopen}bit\ list{\isachardoublequoteclose}\ arexp\isanewline
-\isanewline
-\isacommand{abbreviation}\isamarkupfalse%
-\isanewline
-\ \ {\isachardoublequoteopen}AALT\ bs\ r{\isadigit{1}}\ r{\isadigit{2}}\ {\isasymequiv}\ AALTs\ bs\ {\isacharbrackleft}{\kern0pt}r{\isadigit{1}}{\isacharcomma}{\kern0pt}\ r{\isadigit{2}}{\isacharbrackright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{fun}\isamarkupfalse%
-\ asize\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}arexp\ {\isasymRightarrow}\ nat{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}asize\ AZERO\ {\isacharequal}{\kern0pt}\ {\isadigit{1}}{\isachardoublequoteclose}\isanewline
-{\isacharbar}{\kern0pt}\ {\isachardoublequoteopen}asize\ {\isacharparenleft}{\kern0pt}AONE\ cs{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isadigit{1}}{\isachardoublequoteclose}\ \isanewline
-{\isacharbar}{\kern0pt}\ {\isachardoublequoteopen}asize\ {\isacharparenleft}{\kern0pt}ACHAR\ cs\ c{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isadigit{1}}{\isachardoublequoteclose}\isanewline
-{\isacharbar}{\kern0pt}\ {\isachardoublequoteopen}asize\ {\isacharparenleft}{\kern0pt}AALTs\ cs\ rs{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ Suc\ {\isacharparenleft}{\kern0pt}sum{\isacharunderscore}{\kern0pt}list\ {\isacharparenleft}{\kern0pt}map\ asize\ rs{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-{\isacharbar}{\kern0pt}\ {\isachardoublequoteopen}asize\ {\isacharparenleft}{\kern0pt}ASEQ\ cs\ r{\isadigit{1}}\ r{\isadigit{2}}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ Suc\ {\isacharparenleft}{\kern0pt}asize\ r{\isadigit{1}}\ {\isacharplus}{\kern0pt}\ asize\ r{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-{\isacharbar}{\kern0pt}\ {\isachardoublequoteopen}asize\ {\isacharparenleft}{\kern0pt}ASTAR\ cs\ r{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ Suc\ {\isacharparenleft}{\kern0pt}asize\ r{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{fun}\isamarkupfalse%
-\ \isanewline
-\ \ erase\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}arexp\ {\isasymRightarrow}\ rexp{\isachardoublequoteclose}\isanewline
-\isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}erase\ AZERO\ {\isacharequal}{\kern0pt}\ ZERO{\isachardoublequoteclose}\isanewline
-{\isacharbar}{\kern0pt}\ {\isachardoublequoteopen}erase\ {\isacharparenleft}{\kern0pt}AONE\ {\isacharunderscore}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ ONE{\isachardoublequoteclose}\isanewline
-{\isacharbar}{\kern0pt}\ {\isachardoublequoteopen}erase\ {\isacharparenleft}{\kern0pt}ACHAR\ {\isacharunderscore}{\kern0pt}\ c{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ CH\ c{\isachardoublequoteclose}\isanewline
-{\isacharbar}{\kern0pt}\ {\isachardoublequoteopen}erase\ {\isacharparenleft}{\kern0pt}AALTs\ {\isacharunderscore}{\kern0pt}\ {\isacharbrackleft}{\kern0pt}{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ ZERO{\isachardoublequoteclose}\isanewline
-{\isacharbar}{\kern0pt}\ {\isachardoublequoteopen}erase\ {\isacharparenleft}{\kern0pt}AALTs\ {\isacharunderscore}{\kern0pt}\ {\isacharbrackleft}{\kern0pt}r{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}erase\ r{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-{\isacharbar}{\kern0pt}\ {\isachardoublequoteopen}erase\ {\isacharparenleft}{\kern0pt}AALTs\ bs\ {\isacharparenleft}{\kern0pt}r{\isacharhash}{\kern0pt}rs{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ ALT\ {\isacharparenleft}{\kern0pt}erase\ r{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}erase\ {\isacharparenleft}{\kern0pt}AALTs\ bs\ rs{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-{\isacharbar}{\kern0pt}\ {\isachardoublequoteopen}erase\ {\isacharparenleft}{\kern0pt}ASEQ\ {\isacharunderscore}{\kern0pt}\ r{\isadigit{1}}\ r{\isadigit{2}}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ SEQ\ {\isacharparenleft}{\kern0pt}erase\ r{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}erase\ r{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-{\isacharbar}{\kern0pt}\ {\isachardoublequoteopen}erase\ {\isacharparenleft}{\kern0pt}ASTAR\ {\isacharunderscore}{\kern0pt}\ r{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ STAR\ {\isacharparenleft}{\kern0pt}erase\ r{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isanewline
-\isanewline
-\isanewline
-\isacommand{fun}\isamarkupfalse%
-\ nonalt\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}arexp\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}nonalt\ {\isacharparenleft}{\kern0pt}AALTs\ bs{\isadigit{2}}\ rs{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ False{\isachardoublequoteclose}\isanewline
-{\isacharbar}{\kern0pt}\ {\isachardoublequoteopen}nonalt\ r\ {\isacharequal}{\kern0pt}\ True{\isachardoublequoteclose}\isanewline
-\isanewline
-\isanewline
-\isacommand{fun}\isamarkupfalse%
-\ good\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}arexp\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}good\ AZERO\ {\isacharequal}{\kern0pt}\ False{\isachardoublequoteclose}\isanewline
-{\isacharbar}{\kern0pt}\ {\isachardoublequoteopen}good\ {\isacharparenleft}{\kern0pt}AONE\ cs{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ True{\isachardoublequoteclose}\ \isanewline
-{\isacharbar}{\kern0pt}\ {\isachardoublequoteopen}good\ {\isacharparenleft}{\kern0pt}ACHAR\ cs\ c{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ True{\isachardoublequoteclose}\isanewline
-{\isacharbar}{\kern0pt}\ {\isachardoublequoteopen}good\ {\isacharparenleft}{\kern0pt}AALTs\ cs\ {\isacharbrackleft}{\kern0pt}{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ False{\isachardoublequoteclose}\isanewline
-{\isacharbar}{\kern0pt}\ {\isachardoublequoteopen}good\ {\isacharparenleft}{\kern0pt}AALTs\ cs\ {\isacharbrackleft}{\kern0pt}r{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ False{\isachardoublequoteclose}\isanewline
-{\isacharbar}{\kern0pt}\ {\isachardoublequoteopen}good\ {\isacharparenleft}{\kern0pt}AALTs\ cs\ {\isacharparenleft}{\kern0pt}r{\isadigit{1}}{\isacharhash}{\kern0pt}r{\isadigit{2}}{\isacharhash}{\kern0pt}rs{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymforall}r{\isacharprime}{\kern0pt}\ {\isasymin}\ set\ {\isacharparenleft}{\kern0pt}r{\isadigit{1}}{\isacharhash}{\kern0pt}r{\isadigit{2}}{\isacharhash}{\kern0pt}rs{\isacharparenright}{\kern0pt}{\isachardot}{\kern0pt}\ good\ r{\isacharprime}{\kern0pt}\ {\isasymand}\ nonalt\ r{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-{\isacharbar}{\kern0pt}\ {\isachardoublequoteopen}good\ {\isacharparenleft}{\kern0pt}ASEQ\ {\isacharunderscore}{\kern0pt}\ AZERO\ {\isacharunderscore}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ False{\isachardoublequoteclose}\isanewline
-{\isacharbar}{\kern0pt}\ {\isachardoublequoteopen}good\ {\isacharparenleft}{\kern0pt}ASEQ\ {\isacharunderscore}{\kern0pt}\ {\isacharparenleft}{\kern0pt}AONE\ {\isacharunderscore}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharunderscore}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ False{\isachardoublequoteclose}\isanewline
-{\isacharbar}{\kern0pt}\ {\isachardoublequoteopen}good\ {\isacharparenleft}{\kern0pt}ASEQ\ {\isacharunderscore}{\kern0pt}\ {\isacharunderscore}{\kern0pt}\ AZERO{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ False{\isachardoublequoteclose}\isanewline
-{\isacharbar}{\kern0pt}\ {\isachardoublequoteopen}good\ {\isacharparenleft}{\kern0pt}ASEQ\ cs\ r{\isadigit{1}}\ r{\isadigit{2}}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}good\ r{\isadigit{1}}\ {\isasymand}\ good\ r{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-{\isacharbar}{\kern0pt}\ {\isachardoublequoteopen}good\ {\isacharparenleft}{\kern0pt}ASTAR\ cs\ r{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ True{\isachardoublequoteclose}\isanewline
-\isanewline
-\isanewline
-\isanewline
-\isanewline
-\isacommand{fun}\isamarkupfalse%
-\ fuse\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}bit\ list\ {\isasymRightarrow}\ arexp\ {\isasymRightarrow}\ arexp{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}fuse\ bs\ AZERO\ {\isacharequal}{\kern0pt}\ AZERO{\isachardoublequoteclose}\isanewline
-{\isacharbar}{\kern0pt}\ {\isachardoublequoteopen}fuse\ bs\ {\isacharparenleft}{\kern0pt}AONE\ cs{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ AONE\ {\isacharparenleft}{\kern0pt}bs\ {\isacharat}{\kern0pt}\ cs{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isanewline
-{\isacharbar}{\kern0pt}\ {\isachardoublequoteopen}fuse\ bs\ {\isacharparenleft}{\kern0pt}ACHAR\ cs\ c{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ ACHAR\ {\isacharparenleft}{\kern0pt}bs\ {\isacharat}{\kern0pt}\ cs{\isacharparenright}{\kern0pt}\ c{\isachardoublequoteclose}\isanewline
-{\isacharbar}{\kern0pt}\ {\isachardoublequoteopen}fuse\ bs\ {\isacharparenleft}{\kern0pt}AALTs\ cs\ rs{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ AALTs\ {\isacharparenleft}{\kern0pt}bs\ {\isacharat}{\kern0pt}\ cs{\isacharparenright}{\kern0pt}\ rs{\isachardoublequoteclose}\isanewline
-{\isacharbar}{\kern0pt}\ {\isachardoublequoteopen}fuse\ bs\ {\isacharparenleft}{\kern0pt}ASEQ\ cs\ r{\isadigit{1}}\ r{\isadigit{2}}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ ASEQ\ {\isacharparenleft}{\kern0pt}bs\ {\isacharat}{\kern0pt}\ cs{\isacharparenright}{\kern0pt}\ r{\isadigit{1}}\ r{\isadigit{2}}{\isachardoublequoteclose}\isanewline
-{\isacharbar}{\kern0pt}\ {\isachardoublequoteopen}fuse\ bs\ {\isacharparenleft}{\kern0pt}ASTAR\ cs\ r{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ ASTAR\ {\isacharparenleft}{\kern0pt}bs\ {\isacharat}{\kern0pt}\ cs{\isacharparenright}{\kern0pt}\ r{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ fuse{\isacharunderscore}{\kern0pt}append{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}fuse\ {\isacharparenleft}{\kern0pt}bs{\isadigit{1}}\ {\isacharat}{\kern0pt}\ bs{\isadigit{2}}{\isacharparenright}{\kern0pt}\ r\ {\isacharequal}{\kern0pt}\ fuse\ bs{\isadigit{1}}\ {\isacharparenleft}{\kern0pt}fuse\ bs{\isadigit{2}}\ r{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}induct\ r{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}auto{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{done}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{fun}\isamarkupfalse%
-\ intern\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}rexp\ {\isasymRightarrow}\ arexp{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}intern\ ZERO\ {\isacharequal}{\kern0pt}\ AZERO{\isachardoublequoteclose}\isanewline
-{\isacharbar}{\kern0pt}\ {\isachardoublequoteopen}intern\ ONE\ {\isacharequal}{\kern0pt}\ AONE\ {\isacharbrackleft}{\kern0pt}{\isacharbrackright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-{\isacharbar}{\kern0pt}\ {\isachardoublequoteopen}intern\ {\isacharparenleft}{\kern0pt}CH\ c{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ ACHAR\ {\isacharbrackleft}{\kern0pt}{\isacharbrackright}{\kern0pt}\ c{\isachardoublequoteclose}\isanewline
-{\isacharbar}{\kern0pt}\ {\isachardoublequoteopen}intern\ {\isacharparenleft}{\kern0pt}ALT\ r{\isadigit{1}}\ r{\isadigit{2}}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ AALT\ {\isacharbrackleft}{\kern0pt}{\isacharbrackright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}fuse\ {\isacharbrackleft}{\kern0pt}Z{\isacharbrackright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}intern\ r{\isadigit{1}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ \isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}fuse\ {\isacharbrackleft}{\kern0pt}S{\isacharbrackright}{\kern0pt}\ \ {\isacharparenleft}{\kern0pt}intern\ r{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-{\isacharbar}{\kern0pt}\ {\isachardoublequoteopen}intern\ {\isacharparenleft}{\kern0pt}SEQ\ r{\isadigit{1}}\ r{\isadigit{2}}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ ASEQ\ {\isacharbrackleft}{\kern0pt}{\isacharbrackright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}intern\ r{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}intern\ r{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-{\isacharbar}{\kern0pt}\ {\isachardoublequoteopen}intern\ {\isacharparenleft}{\kern0pt}STAR\ r{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ ASTAR\ {\isacharbrackleft}{\kern0pt}{\isacharbrackright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}intern\ r{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isanewline
-\isacommand{fun}\isamarkupfalse%
-\ retrieve\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}arexp\ {\isasymRightarrow}\ val\ {\isasymRightarrow}\ bit\ list{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}retrieve\ {\isacharparenleft}{\kern0pt}AONE\ bs{\isacharparenright}{\kern0pt}\ Void\ {\isacharequal}{\kern0pt}\ bs{\isachardoublequoteclose}\isanewline
-{\isacharbar}{\kern0pt}\ {\isachardoublequoteopen}retrieve\ {\isacharparenleft}{\kern0pt}ACHAR\ bs\ c{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}Char\ d{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ bs{\isachardoublequoteclose}\isanewline
-{\isacharbar}{\kern0pt}\ {\isachardoublequoteopen}retrieve\ {\isacharparenleft}{\kern0pt}AALTs\ bs\ {\isacharbrackleft}{\kern0pt}r{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\ v\ {\isacharequal}{\kern0pt}\ bs\ {\isacharat}{\kern0pt}\ retrieve\ r\ v{\isachardoublequoteclose}\ \isanewline
-{\isacharbar}{\kern0pt}\ {\isachardoublequoteopen}retrieve\ {\isacharparenleft}{\kern0pt}AALTs\ bs\ {\isacharparenleft}{\kern0pt}r{\isacharhash}{\kern0pt}rs{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}Left\ v{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ bs\ {\isacharat}{\kern0pt}\ retrieve\ r\ v{\isachardoublequoteclose}\isanewline
-{\isacharbar}{\kern0pt}\ {\isachardoublequoteopen}retrieve\ {\isacharparenleft}{\kern0pt}AALTs\ bs\ {\isacharparenleft}{\kern0pt}r{\isacharhash}{\kern0pt}rs{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}Right\ v{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ bs\ {\isacharat}{\kern0pt}\ retrieve\ {\isacharparenleft}{\kern0pt}AALTs\ {\isacharbrackleft}{\kern0pt}{\isacharbrackright}{\kern0pt}\ rs{\isacharparenright}{\kern0pt}\ v{\isachardoublequoteclose}\isanewline
-{\isacharbar}{\kern0pt}\ {\isachardoublequoteopen}retrieve\ {\isacharparenleft}{\kern0pt}ASEQ\ bs\ r{\isadigit{1}}\ r{\isadigit{2}}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}Seq\ v{\isadigit{1}}\ v{\isadigit{2}}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ bs\ {\isacharat}{\kern0pt}\ retrieve\ r{\isadigit{1}}\ v{\isadigit{1}}\ {\isacharat}{\kern0pt}\ retrieve\ r{\isadigit{2}}\ v{\isadigit{2}}{\isachardoublequoteclose}\isanewline
-{\isacharbar}{\kern0pt}\ {\isachardoublequoteopen}retrieve\ {\isacharparenleft}{\kern0pt}ASTAR\ bs\ r{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}Stars\ {\isacharbrackleft}{\kern0pt}{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ bs\ {\isacharat}{\kern0pt}\ {\isacharbrackleft}{\kern0pt}S{\isacharbrackright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-{\isacharbar}{\kern0pt}\ {\isachardoublequoteopen}retrieve\ {\isacharparenleft}{\kern0pt}ASTAR\ bs\ r{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}Stars\ {\isacharparenleft}{\kern0pt}v{\isacharhash}{\kern0pt}vs{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ \isanewline
-\ \ \ \ \ bs\ {\isacharat}{\kern0pt}\ {\isacharbrackleft}{\kern0pt}Z{\isacharbrackright}{\kern0pt}\ {\isacharat}{\kern0pt}\ retrieve\ r\ v\ {\isacharat}{\kern0pt}\ retrieve\ {\isacharparenleft}{\kern0pt}ASTAR\ {\isacharbrackleft}{\kern0pt}{\isacharbrackright}{\kern0pt}\ r{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}Stars\ vs{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isanewline
-\isanewline
-\isacommand{fun}\isamarkupfalse%
-\isanewline
-\ bnullable\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}arexp\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\isanewline
-\isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}bnullable\ {\isacharparenleft}{\kern0pt}AZERO{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ False{\isachardoublequoteclose}\isanewline
-{\isacharbar}{\kern0pt}\ {\isachardoublequoteopen}bnullable\ {\isacharparenleft}{\kern0pt}AONE\ bs{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ True{\isachardoublequoteclose}\isanewline
-{\isacharbar}{\kern0pt}\ {\isachardoublequoteopen}bnullable\ {\isacharparenleft}{\kern0pt}ACHAR\ bs\ c{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ False{\isachardoublequoteclose}\isanewline
-{\isacharbar}{\kern0pt}\ {\isachardoublequoteopen}bnullable\ {\isacharparenleft}{\kern0pt}AALTs\ bs\ rs{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymexists}r\ {\isasymin}\ set\ rs{\isachardot}{\kern0pt}\ bnullable\ r{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-{\isacharbar}{\kern0pt}\ {\isachardoublequoteopen}bnullable\ {\isacharparenleft}{\kern0pt}ASEQ\ bs\ r{\isadigit{1}}\ r{\isadigit{2}}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}bnullable\ r{\isadigit{1}}\ {\isasymand}\ bnullable\ r{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-{\isacharbar}{\kern0pt}\ {\isachardoublequoteopen}bnullable\ {\isacharparenleft}{\kern0pt}ASTAR\ bs\ r{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ True{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{fun}\isamarkupfalse%
-\ \isanewline
-\ \ bmkeps\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}arexp\ {\isasymRightarrow}\ bit\ list{\isachardoublequoteclose}\isanewline
-\isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}bmkeps{\isacharparenleft}{\kern0pt}AONE\ bs{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ bs{\isachardoublequoteclose}\isanewline
-{\isacharbar}{\kern0pt}\ {\isachardoublequoteopen}bmkeps{\isacharparenleft}{\kern0pt}ASEQ\ bs\ r{\isadigit{1}}\ r{\isadigit{2}}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ bs\ {\isacharat}{\kern0pt}\ {\isacharparenleft}{\kern0pt}bmkeps\ r{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isacharat}{\kern0pt}\ {\isacharparenleft}{\kern0pt}bmkeps\ r{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-{\isacharbar}{\kern0pt}\ {\isachardoublequoteopen}bmkeps{\isacharparenleft}{\kern0pt}AALTs\ bs\ {\isacharbrackleft}{\kern0pt}r{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ bs\ {\isacharat}{\kern0pt}\ {\isacharparenleft}{\kern0pt}bmkeps\ r{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-{\isacharbar}{\kern0pt}\ {\isachardoublequoteopen}bmkeps{\isacharparenleft}{\kern0pt}AALTs\ bs\ {\isacharparenleft}{\kern0pt}r{\isacharhash}{\kern0pt}rs{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}if\ bnullable{\isacharparenleft}{\kern0pt}r{\isacharparenright}{\kern0pt}\ then\ bs\ {\isacharat}{\kern0pt}\ {\isacharparenleft}{\kern0pt}bmkeps\ r{\isacharparenright}{\kern0pt}\ else\ {\isacharparenleft}{\kern0pt}bmkeps\ {\isacharparenleft}{\kern0pt}AALTs\ bs\ rs{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-{\isacharbar}{\kern0pt}\ {\isachardoublequoteopen}bmkeps{\isacharparenleft}{\kern0pt}ASTAR\ bs\ r{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ bs\ {\isacharat}{\kern0pt}\ {\isacharbrackleft}{\kern0pt}S{\isacharbrackright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isanewline
-\isacommand{fun}\isamarkupfalse%
-\isanewline
-\ bder\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}char\ {\isasymRightarrow}\ arexp\ {\isasymRightarrow}\ arexp{\isachardoublequoteclose}\isanewline
-\isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}bder\ c\ {\isacharparenleft}{\kern0pt}AZERO{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ AZERO{\isachardoublequoteclose}\isanewline
-{\isacharbar}{\kern0pt}\ {\isachardoublequoteopen}bder\ c\ {\isacharparenleft}{\kern0pt}AONE\ bs{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ AZERO{\isachardoublequoteclose}\isanewline
-{\isacharbar}{\kern0pt}\ {\isachardoublequoteopen}bder\ c\ {\isacharparenleft}{\kern0pt}ACHAR\ bs\ d{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}if\ c\ {\isacharequal}{\kern0pt}\ d\ then\ AONE\ bs\ else\ AZERO{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-{\isacharbar}{\kern0pt}\ {\isachardoublequoteopen}bder\ c\ {\isacharparenleft}{\kern0pt}AALTs\ bs\ rs{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ AALTs\ bs\ {\isacharparenleft}{\kern0pt}map\ {\isacharparenleft}{\kern0pt}bder\ c{\isacharparenright}{\kern0pt}\ rs{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-{\isacharbar}{\kern0pt}\ {\isachardoublequoteopen}bder\ c\ {\isacharparenleft}{\kern0pt}ASEQ\ bs\ r{\isadigit{1}}\ r{\isadigit{2}}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ \isanewline
-\ \ \ \ \ {\isacharparenleft}{\kern0pt}if\ bnullable\ r{\isadigit{1}}\isanewline
-\ \ \ \ \ \ then\ AALT\ bs\ {\isacharparenleft}{\kern0pt}ASEQ\ {\isacharbrackleft}{\kern0pt}{\isacharbrackright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}bder\ c\ r{\isadigit{1}}{\isacharparenright}{\kern0pt}\ r{\isadigit{2}}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}fuse\ {\isacharparenleft}{\kern0pt}bmkeps\ r{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}bder\ c\ r{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ else\ ASEQ\ bs\ {\isacharparenleft}{\kern0pt}bder\ c\ r{\isadigit{1}}{\isacharparenright}{\kern0pt}\ r{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-{\isacharbar}{\kern0pt}\ {\isachardoublequoteopen}bder\ c\ {\isacharparenleft}{\kern0pt}ASTAR\ bs\ r{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ ASEQ\ bs\ {\isacharparenleft}{\kern0pt}fuse\ {\isacharbrackleft}{\kern0pt}Z{\isacharbrackright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}bder\ c\ r{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}ASTAR\ {\isacharbrackleft}{\kern0pt}{\isacharbrackright}{\kern0pt}\ r{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isanewline
-\isacommand{fun}\isamarkupfalse%
-\ \isanewline
-\ \ bders\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}arexp\ {\isasymRightarrow}\ string\ {\isasymRightarrow}\ arexp{\isachardoublequoteclose}\isanewline
-\isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}bders\ r\ {\isacharbrackleft}{\kern0pt}{\isacharbrackright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ r{\isachardoublequoteclose}\isanewline
-{\isacharbar}{\kern0pt}\ {\isachardoublequoteopen}bders\ r\ {\isacharparenleft}{\kern0pt}c{\isacharhash}{\kern0pt}s{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ bders\ {\isacharparenleft}{\kern0pt}bder\ c\ r{\isacharparenright}{\kern0pt}\ s{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ bders{\isacharunderscore}{\kern0pt}append{\isacharcolon}{\kern0pt}\isanewline
-\ \ {\isachardoublequoteopen}bders\ r\ {\isacharparenleft}{\kern0pt}s{\isadigit{1}}\ {\isacharat}{\kern0pt}\ s{\isadigit{2}}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ bders\ {\isacharparenleft}{\kern0pt}bders\ r\ s{\isadigit{1}}{\isacharparenright}{\kern0pt}\ s{\isadigit{2}}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}induct\ s{\isadigit{1}}\ arbitrary{\isacharcolon}{\kern0pt}\ r\ s{\isadigit{2}}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp{\isacharunderscore}{\kern0pt}all{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{done}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ bnullable{\isacharunderscore}{\kern0pt}correctness{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}nullable\ {\isacharparenleft}{\kern0pt}erase\ r{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ bnullable\ r{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}induct\ r\ rule{\isacharcolon}{\kern0pt}\ erase{\isachardot}{\kern0pt}induct{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp{\isacharunderscore}{\kern0pt}all{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{done}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ erase{\isacharunderscore}{\kern0pt}fuse{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}erase\ {\isacharparenleft}{\kern0pt}fuse\ bs\ r{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ erase\ r{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}induct\ r\ rule{\isacharcolon}{\kern0pt}\ erase{\isachardot}{\kern0pt}induct{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp{\isacharunderscore}{\kern0pt}all{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{done}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{thm}\isamarkupfalse%
-\ Posix{\isachardot}{\kern0pt}induct\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ erase{\isacharunderscore}{\kern0pt}intern\ {\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}erase\ {\isacharparenleft}{\kern0pt}intern\ r{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ r{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}induct\ r{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp{\isacharunderscore}{\kern0pt}all\ add{\isacharcolon}{\kern0pt}\ erase{\isacharunderscore}{\kern0pt}fuse{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{done}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ erase{\isacharunderscore}{\kern0pt}bder\ {\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}erase\ {\isacharparenleft}{\kern0pt}bder\ a\ r{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ der\ a\ {\isacharparenleft}{\kern0pt}erase\ r{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}induct\ r\ rule{\isacharcolon}{\kern0pt}\ erase{\isachardot}{\kern0pt}induct{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp{\isacharunderscore}{\kern0pt}all\ add{\isacharcolon}{\kern0pt}\ erase{\isacharunderscore}{\kern0pt}fuse\ bnullable{\isacharunderscore}{\kern0pt}correctness{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{done}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ erase{\isacharunderscore}{\kern0pt}bders\ {\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}erase\ {\isacharparenleft}{\kern0pt}bders\ r\ s{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ ders\ s\ {\isacharparenleft}{\kern0pt}erase\ r{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}induct\ s\ arbitrary{\isacharcolon}{\kern0pt}\ r\ {\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp{\isacharunderscore}{\kern0pt}all{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{done}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ retrieve{\isacharunderscore}{\kern0pt}encode{\isacharunderscore}{\kern0pt}STARS{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}{\isasymforall}v{\isasymin}set\ vs{\isachardot}{\kern0pt}\ {\isasymTurnstile}\ v\ {\isacharcolon}{\kern0pt}\ r\ {\isasymand}\ code\ v\ {\isacharequal}{\kern0pt}\ retrieve\ {\isacharparenleft}{\kern0pt}intern\ r{\isacharparenright}{\kern0pt}\ v{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}code\ {\isacharparenleft}{\kern0pt}Stars\ vs{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ retrieve\ {\isacharparenleft}{\kern0pt}ASTAR\ {\isacharbrackleft}{\kern0pt}{\isacharbrackright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}intern\ r{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}Stars\ vs{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ assms\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}induct\ vs{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp{\isacharunderscore}{\kern0pt}all{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{done}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ retrieve{\isacharunderscore}{\kern0pt}fuse{\isadigit{2}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}{\isasymTurnstile}\ v\ {\isacharcolon}{\kern0pt}\ {\isacharparenleft}{\kern0pt}erase\ r{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}retrieve\ {\isacharparenleft}{\kern0pt}fuse\ bs\ r{\isacharparenright}{\kern0pt}\ v\ {\isacharequal}{\kern0pt}\ bs\ {\isacharat}{\kern0pt}\ retrieve\ r\ v{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ assms\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}induct\ r\ arbitrary{\isacharcolon}{\kern0pt}\ v\ bs{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}auto\ elim{\isacharcolon}{\kern0pt}\ Prf{\isacharunderscore}{\kern0pt}elims{\isacharparenright}{\kern0pt}{\isacharbrackleft}{\kern0pt}{\isadigit{4}}{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \ \isacommand{defer}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{using}\isamarkupfalse%
-\ retrieve{\isacharunderscore}{\kern0pt}encode{\isacharunderscore}{\kern0pt}STARS\isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}auto\ elim{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ Prf{\isacharunderscore}{\kern0pt}elims{\isacharparenright}{\kern0pt}{\isacharbrackleft}{\kern0pt}{\isadigit{1}}{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}case{\isacharunderscore}{\kern0pt}tac\ vs{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}case{\isacharunderscore}{\kern0pt}tac\ x{\isadigit{2}}a{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}auto\ elim{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ Prf{\isacharunderscore}{\kern0pt}elims{\isacharparenright}{\kern0pt}{\isacharbrackleft}{\kern0pt}{\isadigit{1}}{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}case{\isacharunderscore}{\kern0pt}tac\ list{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}auto{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}auto\ elim{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ Prf{\isacharunderscore}{\kern0pt}elims{\isacharparenright}{\kern0pt}{\isacharbrackleft}{\kern0pt}{\isadigit{1}}{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \isacommand{done}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ retrieve{\isacharunderscore}{\kern0pt}fuse{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}{\isasymTurnstile}\ v\ {\isacharcolon}{\kern0pt}\ r{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}retrieve\ {\isacharparenleft}{\kern0pt}fuse\ bs\ {\isacharparenleft}{\kern0pt}intern\ r{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ v\ {\isacharequal}{\kern0pt}\ bs\ {\isacharat}{\kern0pt}\ retrieve\ {\isacharparenleft}{\kern0pt}intern\ r{\isacharparenright}{\kern0pt}\ v{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ assms\ \isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp{\isacharunderscore}{\kern0pt}all\ add{\isacharcolon}{\kern0pt}\ retrieve{\isacharunderscore}{\kern0pt}fuse{\isadigit{2}}{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ retrieve{\isacharunderscore}{\kern0pt}code{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}{\isasymTurnstile}\ v\ {\isacharcolon}{\kern0pt}\ r{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}code\ v\ {\isacharequal}{\kern0pt}\ retrieve\ {\isacharparenleft}{\kern0pt}intern\ r{\isacharparenright}{\kern0pt}\ v{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ assms\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}induct\ v\ r\ {\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp{\isacharunderscore}{\kern0pt}all\ add{\isacharcolon}{\kern0pt}\ retrieve{\isacharunderscore}{\kern0pt}fuse\ retrieve{\isacharunderscore}{\kern0pt}encode{\isacharunderscore}{\kern0pt}STARS{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{done}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ bnullable{\isacharunderscore}{\kern0pt}Hdbmkeps{\isacharunderscore}{\kern0pt}Hd{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}bnullable\ a{\isachardoublequoteclose}\ \isanewline
-\ \ \isakeyword{shows}\ \ {\isachardoublequoteopen}bmkeps\ {\isacharparenleft}{\kern0pt}AALTs\ bs\ {\isacharparenleft}{\kern0pt}a\ {\isacharhash}{\kern0pt}\ rs{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ bs\ {\isacharat}{\kern0pt}\ {\isacharparenleft}{\kern0pt}bmkeps\ a{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ assms\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ bmkeps{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{3}}{\isacharparenright}{\kern0pt}\ bmkeps{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{4}}{\isacharparenright}{\kern0pt}\ list{\isachardot}{\kern0pt}exhaust{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ r{\isadigit{1}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}{\isasymnot}\ bnullable\ a{\isachardoublequoteclose}\ {\isachardoublequoteopen}bnullable\ {\isacharparenleft}{\kern0pt}AALTs\ bs\ rs{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ \ {\isachardoublequoteopen}bmkeps\ {\isacharparenleft}{\kern0pt}AALTs\ bs\ {\isacharparenleft}{\kern0pt}a\ {\isacharhash}{\kern0pt}\ rs{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ bmkeps\ {\isacharparenleft}{\kern0pt}AALTs\ bs\ rs{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ assms\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}induct\ rs{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}auto{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{done}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ r{\isadigit{2}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}x\ {\isasymin}\ set\ rs{\isachardoublequoteclose}\ {\isachardoublequoteopen}bnullable\ x{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}bnullable\ {\isacharparenleft}{\kern0pt}AALTs\ bs\ rs{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ assms\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}induct\ rs{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}auto{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{done}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ \ r{\isadigit{3}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}{\isasymnot}\ bnullable\ r{\isachardoublequoteclose}\ \isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}\ {\isasymexists}\ x\ {\isasymin}\ set\ rs{\isachardot}{\kern0pt}\ bnullable\ x{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}retrieve\ {\isacharparenleft}{\kern0pt}AALTs\ bs\ rs{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}mkeps\ {\isacharparenleft}{\kern0pt}erase\ {\isacharparenleft}{\kern0pt}AALTs\ bs\ rs{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ retrieve\ {\isacharparenleft}{\kern0pt}AALTs\ bs\ {\isacharparenleft}{\kern0pt}r\ {\isacharhash}{\kern0pt}\ rs{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}mkeps\ {\isacharparenleft}{\kern0pt}erase\ {\isacharparenleft}{\kern0pt}AALTs\ bs\ {\isacharparenleft}{\kern0pt}r\ {\isacharhash}{\kern0pt}\ rs{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ assms\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}induct\ rs\ arbitrary{\isacharcolon}{\kern0pt}\ r\ bs{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}auto{\isacharparenright}{\kern0pt}{\isacharbrackleft}{\kern0pt}{\isadigit{1}}{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}auto{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{using}\isamarkupfalse%
-\ bnullable{\isacharunderscore}{\kern0pt}correctness\ \isacommand{apply}\isamarkupfalse%
-\ blast\isanewline
-\ \ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}auto\ simp\ add{\isacharcolon}{\kern0pt}\ bnullable{\isacharunderscore}{\kern0pt}correctness\ mkeps{\isacharunderscore}{\kern0pt}nullable\ retrieve{\isacharunderscore}{\kern0pt}fuse{\isadigit{2}}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}subst\ retrieve{\isacharunderscore}{\kern0pt}fuse{\isadigit{2}}{\isacharbrackleft}{\kern0pt}symmetric{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}smt\ bnullable{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{4}}{\isacharparenright}{\kern0pt}\ bnullable{\isacharunderscore}{\kern0pt}correctness\ erase{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{5}}{\isacharparenright}{\kern0pt}\ erase{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{6}}{\isacharparenright}{\kern0pt}\ insert{\isacharunderscore}{\kern0pt}iff\ list{\isachardot}{\kern0pt}exhaust\ list{\isachardot}{\kern0pt}set{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}\ mkeps{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{3}}{\isacharparenright}{\kern0pt}\ mkeps{\isacharunderscore}{\kern0pt}nullable{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}case{\isacharunderscore}{\kern0pt}tac\ {\isachardoublequoteopen}bnullable\ a{\isachardoublequoteclose}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}smt\ append{\isacharunderscore}{\kern0pt}Nil{\isadigit{2}}\ bnullable{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{4}}{\isacharparenright}{\kern0pt}\ bnullable{\isacharunderscore}{\kern0pt}correctness\ erase{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{5}}{\isacharparenright}{\kern0pt}\ erase{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{6}}{\isacharparenright}{\kern0pt}\ fuse{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{4}}{\isacharparenright}{\kern0pt}\ insert{\isacharunderscore}{\kern0pt}iff\ list{\isachardot}{\kern0pt}exhaust\ list{\isachardot}{\kern0pt}set{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}\ mkeps{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{3}}{\isacharparenright}{\kern0pt}\ mkeps{\isacharunderscore}{\kern0pt}nullable\ retrieve{\isacharunderscore}{\kern0pt}fuse{\isadigit{2}}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}drule{\isacharunderscore}{\kern0pt}tac\ x{\isacharequal}{\kern0pt}{\isachardoublequoteopen}a{\isachardoublequoteclose}\ \isakeyword{in}\ meta{\isacharunderscore}{\kern0pt}spec{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}drule{\isacharunderscore}{\kern0pt}tac\ x{\isacharequal}{\kern0pt}{\isachardoublequoteopen}bs{\isachardoublequoteclose}\ \isakeyword{in}\ meta{\isacharunderscore}{\kern0pt}spec{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}drule\ meta{\isacharunderscore}{\kern0pt}mp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}drule\ meta{\isacharunderscore}{\kern0pt}mp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}auto{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}subst\ retrieve{\isacharunderscore}{\kern0pt}fuse{\isadigit{2}}{\isacharbrackleft}{\kern0pt}symmetric{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}case{\isacharunderscore}{\kern0pt}tac\ rs{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}auto{\isacharparenright}{\kern0pt}{\isacharbrackleft}{\kern0pt}{\isadigit{1}}{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ bnullable{\isacharunderscore}{\kern0pt}correctness{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ append{\isacharunderscore}{\kern0pt}Nil{\isadigit{2}}\ bnullable{\isacharunderscore}{\kern0pt}correctness\ erase{\isacharunderscore}{\kern0pt}fuse\ fuse{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{4}}{\isacharparenright}{\kern0pt}\ list{\isachardot}{\kern0pt}set{\isacharunderscore}{\kern0pt}intros{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}\ mkeps{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{3}}{\isacharparenright}{\kern0pt}\ mkeps{\isacharunderscore}{\kern0pt}nullable\ nullable{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{4}}{\isacharparenright}{\kern0pt}\ r{\isadigit{2}}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{apply}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ bnullable{\isacharunderscore}{\kern0pt}correctness{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ append{\isacharunderscore}{\kern0pt}Nil{\isadigit{2}}\ bnullable{\isacharunderscore}{\kern0pt}correctness\ erase{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{6}}{\isacharparenright}{\kern0pt}\ erase{\isacharunderscore}{\kern0pt}fuse\ fuse{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{4}}{\isacharparenright}{\kern0pt}\ list{\isachardot}{\kern0pt}set{\isacharunderscore}{\kern0pt}intros{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}\ mkeps{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{3}}{\isacharparenright}{\kern0pt}\ mkeps{\isacharunderscore}{\kern0pt}nullable\ r{\isadigit{2}}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{done}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ t{\isacharcolon}{\kern0pt}\ \isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}{\isasymforall}r\ {\isasymin}\ set\ rs{\isachardot}{\kern0pt}\ nullable\ {\isacharparenleft}{\kern0pt}erase\ r{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\ bmkeps\ r\ {\isacharequal}{\kern0pt}\ retrieve\ r\ {\isacharparenleft}{\kern0pt}mkeps\ {\isacharparenleft}{\kern0pt}erase\ r{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}nullable\ {\isacharparenleft}{\kern0pt}erase\ {\isacharparenleft}{\kern0pt}AALTs\ bs\ rs{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}\ bmkeps\ {\isacharparenleft}{\kern0pt}AALTs\ bs\ rs{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ retrieve\ {\isacharparenleft}{\kern0pt}AALTs\ bs\ rs{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}mkeps\ {\isacharparenleft}{\kern0pt}erase\ {\isacharparenleft}{\kern0pt}AALTs\ bs\ rs{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ assms\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}induct\ rs\ arbitrary{\isacharcolon}{\kern0pt}\ bs{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}auto\ simp\ add{\isacharcolon}{\kern0pt}\ bnullable{\isacharunderscore}{\kern0pt}correctness{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}case{\isacharunderscore}{\kern0pt}tac\ rs{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}auto\ simp\ add{\isacharcolon}{\kern0pt}\ bnullable{\isacharunderscore}{\kern0pt}correctness{\isacharparenright}{\kern0pt}{\isacharbrackleft}{\kern0pt}{\isadigit{2}}{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}subst\ r{\isadigit{1}}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}rule\ r{\isadigit{2}}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}assumption{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}drule{\isacharunderscore}{\kern0pt}tac\ x{\isacharequal}{\kern0pt}{\isachardoublequoteopen}bs{\isachardoublequoteclose}\ \isakeyword{in}\ meta{\isacharunderscore}{\kern0pt}spec{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}drule\ meta{\isacharunderscore}{\kern0pt}mp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}auto{\isacharparenright}{\kern0pt}{\isacharbrackleft}{\kern0pt}{\isadigit{1}}{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \ \isacommand{prefer}\isamarkupfalse%
-\ {\isadigit{2}}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}case{\isacharunderscore}{\kern0pt}tac\ {\isachardoublequoteopen}bnullable\ a{\isachardoublequoteclose}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}subst\ bnullable{\isacharunderscore}{\kern0pt}Hdbmkeps{\isacharunderscore}{\kern0pt}Hd{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-\ blast\isanewline
-\ \ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}subgoal{\isacharunderscore}{\kern0pt}tac\ {\isachardoublequoteopen}nullable\ {\isacharparenleft}{\kern0pt}erase\ a{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{prefer}\isamarkupfalse%
-\ {\isadigit{2}}\isanewline
-\ \ \isacommand{using}\isamarkupfalse%
-\ bnullable{\isacharunderscore}{\kern0pt}correctness\ \isacommand{apply}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharcomma}{\kern0pt}\ lifting{\isacharparenright}{\kern0pt}\ erase{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{5}}{\isacharparenright}{\kern0pt}\ erase{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{6}}{\isacharparenright}{\kern0pt}\ list{\isachardot}{\kern0pt}exhaust\ mkeps{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{3}}{\isacharparenright}{\kern0pt}\ retrieve{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{3}}{\isacharparenright}{\kern0pt}\ retrieve{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{4}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}subst\ r{\isadigit{1}}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{using}\isamarkupfalse%
-\ r{\isadigit{2}}\ \isacommand{apply}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}drule{\isacharunderscore}{\kern0pt}tac\ x{\isacharequal}{\kern0pt}{\isachardoublequoteopen}bs{\isachardoublequoteclose}\ \isakeyword{in}\ meta{\isacharunderscore}{\kern0pt}spec{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}drule\ meta{\isacharunderscore}{\kern0pt}mp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}auto{\isacharparenright}{\kern0pt}{\isacharbrackleft}{\kern0pt}{\isadigit{1}}{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{using}\isamarkupfalse%
-\ r{\isadigit{3}}\ \isacommand{apply}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}auto{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{using}\isamarkupfalse%
-\ r{\isadigit{3}}\ \isacommand{by}\isamarkupfalse%
-\ blast%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ bmkeps{\isacharunderscore}{\kern0pt}retrieve{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}nullable\ {\isacharparenleft}{\kern0pt}erase\ r{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}bmkeps\ r\ {\isacharequal}{\kern0pt}\ retrieve\ r\ {\isacharparenleft}{\kern0pt}mkeps\ {\isacharparenleft}{\kern0pt}erase\ r{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ assms\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}induct\ r{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \isacommand{defer}\isamarkupfalse%
-\isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}rule\ t{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}auto{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{done}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ bder{\isacharunderscore}{\kern0pt}retrieve{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}{\isasymTurnstile}\ v\ {\isacharcolon}{\kern0pt}\ der\ c\ {\isacharparenleft}{\kern0pt}erase\ r{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}retrieve\ {\isacharparenleft}{\kern0pt}bder\ c\ r{\isacharparenright}{\kern0pt}\ v\ {\isacharequal}{\kern0pt}\ retrieve\ r\ {\isacharparenleft}{\kern0pt}injval\ {\isacharparenleft}{\kern0pt}erase\ r{\isacharparenright}{\kern0pt}\ c\ v{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ assms\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}induct\ r\ arbitrary{\isacharcolon}{\kern0pt}\ v\ rule{\isacharcolon}{\kern0pt}\ erase{\isachardot}{\kern0pt}induct{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}erule\ Prf{\isacharunderscore}{\kern0pt}elims{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}erule\ Prf{\isacharunderscore}{\kern0pt}elims{\isacharparenright}{\kern0pt}\ \isanewline
-\ \ \ \ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}case{\isacharunderscore}{\kern0pt}tac\ {\isachardoublequoteopen}c\ {\isacharequal}{\kern0pt}\ ca{\isachardoublequoteclose}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}erule\ Prf{\isacharunderscore}{\kern0pt}elims{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}erule\ Prf{\isacharunderscore}{\kern0pt}elims{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}erule\ Prf{\isacharunderscore}{\kern0pt}elims{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}rename{\isacharunderscore}{\kern0pt}tac\ {\isachardoublequoteopen}r\isactrlsub {\isadigit{1}}{\isachardoublequoteclose}\ {\isachardoublequoteopen}r\isactrlsub {\isadigit{2}}{\isachardoublequoteclose}\ rs\ v{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}erule\ Prf{\isacharunderscore}{\kern0pt}elims{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}case{\isacharunderscore}{\kern0pt}tac\ rs{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}smt\ Prf{\isacharunderscore}{\kern0pt}elims{\isacharparenleft}{\kern0pt}{\isadigit{3}}{\isacharparenright}{\kern0pt}\ injval{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}\ injval{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{3}}{\isacharparenright}{\kern0pt}\ retrieve{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{4}}{\isacharparenright}{\kern0pt}\ retrieve{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{5}}{\isacharparenright}{\kern0pt}\ same{\isacharunderscore}{\kern0pt}append{\isacharunderscore}{\kern0pt}eq{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}case{\isacharunderscore}{\kern0pt}tac\ {\isachardoublequoteopen}nullable\ {\isacharparenleft}{\kern0pt}erase\ r{\isadigit{1}}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}erule\ Prf{\isacharunderscore}{\kern0pt}elims{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}subgoal{\isacharunderscore}{\kern0pt}tac\ {\isachardoublequoteopen}bnullable\ r{\isadigit{1}}{\isachardoublequoteclose}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{prefer}\isamarkupfalse%
-\ {\isadigit{2}}\isanewline
-\ \ \isacommand{using}\isamarkupfalse%
-\ bnullable{\isacharunderscore}{\kern0pt}correctness\ \isacommand{apply}\isamarkupfalse%
-\ blast\isanewline
-\ \ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}erule\ Prf{\isacharunderscore}{\kern0pt}elims{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}subgoal{\isacharunderscore}{\kern0pt}tac\ {\isachardoublequoteopen}bnullable\ r{\isadigit{1}}{\isachardoublequoteclose}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{prefer}\isamarkupfalse%
-\ {\isadigit{2}}\isanewline
-\ \ \isacommand{using}\isamarkupfalse%
-\ bnullable{\isacharunderscore}{\kern0pt}correctness\ \isacommand{apply}\isamarkupfalse%
-\ blast\isanewline
-\ \ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ retrieve{\isacharunderscore}{\kern0pt}fuse{\isadigit{2}}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ bmkeps{\isacharunderscore}{\kern0pt}retrieve{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}erule\ Prf{\isacharunderscore}{\kern0pt}elims{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{using}\isamarkupfalse%
-\ bnullable{\isacharunderscore}{\kern0pt}correctness\ \isacommand{apply}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}rename{\isacharunderscore}{\kern0pt}tac\ bs\ r\ v{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}erule\ Prf{\isacharunderscore}{\kern0pt}elims{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}clarify{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}erule\ Prf{\isacharunderscore}{\kern0pt}elims{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}clarify{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}subst\ injval{\isachardot}{\kern0pt}simps{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp\ del{\isacharcolon}{\kern0pt}\ retrieve{\isachardot}{\kern0pt}simps{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}subst\ retrieve{\isachardot}{\kern0pt}simps{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}subst\ retrieve{\isachardot}{\kern0pt}simps{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ retrieve{\isacharunderscore}{\kern0pt}fuse{\isadigit{2}}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{done}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\ \ \isanewline
-\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ MAIN{\isacharunderscore}{\kern0pt}decode{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}{\isasymTurnstile}\ v\ {\isacharcolon}{\kern0pt}\ ders\ s\ r{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}Some\ {\isacharparenleft}{\kern0pt}flex\ r\ id\ s\ v{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ decode\ {\isacharparenleft}{\kern0pt}retrieve\ {\isacharparenleft}{\kern0pt}bders\ {\isacharparenleft}{\kern0pt}intern\ r{\isacharparenright}{\kern0pt}\ s{\isacharparenright}{\kern0pt}\ v{\isacharparenright}{\kern0pt}\ r{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ assms\isanewline
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}induct\ s\ arbitrary{\isacharcolon}{\kern0pt}\ v\ rule{\isacharcolon}{\kern0pt}\ rev{\isacharunderscore}{\kern0pt}induct{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{case}\isamarkupfalse%
-\ Nil\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymTurnstile}\ v\ {\isacharcolon}{\kern0pt}\ ders\ {\isacharbrackleft}{\kern0pt}{\isacharbrackright}{\kern0pt}\ r{\isachardoublequoteclose}\ \isacommand{by}\isamarkupfalse%
-\ fact\isanewline
-\ \ \isacommand{then}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymTurnstile}\ v\ {\isacharcolon}{\kern0pt}\ r{\isachardoublequoteclose}\ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{then}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}Some\ v\ {\isacharequal}{\kern0pt}\ decode\ {\isacharparenleft}{\kern0pt}retrieve\ {\isacharparenleft}{\kern0pt}intern\ r{\isacharparenright}{\kern0pt}\ v{\isacharparenright}{\kern0pt}\ r{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ decode{\isacharunderscore}{\kern0pt}code\ retrieve{\isacharunderscore}{\kern0pt}code\ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{then}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}Some\ {\isacharparenleft}{\kern0pt}flex\ r\ id\ {\isacharbrackleft}{\kern0pt}{\isacharbrackright}{\kern0pt}\ v{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ decode\ {\isacharparenleft}{\kern0pt}retrieve\ {\isacharparenleft}{\kern0pt}bders\ {\isacharparenleft}{\kern0pt}intern\ r{\isacharparenright}{\kern0pt}\ {\isacharbrackleft}{\kern0pt}{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\ v{\isacharparenright}{\kern0pt}\ r{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{case}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}snoc\ c\ s\ v{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ IH{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymAnd}v{\isachardot}{\kern0pt}\ {\isasymTurnstile}\ v\ {\isacharcolon}{\kern0pt}\ ders\ s\ r\ {\isasymLongrightarrow}\ \isanewline
-\ \ \ \ \ Some\ {\isacharparenleft}{\kern0pt}flex\ r\ id\ s\ v{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ decode\ {\isacharparenleft}{\kern0pt}retrieve\ {\isacharparenleft}{\kern0pt}bders\ {\isacharparenleft}{\kern0pt}intern\ r{\isacharparenright}{\kern0pt}\ s{\isacharparenright}{\kern0pt}\ v{\isacharparenright}{\kern0pt}\ r{\isachardoublequoteclose}\ \isacommand{by}\isamarkupfalse%
-\ fact\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ asm{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymTurnstile}\ v\ {\isacharcolon}{\kern0pt}\ ders\ {\isacharparenleft}{\kern0pt}s\ {\isacharat}{\kern0pt}\ {\isacharbrackleft}{\kern0pt}c{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\ r{\isachardoublequoteclose}\ \isacommand{by}\isamarkupfalse%
-\ fact\isanewline
-\ \ \isacommand{then}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ asm{\isadigit{2}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymTurnstile}\ injval\ {\isacharparenleft}{\kern0pt}ders\ s\ r{\isacharparenright}{\kern0pt}\ c\ v\ {\isacharcolon}{\kern0pt}\ ders\ s\ r{\isachardoublequoteclose}\ \isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ Prf{\isacharunderscore}{\kern0pt}injval\ ders{\isacharunderscore}{\kern0pt}append{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}Some\ {\isacharparenleft}{\kern0pt}flex\ r\ id\ {\isacharparenleft}{\kern0pt}s\ {\isacharat}{\kern0pt}\ {\isacharbrackleft}{\kern0pt}c{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\ v{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ Some\ {\isacharparenleft}{\kern0pt}flex\ r\ id\ s\ {\isacharparenleft}{\kern0pt}injval\ {\isacharparenleft}{\kern0pt}ders\ s\ r{\isacharparenright}{\kern0pt}\ c\ v{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ flex{\isacharunderscore}{\kern0pt}append{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}\ {\isacharequal}{\kern0pt}\ decode\ {\isacharparenleft}{\kern0pt}retrieve\ {\isacharparenleft}{\kern0pt}bders\ {\isacharparenleft}{\kern0pt}intern\ r{\isacharparenright}{\kern0pt}\ s{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}injval\ {\isacharparenleft}{\kern0pt}ders\ s\ r{\isacharparenright}{\kern0pt}\ c\ v{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ r{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ asm{\isadigit{2}}\ IH\ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}\ {\isacharequal}{\kern0pt}\ decode\ {\isacharparenleft}{\kern0pt}retrieve\ {\isacharparenleft}{\kern0pt}bder\ c\ {\isacharparenleft}{\kern0pt}bders\ {\isacharparenleft}{\kern0pt}intern\ r{\isacharparenright}{\kern0pt}\ s{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ v{\isacharparenright}{\kern0pt}\ r{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ asm\ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp{\isacharunderscore}{\kern0pt}all\ add{\isacharcolon}{\kern0pt}\ bder{\isacharunderscore}{\kern0pt}retrieve\ ders{\isacharunderscore}{\kern0pt}append{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{finally}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}Some\ {\isacharparenleft}{\kern0pt}flex\ r\ id\ {\isacharparenleft}{\kern0pt}s\ {\isacharat}{\kern0pt}\ {\isacharbrackleft}{\kern0pt}c{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\ v{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ \isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ decode\ {\isacharparenleft}{\kern0pt}retrieve\ {\isacharparenleft}{\kern0pt}bders\ {\isacharparenleft}{\kern0pt}intern\ r{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}s\ {\isacharat}{\kern0pt}\ {\isacharbrackleft}{\kern0pt}c{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ v{\isacharparenright}{\kern0pt}\ r{\isachardoublequoteclose}\ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ bders{\isacharunderscore}{\kern0pt}append{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ blex\ \isakeyword{where}\isanewline
-\ {\isachardoublequoteopen}blex\ a\ s\ {\isasymequiv}\ if\ bnullable\ {\isacharparenleft}{\kern0pt}bders\ a\ s{\isacharparenright}{\kern0pt}\ then\ Some\ {\isacharparenleft}{\kern0pt}bmkeps\ {\isacharparenleft}{\kern0pt}bders\ a\ s{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ else\ None{\isachardoublequoteclose}\isanewline
-\isanewline
-\isanewline
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ blexer\ \isakeyword{where}\isanewline
-\ {\isachardoublequoteopen}blexer\ r\ s\ {\isasymequiv}\ if\ bnullable\ {\isacharparenleft}{\kern0pt}bders\ {\isacharparenleft}{\kern0pt}intern\ r{\isacharparenright}{\kern0pt}\ s{\isacharparenright}{\kern0pt}\ then\ \isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ decode\ {\isacharparenleft}{\kern0pt}bmkeps\ {\isacharparenleft}{\kern0pt}bders\ {\isacharparenleft}{\kern0pt}intern\ r{\isacharparenright}{\kern0pt}\ s{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ r\ else\ None{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ blexer{\isacharunderscore}{\kern0pt}correctness{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}blexer\ r\ s\ {\isacharequal}{\kern0pt}\ lexer\ r\ s{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{{\isacharbraceleft}{\kern0pt}}\isamarkupfalse%
-\ \isacommand{define}\isamarkupfalse%
-\ bds\ \isakeyword{where}\ {\isachardoublequoteopen}bds\ {\isasymequiv}\ bders\ {\isacharparenleft}{\kern0pt}intern\ r{\isacharparenright}{\kern0pt}\ s{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{define}\isamarkupfalse%
-\ ds\ \ \isakeyword{where}\ {\isachardoublequoteopen}ds\ {\isasymequiv}\ ders\ s\ r{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{assume}\isamarkupfalse%
-\ asm{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}nullable\ ds{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ era{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}erase\ bds\ {\isacharequal}{\kern0pt}\ ds{\isachardoublequoteclose}\ \isanewline
-\ \ \ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ ds{\isacharunderscore}{\kern0pt}def\ bds{\isacharunderscore}{\kern0pt}def\ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ mke{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymTurnstile}\ mkeps\ ds\ {\isacharcolon}{\kern0pt}\ ds{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ asm\ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ mkeps{\isacharunderscore}{\kern0pt}nullable{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}decode\ {\isacharparenleft}{\kern0pt}bmkeps\ bds{\isacharparenright}{\kern0pt}\ r\ {\isacharequal}{\kern0pt}\ decode\ {\isacharparenleft}{\kern0pt}retrieve\ bds\ {\isacharparenleft}{\kern0pt}mkeps\ ds{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ r{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ bmkeps{\isacharunderscore}{\kern0pt}retrieve\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ asm\ era\ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ bmkeps{\isacharunderscore}{\kern0pt}retrieve{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}\ {\isacharequal}{\kern0pt}\ \ Some\ {\isacharparenleft}{\kern0pt}flex\ r\ id\ s\ {\isacharparenleft}{\kern0pt}mkeps\ ds{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ mke\ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp{\isacharunderscore}{\kern0pt}all\ add{\isacharcolon}{\kern0pt}\ MAIN{\isacharunderscore}{\kern0pt}decode\ ds{\isacharunderscore}{\kern0pt}def\ bds{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{finally}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}decode\ {\isacharparenleft}{\kern0pt}bmkeps\ bds{\isacharparenright}{\kern0pt}\ r\ {\isacharequal}{\kern0pt}\ Some\ {\isacharparenleft}{\kern0pt}flex\ r\ id\ s\ {\isacharparenleft}{\kern0pt}mkeps\ ds{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isanewline
-\ \ \ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ bds{\isacharunderscore}{\kern0pt}def\ ds{\isacharunderscore}{\kern0pt}def\ \isacommand{{\isachardot}{\kern0pt}}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{{\isacharbraceright}{\kern0pt}}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{then}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}blexer\ r\ s\ {\isacharequal}{\kern0pt}\ lexer\ r\ s{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ blexer{\isacharunderscore}{\kern0pt}def\ lexer{\isacharunderscore}{\kern0pt}flex\isanewline
-\ \ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}subst\ bnullable{\isacharunderscore}{\kern0pt}correctness{\isacharbrackleft}{\kern0pt}symmetric{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{done}\isamarkupfalse%
-\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{fun}\isamarkupfalse%
-\ distinctBy\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ list\ {\isasymRightarrow}\ {\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}a\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}b{\isacharparenright}{\kern0pt}\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}b\ set\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ list{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}distinctBy\ {\isacharbrackleft}{\kern0pt}{\isacharbrackright}{\kern0pt}\ f\ acc\ {\isacharequal}{\kern0pt}\ {\isacharbrackleft}{\kern0pt}{\isacharbrackright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-{\isacharbar}{\kern0pt}\ {\isachardoublequoteopen}distinctBy\ {\isacharparenleft}{\kern0pt}x{\isacharhash}{\kern0pt}xs{\isacharparenright}{\kern0pt}\ f\ acc\ {\isacharequal}{\kern0pt}\ \isanewline
-\ \ \ \ \ {\isacharparenleft}{\kern0pt}if\ {\isacharparenleft}{\kern0pt}f\ x{\isacharparenright}{\kern0pt}\ {\isasymin}\ acc\ then\ distinctBy\ xs\ f\ acc\ \isanewline
-\ \ \ \ \ \ else\ x\ {\isacharhash}{\kern0pt}\ {\isacharparenleft}{\kern0pt}distinctBy\ xs\ f\ {\isacharparenleft}{\kern0pt}{\isacharbraceleft}{\kern0pt}f\ x{\isacharbraceright}{\kern0pt}\ {\isasymunion}\ acc{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isanewline
-\isanewline
-\isanewline
-\isacommand{fun}\isamarkupfalse%
-\ flts\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}arexp\ list\ {\isasymRightarrow}\ arexp\ list{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{where}\ \isanewline
-\ \ {\isachardoublequoteopen}flts\ {\isacharbrackleft}{\kern0pt}{\isacharbrackright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharbrackleft}{\kern0pt}{\isacharbrackright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-{\isacharbar}{\kern0pt}\ {\isachardoublequoteopen}flts\ {\isacharparenleft}{\kern0pt}AZERO\ {\isacharhash}{\kern0pt}\ rs{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ flts\ rs{\isachardoublequoteclose}\isanewline
-{\isacharbar}{\kern0pt}\ {\isachardoublequoteopen}flts\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}AALTs\ bs\ \ rs{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isacharhash}{\kern0pt}\ rs{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}map\ {\isacharparenleft}{\kern0pt}fuse\ bs{\isacharparenright}{\kern0pt}\ rs{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isacharat}{\kern0pt}\ flts\ rs{\isachardoublequoteclose}\isanewline
-{\isacharbar}{\kern0pt}\ {\isachardoublequoteopen}flts\ {\isacharparenleft}{\kern0pt}r{\isadigit{1}}\ {\isacharhash}{\kern0pt}\ rs{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ r{\isadigit{1}}\ {\isacharhash}{\kern0pt}\ flts\ rs{\isachardoublequoteclose}\isanewline
-\isanewline
-\isanewline
-\isanewline
-\isanewline
-\isacommand{fun}\isamarkupfalse%
-\ li\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}bit\ list\ {\isasymRightarrow}\ arexp\ list\ {\isasymRightarrow}\ arexp{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}li\ {\isacharunderscore}{\kern0pt}\ {\isacharbrackleft}{\kern0pt}{\isacharbrackright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ AZERO{\isachardoublequoteclose}\isanewline
-{\isacharbar}{\kern0pt}\ {\isachardoublequoteopen}li\ bs\ {\isacharbrackleft}{\kern0pt}a{\isacharbrackright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ fuse\ bs\ a{\isachardoublequoteclose}\isanewline
-{\isacharbar}{\kern0pt}\ {\isachardoublequoteopen}li\ bs\ as\ {\isacharequal}{\kern0pt}\ AALTs\ bs\ as{\isachardoublequoteclose}\isanewline
-\isanewline
-\isanewline
-\isanewline
-\isanewline
-\isacommand{fun}\isamarkupfalse%
-\ bsimp{\isacharunderscore}{\kern0pt}ASEQ\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}bit\ list\ {\isasymRightarrow}\ arexp\ {\isasymRightarrow}\ arexp\ {\isasymRightarrow}\ arexp{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}bsimp{\isacharunderscore}{\kern0pt}ASEQ\ {\isacharunderscore}{\kern0pt}\ AZERO\ {\isacharunderscore}{\kern0pt}\ {\isacharequal}{\kern0pt}\ AZERO{\isachardoublequoteclose}\isanewline
-{\isacharbar}{\kern0pt}\ {\isachardoublequoteopen}bsimp{\isacharunderscore}{\kern0pt}ASEQ\ {\isacharunderscore}{\kern0pt}\ {\isacharunderscore}{\kern0pt}\ AZERO\ {\isacharequal}{\kern0pt}\ AZERO{\isachardoublequoteclose}\isanewline
-{\isacharbar}{\kern0pt}\ {\isachardoublequoteopen}bsimp{\isacharunderscore}{\kern0pt}ASEQ\ bs{\isadigit{1}}\ {\isacharparenleft}{\kern0pt}AONE\ bs{\isadigit{2}}{\isacharparenright}{\kern0pt}\ r{\isadigit{2}}\ {\isacharequal}{\kern0pt}\ fuse\ {\isacharparenleft}{\kern0pt}bs{\isadigit{1}}\ {\isacharat}{\kern0pt}\ bs{\isadigit{2}}{\isacharparenright}{\kern0pt}\ r{\isadigit{2}}{\isachardoublequoteclose}\isanewline
-{\isacharbar}{\kern0pt}\ {\isachardoublequoteopen}bsimp{\isacharunderscore}{\kern0pt}ASEQ\ bs{\isadigit{1}}\ r{\isadigit{1}}\ r{\isadigit{2}}\ {\isacharequal}{\kern0pt}\ ASEQ\ \ bs{\isadigit{1}}\ r{\isadigit{1}}\ r{\isadigit{2}}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isanewline
-\isacommand{fun}\isamarkupfalse%
-\ bsimp{\isacharunderscore}{\kern0pt}AALTs\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}bit\ list\ {\isasymRightarrow}\ arexp\ list\ {\isasymRightarrow}\ arexp{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}bsimp{\isacharunderscore}{\kern0pt}AALTs\ {\isacharunderscore}{\kern0pt}\ {\isacharbrackleft}{\kern0pt}{\isacharbrackright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ AZERO{\isachardoublequoteclose}\isanewline
-{\isacharbar}{\kern0pt}\ {\isachardoublequoteopen}bsimp{\isacharunderscore}{\kern0pt}AALTs\ bs{\isadigit{1}}\ {\isacharbrackleft}{\kern0pt}r{\isacharbrackright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ fuse\ bs{\isadigit{1}}\ r{\isachardoublequoteclose}\isanewline
-{\isacharbar}{\kern0pt}\ {\isachardoublequoteopen}bsimp{\isacharunderscore}{\kern0pt}AALTs\ bs{\isadigit{1}}\ rs\ {\isacharequal}{\kern0pt}\ AALTs\ bs{\isadigit{1}}\ rs{\isachardoublequoteclose}\isanewline
-\isanewline
-\isanewline
-\isacommand{fun}\isamarkupfalse%
-\ bsimp\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}arexp\ {\isasymRightarrow}\ arexp{\isachardoublequoteclose}\ \isanewline
-\ \ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}bsimp\ {\isacharparenleft}{\kern0pt}ASEQ\ bs{\isadigit{1}}\ r{\isadigit{1}}\ r{\isadigit{2}}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ bsimp{\isacharunderscore}{\kern0pt}ASEQ\ bs{\isadigit{1}}\ {\isacharparenleft}{\kern0pt}bsimp\ r{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}bsimp\ r{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-{\isacharbar}{\kern0pt}\ {\isachardoublequoteopen}bsimp\ {\isacharparenleft}{\kern0pt}AALTs\ bs{\isadigit{1}}\ rs{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ bsimp{\isacharunderscore}{\kern0pt}AALTs\ bs{\isadigit{1}}\ {\isacharparenleft}{\kern0pt}distinctBy\ \ {\isacharparenleft}{\kern0pt}flts\ {\isacharparenleft}{\kern0pt}map\ bsimp\ rs{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ erase\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharparenright}{\kern0pt}\ {\isachardoublequoteclose}\isanewline
-{\isacharbar}{\kern0pt}\ {\isachardoublequoteopen}bsimp\ r\ {\isacharequal}{\kern0pt}\ r{\isachardoublequoteclose}\isanewline
-\isanewline
-\isanewline
-\isanewline
-\isanewline
-\isacommand{fun}\isamarkupfalse%
-\ \isanewline
-\ \ bders{\isacharunderscore}{\kern0pt}simp\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}arexp\ {\isasymRightarrow}\ string\ {\isasymRightarrow}\ arexp{\isachardoublequoteclose}\isanewline
-\isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}bders{\isacharunderscore}{\kern0pt}simp\ r\ {\isacharbrackleft}{\kern0pt}{\isacharbrackright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ r{\isachardoublequoteclose}\isanewline
-{\isacharbar}{\kern0pt}\ {\isachardoublequoteopen}bders{\isacharunderscore}{\kern0pt}simp\ r\ {\isacharparenleft}{\kern0pt}c\ {\isacharhash}{\kern0pt}\ s{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ bders{\isacharunderscore}{\kern0pt}simp\ {\isacharparenleft}{\kern0pt}bsimp\ {\isacharparenleft}{\kern0pt}bder\ c\ r{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ s{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ blexer{\isacharunderscore}{\kern0pt}simp\ \isakeyword{where}\isanewline
-\ {\isachardoublequoteopen}blexer{\isacharunderscore}{\kern0pt}simp\ r\ s\ {\isasymequiv}\ if\ bnullable\ {\isacharparenleft}{\kern0pt}bders{\isacharunderscore}{\kern0pt}simp\ {\isacharparenleft}{\kern0pt}intern\ r{\isacharparenright}{\kern0pt}\ s{\isacharparenright}{\kern0pt}\ then\ \isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ decode\ {\isacharparenleft}{\kern0pt}bmkeps\ {\isacharparenleft}{\kern0pt}bders{\isacharunderscore}{\kern0pt}simp\ {\isacharparenleft}{\kern0pt}intern\ r{\isacharparenright}{\kern0pt}\ s{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ r\ else\ None{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{export{\isacharunderscore}{\kern0pt}code}\isamarkupfalse%
-\ bders{\isacharunderscore}{\kern0pt}simp\ \isakeyword{in}\ Scala\ \isakeyword{module{\isacharunderscore}{\kern0pt}name}\ Example\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ bders{\isacharunderscore}{\kern0pt}simp{\isacharunderscore}{\kern0pt}append{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}bders{\isacharunderscore}{\kern0pt}simp\ r\ {\isacharparenleft}{\kern0pt}s{\isadigit{1}}\ {\isacharat}{\kern0pt}\ s{\isadigit{2}}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ bders{\isacharunderscore}{\kern0pt}simp\ {\isacharparenleft}{\kern0pt}bders{\isacharunderscore}{\kern0pt}simp\ r\ s{\isadigit{1}}{\isacharparenright}{\kern0pt}\ s{\isadigit{2}}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}induct\ s{\isadigit{1}}\ arbitrary{\isacharcolon}{\kern0pt}\ r\ s{\isadigit{2}}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{done}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isanewline
-\isanewline
-\isanewline
-\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ L{\isacharunderscore}{\kern0pt}bsimp{\isacharunderscore}{\kern0pt}ASEQ{\isacharcolon}{\kern0pt}\isanewline
-\ \ {\isachardoublequoteopen}L\ {\isacharparenleft}{\kern0pt}SEQ\ {\isacharparenleft}{\kern0pt}erase\ r{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}erase\ r{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ L\ {\isacharparenleft}{\kern0pt}erase\ {\isacharparenleft}{\kern0pt}bsimp{\isacharunderscore}{\kern0pt}ASEQ\ bs\ r{\isadigit{1}}\ r{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}induct\ bs\ r{\isadigit{1}}\ r{\isadigit{2}}\ rule{\isacharcolon}{\kern0pt}\ bsimp{\isacharunderscore}{\kern0pt}ASEQ{\isachardot}{\kern0pt}induct{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp{\isacharunderscore}{\kern0pt}all{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ erase{\isacharunderscore}{\kern0pt}fuse\ fuse{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{4}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ L{\isacharunderscore}{\kern0pt}bsimp{\isacharunderscore}{\kern0pt}AALTs{\isacharcolon}{\kern0pt}\isanewline
-\ \ {\isachardoublequoteopen}L\ {\isacharparenleft}{\kern0pt}erase\ {\isacharparenleft}{\kern0pt}AALTs\ bs\ rs{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ L\ {\isacharparenleft}{\kern0pt}erase\ {\isacharparenleft}{\kern0pt}bsimp{\isacharunderscore}{\kern0pt}AALTs\ bs\ rs{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}induct\ bs\ rs\ rule{\isacharcolon}{\kern0pt}\ bsimp{\isacharunderscore}{\kern0pt}AALTs{\isachardot}{\kern0pt}induct{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp{\isacharunderscore}{\kern0pt}all\ add{\isacharcolon}{\kern0pt}\ erase{\isacharunderscore}{\kern0pt}fuse{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{done}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ L{\isacharunderscore}{\kern0pt}erase{\isacharunderscore}{\kern0pt}AALTs{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}L\ {\isacharparenleft}{\kern0pt}erase\ {\isacharparenleft}{\kern0pt}AALTs\ bs\ rs{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isasymUnion}\ {\isacharparenleft}{\kern0pt}L\ {\isacharbackquote}{\kern0pt}\ erase\ {\isacharbackquote}{\kern0pt}\ {\isacharparenleft}{\kern0pt}set\ rs{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}induct\ rs{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}case{\isacharunderscore}{\kern0pt}tac\ rs{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{done}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ L{\isacharunderscore}{\kern0pt}erase{\isacharunderscore}{\kern0pt}flts{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}{\isasymUnion}\ {\isacharparenleft}{\kern0pt}L\ {\isacharbackquote}{\kern0pt}\ erase\ {\isacharbackquote}{\kern0pt}\ {\isacharparenleft}{\kern0pt}set\ {\isacharparenleft}{\kern0pt}flts\ rs{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isasymUnion}\ {\isacharparenleft}{\kern0pt}L\ {\isacharbackquote}{\kern0pt}\ erase\ {\isacharbackquote}{\kern0pt}\ {\isacharparenleft}{\kern0pt}set\ rs{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}induct\ rs\ rule{\isacharcolon}{\kern0pt}\ flts{\isachardot}{\kern0pt}induct{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp{\isacharunderscore}{\kern0pt}all{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}auto{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{using}\isamarkupfalse%
-\ L{\isacharunderscore}{\kern0pt}erase{\isacharunderscore}{\kern0pt}AALTs\ erase{\isacharunderscore}{\kern0pt}fuse\ \isacommand{apply}\isamarkupfalse%
-\ auto{\isacharbrackleft}{\kern0pt}{\isadigit{1}}{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ L{\isacharunderscore}{\kern0pt}erase{\isacharunderscore}{\kern0pt}AALTs\ erase{\isacharunderscore}{\kern0pt}fuse{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ L{\isacharunderscore}{\kern0pt}erase{\isacharunderscore}{\kern0pt}dB{\isacharunderscore}{\kern0pt}acc{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}\ {\isasymUnion}{\isacharparenleft}{\kern0pt}L\ {\isacharbackquote}{\kern0pt}\ acc{\isacharparenright}{\kern0pt}\ {\isasymunion}\ {\isacharparenleft}{\kern0pt}\ {\isasymUnion}\ {\isacharparenleft}{\kern0pt}L\ {\isacharbackquote}{\kern0pt}\ erase\ {\isacharbackquote}{\kern0pt}\ {\isacharparenleft}{\kern0pt}set\ {\isacharparenleft}{\kern0pt}distinctBy\ rs\ erase\ acc{\isacharparenright}{\kern0pt}\ {\isacharparenright}{\kern0pt}\ {\isacharparenright}{\kern0pt}\ {\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isasymUnion}{\isacharparenleft}{\kern0pt}L\ {\isacharbackquote}{\kern0pt}\ acc{\isacharparenright}{\kern0pt}\ {\isasymunion}\ \ {\isasymUnion}\ {\isacharparenleft}{\kern0pt}L\ {\isacharbackquote}{\kern0pt}\ erase\ {\isacharbackquote}{\kern0pt}\ {\isacharparenleft}{\kern0pt}set\ rs{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}induction\ rs\ arbitrary{\isacharcolon}{\kern0pt}\ acc{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}smt\ {\isacharparenleft}{\kern0pt}z{\isadigit{3}}{\isacharparenright}{\kern0pt}\ SUP{\isacharunderscore}{\kern0pt}absorb\ UN{\isacharunderscore}{\kern0pt}insert\ sup{\isacharunderscore}{\kern0pt}assoc\ sup{\isacharunderscore}{\kern0pt}commute{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ L{\isacharunderscore}{\kern0pt}erase{\isacharunderscore}{\kern0pt}dB{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}\ {\isacharparenleft}{\kern0pt}\ {\isasymUnion}\ {\isacharparenleft}{\kern0pt}L\ {\isacharbackquote}{\kern0pt}\ erase\ {\isacharbackquote}{\kern0pt}\ {\isacharparenleft}{\kern0pt}set\ {\isacharparenleft}{\kern0pt}distinctBy\ rs\ erase\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenright}{\kern0pt}\ {\isacharparenright}{\kern0pt}\ {\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isasymUnion}\ {\isacharparenleft}{\kern0pt}L\ {\isacharbackquote}{\kern0pt}\ erase\ {\isacharbackquote}{\kern0pt}\ {\isacharparenleft}{\kern0pt}set\ rs{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ L{\isacharunderscore}{\kern0pt}erase{\isacharunderscore}{\kern0pt}dB{\isacharunderscore}{\kern0pt}acc\ Un{\isacharunderscore}{\kern0pt}commute\ Union{\isacharunderscore}{\kern0pt}image{\isacharunderscore}{\kern0pt}empty{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ L{\isacharunderscore}{\kern0pt}bsimp{\isacharunderscore}{\kern0pt}erase{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}L\ {\isacharparenleft}{\kern0pt}erase\ r{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ L\ {\isacharparenleft}{\kern0pt}erase\ {\isacharparenleft}{\kern0pt}bsimp\ r{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}induct\ r{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}auto\ simp\ add{\isacharcolon}{\kern0pt}\ Sequ{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}{\isacharbrackleft}{\kern0pt}{\isadigit{1}}{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}subst\ L{\isacharunderscore}{\kern0pt}bsimp{\isacharunderscore}{\kern0pt}ASEQ{\isacharbrackleft}{\kern0pt}symmetric{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}auto\ simp\ add{\isacharcolon}{\kern0pt}\ Sequ{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}{\isacharbrackleft}{\kern0pt}{\isadigit{1}}{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}subst\ {\isacharparenleft}{\kern0pt}asm{\isacharparenright}{\kern0pt}\ \ L{\isacharunderscore}{\kern0pt}bsimp{\isacharunderscore}{\kern0pt}ASEQ{\isacharbrackleft}{\kern0pt}symmetric{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}auto\ simp\ add{\isacharcolon}{\kern0pt}\ Sequ{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}{\isacharbrackleft}{\kern0pt}{\isadigit{1}}{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}subst\ L{\isacharunderscore}{\kern0pt}bsimp{\isacharunderscore}{\kern0pt}AALTs{\isacharbrackleft}{\kern0pt}symmetric{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \isacommand{defer}\isamarkupfalse%
-\isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}subst\ {\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}L{\isacharunderscore}{\kern0pt}erase{\isacharunderscore}{\kern0pt}AALTs{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}subst\ L{\isacharunderscore}{\kern0pt}erase{\isacharunderscore}{\kern0pt}dB{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}subst\ L{\isacharunderscore}{\kern0pt}erase{\isacharunderscore}{\kern0pt}flts{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}auto{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ L{\isacharunderscore}{\kern0pt}erase{\isacharunderscore}{\kern0pt}AALTs{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{using}\isamarkupfalse%
-\ L{\isacharunderscore}{\kern0pt}erase{\isacharunderscore}{\kern0pt}AALTs\ \isacommand{by}\isamarkupfalse%
-\ blast%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ bsimp{\isacharunderscore}{\kern0pt}ASEQ{\isadigit{0}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}bsimp{\isacharunderscore}{\kern0pt}ASEQ\ bs\ r{\isadigit{1}}\ AZERO\ {\isacharequal}{\kern0pt}\ AZERO{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}induct\ r{\isadigit{1}}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}auto{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{done}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ bsimp{\isacharunderscore}{\kern0pt}ASEQ{\isadigit{1}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}r{\isadigit{1}}\ {\isasymnoteq}\ AZERO{\isachardoublequoteclose}\ {\isachardoublequoteopen}r{\isadigit{2}}\ {\isasymnoteq}\ AZERO{\isachardoublequoteclose}\ {\isachardoublequoteopen}{\isasymforall}bs{\isachardot}{\kern0pt}\ r{\isadigit{1}}\ {\isasymnoteq}\ AONE\ bs{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}bsimp{\isacharunderscore}{\kern0pt}ASEQ\ bs\ r{\isadigit{1}}\ r{\isadigit{2}}\ {\isacharequal}{\kern0pt}\ ASEQ\ bs\ r{\isadigit{1}}\ r{\isadigit{2}}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ assms\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}induct\ bs\ r{\isadigit{1}}\ r{\isadigit{2}}\ rule{\isacharcolon}{\kern0pt}\ bsimp{\isacharunderscore}{\kern0pt}ASEQ{\isachardot}{\kern0pt}induct{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}auto{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{done}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ bsimp{\isacharunderscore}{\kern0pt}ASEQ{\isadigit{2}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}bsimp{\isacharunderscore}{\kern0pt}ASEQ\ bs\ {\isacharparenleft}{\kern0pt}AONE\ bs{\isadigit{1}}{\isacharparenright}{\kern0pt}\ r{\isadigit{2}}\ {\isacharequal}{\kern0pt}\ fuse\ {\isacharparenleft}{\kern0pt}bs\ {\isacharat}{\kern0pt}\ bs{\isadigit{1}}{\isacharparenright}{\kern0pt}\ r{\isadigit{2}}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}induct\ r{\isadigit{2}}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}auto{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{done}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ L{\isacharunderscore}{\kern0pt}bders{\isacharunderscore}{\kern0pt}simp{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}L\ {\isacharparenleft}{\kern0pt}erase\ {\isacharparenleft}{\kern0pt}bders{\isacharunderscore}{\kern0pt}simp\ r\ s{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ L\ {\isacharparenleft}{\kern0pt}erase\ {\isacharparenleft}{\kern0pt}bders\ r\ s{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}induct\ s\ arbitrary{\isacharcolon}{\kern0pt}\ r\ rule{\isacharcolon}{\kern0pt}\ rev{\isacharunderscore}{\kern0pt}induct{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ ders{\isacharunderscore}{\kern0pt}append{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ bders{\isacharunderscore}{\kern0pt}simp{\isacharunderscore}{\kern0pt}append{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ L{\isacharunderscore}{\kern0pt}bsimp{\isacharunderscore}{\kern0pt}erase{\isacharbrackleft}{\kern0pt}symmetric{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ der{\isacharunderscore}{\kern0pt}correctness{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ b{\isadigit{2}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}bnullable\ r{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}bmkeps\ {\isacharparenleft}{\kern0pt}fuse\ bs\ r{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ bs\ {\isacharat}{\kern0pt}\ bmkeps\ r{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ assms\ bmkeps{\isacharunderscore}{\kern0pt}retrieve\ bnullable{\isacharunderscore}{\kern0pt}correctness\ erase{\isacharunderscore}{\kern0pt}fuse\ mkeps{\isacharunderscore}{\kern0pt}nullable\ retrieve{\isacharunderscore}{\kern0pt}fuse{\isadigit{2}}{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ b{\isadigit{4}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}bnullable\ {\isacharparenleft}{\kern0pt}bders{\isacharunderscore}{\kern0pt}simp\ r\ s{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ bnullable\ {\isacharparenleft}{\kern0pt}bders\ r\ s{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ L{\isacharunderscore}{\kern0pt}bders{\isacharunderscore}{\kern0pt}simp\ bnullable{\isacharunderscore}{\kern0pt}correctness\ lexer{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}\ lexer{\isacharunderscore}{\kern0pt}correct{\isacharunderscore}{\kern0pt}None\ option{\isachardot}{\kern0pt}distinct{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ qq{\isadigit{1}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}{\isasymexists}r\ {\isasymin}\ set\ rs{\isachardot}{\kern0pt}\ bnullable\ r{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}bmkeps\ {\isacharparenleft}{\kern0pt}AALTs\ bs\ {\isacharparenleft}{\kern0pt}rs\ {\isacharat}{\kern0pt}\ rs{\isadigit{1}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ bmkeps\ {\isacharparenleft}{\kern0pt}AALTs\ bs\ rs{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ assms\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}induct\ rs\ arbitrary{\isacharcolon}{\kern0pt}\ rs{\isadigit{1}}\ bs{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ Nil{\isacharunderscore}{\kern0pt}is{\isacharunderscore}{\kern0pt}append{\isacharunderscore}{\kern0pt}conv\ bmkeps{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{4}}{\isacharparenright}{\kern0pt}\ neq{\isacharunderscore}{\kern0pt}Nil{\isacharunderscore}{\kern0pt}conv\ bnullable{\isacharunderscore}{\kern0pt}Hdbmkeps{\isacharunderscore}{\kern0pt}Hd\ split{\isacharunderscore}{\kern0pt}list{\isacharunderscore}{\kern0pt}last{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ qq{\isadigit{2}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}{\isasymforall}r\ {\isasymin}\ set\ rs{\isachardot}{\kern0pt}\ {\isasymnot}\ bnullable\ r{\isachardoublequoteclose}\ {\isachardoublequoteopen}{\isasymexists}r\ {\isasymin}\ set\ rs{\isadigit{1}}{\isachardot}{\kern0pt}\ bnullable\ r{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}bmkeps\ {\isacharparenleft}{\kern0pt}AALTs\ bs\ {\isacharparenleft}{\kern0pt}rs\ {\isacharat}{\kern0pt}\ rs{\isadigit{1}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ bmkeps\ {\isacharparenleft}{\kern0pt}AALTs\ bs\ rs{\isadigit{1}}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ assms\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}induct\ rs\ arbitrary{\isacharcolon}{\kern0pt}\ rs{\isadigit{1}}\ bs{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ append{\isacharunderscore}{\kern0pt}assoc\ in{\isacharunderscore}{\kern0pt}set{\isacharunderscore}{\kern0pt}conv{\isacharunderscore}{\kern0pt}decomp\ r{\isadigit{1}}\ r{\isadigit{2}}{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\ \ \isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ qq{\isadigit{3}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}bnullable\ {\isacharparenleft}{\kern0pt}AALTs\ bs\ rs{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymexists}r\ {\isasymin}\ set\ rs{\isachardot}{\kern0pt}\ bnullable\ r{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}induct\ rs\ arbitrary{\isacharcolon}{\kern0pt}\ bs{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{done}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isanewline
-\isanewline
-\isanewline
-\isacommand{fun}\isamarkupfalse%
-\ nonnested\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}arexp\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}nonnested\ {\isacharparenleft}{\kern0pt}AALTs\ bs{\isadigit{2}}\ {\isacharbrackleft}{\kern0pt}{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ True{\isachardoublequoteclose}\isanewline
-{\isacharbar}{\kern0pt}\ {\isachardoublequoteopen}nonnested\ {\isacharparenleft}{\kern0pt}AALTs\ bs{\isadigit{2}}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}AALTs\ bs{\isadigit{1}}\ rs{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isacharhash}{\kern0pt}\ rs{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ False{\isachardoublequoteclose}\isanewline
-{\isacharbar}{\kern0pt}\ {\isachardoublequoteopen}nonnested\ {\isacharparenleft}{\kern0pt}AALTs\ bs{\isadigit{2}}\ {\isacharparenleft}{\kern0pt}r\ {\isacharhash}{\kern0pt}\ rs{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ nonnested\ {\isacharparenleft}{\kern0pt}AALTs\ bs{\isadigit{2}}\ rs{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-{\isacharbar}{\kern0pt}\ {\isachardoublequoteopen}nonnested\ r\ {\isacharequal}{\kern0pt}\ True{\isachardoublequoteclose}\isanewline
-\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ \ k{\isadigit{0}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}flts\ {\isacharparenleft}{\kern0pt}r\ {\isacharhash}{\kern0pt}\ rs{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ flts\ {\isacharbrackleft}{\kern0pt}r{\isacharbrackright}{\kern0pt}\ {\isacharat}{\kern0pt}\ flts\ rs{\isadigit{1}}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}induct\ r\ arbitrary{\isacharcolon}{\kern0pt}\ rs{\isadigit{1}}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}auto{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{done}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ \ k{\isadigit{0}}{\isadigit{0}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}flts\ {\isacharparenleft}{\kern0pt}rs{\isadigit{1}}\ {\isacharat}{\kern0pt}\ rs{\isadigit{2}}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ flts\ rs{\isadigit{1}}\ {\isacharat}{\kern0pt}\ flts\ rs{\isadigit{2}}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}induct\ rs{\isadigit{1}}\ arbitrary{\isacharcolon}{\kern0pt}\ rs{\isadigit{2}}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}auto{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ append{\isachardot}{\kern0pt}assoc\ k{\isadigit{0}}{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ \ k{\isadigit{0}}a{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}flts\ {\isacharbrackleft}{\kern0pt}AALTs\ bs\ rs{\isacharbrackright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ map\ {\isacharparenleft}{\kern0pt}fuse\ bs{\isacharparenright}{\kern0pt}\ \ rs{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{done}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isanewline
-\isanewline
-\isanewline
-\isanewline
-\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ bsimp{\isacharunderscore}{\kern0pt}AALTs{\isacharunderscore}{\kern0pt}qq{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}{\isadigit{1}}\ {\isacharless}{\kern0pt}\ length\ rs{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}bsimp{\isacharunderscore}{\kern0pt}AALTs\ bs\ rs\ {\isacharequal}{\kern0pt}\ AALTs\ bs\ \ rs{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ \ assms\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}case{\isacharunderscore}{\kern0pt}tac\ rs{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}case{\isacharunderscore}{\kern0pt}tac\ list{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp{\isacharunderscore}{\kern0pt}all{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{done}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ bbbbs{\isadigit{1}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}nonalt\ r\ {\isasymor}\ {\isacharparenleft}{\kern0pt}{\isasymexists}bs\ rs{\isachardot}{\kern0pt}\ r\ \ {\isacharequal}{\kern0pt}\ AALTs\ bs\ rs{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ nonalt{\isachardot}{\kern0pt}elims{\isacharparenleft}{\kern0pt}{\isadigit{3}}{\isacharparenright}{\kern0pt}\ \isacommand{by}\isamarkupfalse%
-\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\ \ \isanewline
-\isanewline
-\isanewline
-\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ flts{\isacharunderscore}{\kern0pt}append{\isacharcolon}{\kern0pt}\isanewline
-\ \ {\isachardoublequoteopen}flts\ {\isacharparenleft}{\kern0pt}xs{\isadigit{1}}\ {\isacharat}{\kern0pt}\ xs{\isadigit{2}}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ flts\ xs{\isadigit{1}}\ {\isacharat}{\kern0pt}\ flts\ xs{\isadigit{2}}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}induct\ xs{\isadigit{1}}\ \ arbitrary{\isacharcolon}{\kern0pt}\ xs{\isadigit{2}}\ \ rule{\isacharcolon}{\kern0pt}\ rev{\isacharunderscore}{\kern0pt}induct{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}auto{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}case{\isacharunderscore}{\kern0pt}tac\ xs{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}auto{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}case{\isacharunderscore}{\kern0pt}tac\ x{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}auto{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}case{\isacharunderscore}{\kern0pt}tac\ x{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}auto{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{done}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{fun}\isamarkupfalse%
-\ nonazero\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}arexp\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}nonazero\ AZERO\ {\isacharequal}{\kern0pt}\ False{\isachardoublequoteclose}\isanewline
-{\isacharbar}{\kern0pt}\ {\isachardoublequoteopen}nonazero\ r\ {\isacharequal}{\kern0pt}\ True{\isachardoublequoteclose}\isanewline
-\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ flts{\isacharunderscore}{\kern0pt}single{\isadigit{1}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}nonalt\ r{\isachardoublequoteclose}\ {\isachardoublequoteopen}nonazero\ r{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}flts\ {\isacharbrackleft}{\kern0pt}r{\isacharbrackright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharbrackleft}{\kern0pt}r{\isacharbrackright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ assms\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}induct\ r{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}auto{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{done}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ q{\isadigit{3}}a{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}{\isasymexists}r\ {\isasymin}\ set\ rs{\isachardot}{\kern0pt}\ bnullable\ r{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}bmkeps\ {\isacharparenleft}{\kern0pt}AALTs\ bs\ {\isacharparenleft}{\kern0pt}map\ {\isacharparenleft}{\kern0pt}fuse\ bs{\isadigit{1}}{\isacharparenright}{\kern0pt}\ rs{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ bmkeps\ {\isacharparenleft}{\kern0pt}AALTs\ {\isacharparenleft}{\kern0pt}bs{\isacharat}{\kern0pt}bs{\isadigit{1}}{\isacharparenright}{\kern0pt}\ rs{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ assms\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}induct\ rs\ arbitrary{\isacharcolon}{\kern0pt}\ bs\ bs{\isadigit{1}}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}auto{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ append{\isacharunderscore}{\kern0pt}assoc\ b{\isadigit{2}}\ bnullable{\isacharunderscore}{\kern0pt}correctness\ erase{\isacharunderscore}{\kern0pt}fuse\ bnullable{\isacharunderscore}{\kern0pt}Hdbmkeps{\isacharunderscore}{\kern0pt}Hd{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}case{\isacharunderscore}{\kern0pt}tac\ {\isachardoublequoteopen}bnullable\ a{\isachardoublequoteclose}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ append{\isachardot}{\kern0pt}assoc\ b{\isadigit{2}}\ bnullable{\isacharunderscore}{\kern0pt}correctness\ erase{\isacharunderscore}{\kern0pt}fuse\ bnullable{\isacharunderscore}{\kern0pt}Hdbmkeps{\isacharunderscore}{\kern0pt}Hd{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}case{\isacharunderscore}{\kern0pt}tac\ rs{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}auto{\isacharparenright}{\kern0pt}{\isacharbrackleft}{\kern0pt}{\isadigit{1}}{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ bnullable{\isacharunderscore}{\kern0pt}correctness\ erase{\isacharunderscore}{\kern0pt}fuse{\isacharparenright}{\kern0pt}{\isacharplus}{\kern0pt}\isanewline
-\ \ \isacommand{done}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ qq{\isadigit{4}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}{\isasymexists}x{\isasymin}set\ list{\isachardot}{\kern0pt}\ bnullable\ x{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}{\isasymexists}x{\isasymin}set\ {\isacharparenleft}{\kern0pt}flts\ list{\isacharparenright}{\kern0pt}{\isachardot}{\kern0pt}\ bnullable\ x{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ assms\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}induct\ list\ rule{\isacharcolon}{\kern0pt}\ flts{\isachardot}{\kern0pt}induct{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}auto{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ UnCI\ bnullable{\isacharunderscore}{\kern0pt}correctness\ erase{\isacharunderscore}{\kern0pt}fuse\ imageI{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\ \ \isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ qs{\isadigit{3}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}{\isasymexists}r\ {\isasymin}\ set\ rs{\isachardot}{\kern0pt}\ bnullable\ r{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}bmkeps\ {\isacharparenleft}{\kern0pt}AALTs\ bs\ rs{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ bmkeps\ {\isacharparenleft}{\kern0pt}AALTs\ bs\ {\isacharparenleft}{\kern0pt}flts\ rs{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ assms\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}induct\ rs\ arbitrary{\isacharcolon}{\kern0pt}\ bs\ taking{\isacharcolon}{\kern0pt}\ size\ rule{\isacharcolon}{\kern0pt}\ measure{\isacharunderscore}{\kern0pt}induct{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}case{\isacharunderscore}{\kern0pt}tac\ x{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}case{\isacharunderscore}{\kern0pt}tac\ a{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ r{\isadigit{1}}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ bnullable{\isacharunderscore}{\kern0pt}Hdbmkeps{\isacharunderscore}{\kern0pt}Hd{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}case{\isacharunderscore}{\kern0pt}tac\ {\isachardoublequoteopen}flts\ list{\isachardoublequoteclose}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ L{\isacharunderscore}{\kern0pt}erase{\isacharunderscore}{\kern0pt}AALTs\ L{\isacharunderscore}{\kern0pt}erase{\isacharunderscore}{\kern0pt}flts\ L{\isacharunderscore}{\kern0pt}flat{\isacharunderscore}{\kern0pt}Prf{\isadigit{1}}\ L{\isacharunderscore}{\kern0pt}flat{\isacharunderscore}{\kern0pt}Prf{\isadigit{2}}\ Prf{\isacharunderscore}{\kern0pt}elims{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}\ bnullable{\isacharunderscore}{\kern0pt}correctness\ erase{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{4}}{\isacharparenright}{\kern0pt}\ mkeps{\isacharunderscore}{\kern0pt}nullable\ r{\isadigit{2}}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ r{\isadigit{1}}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{prefer}\isamarkupfalse%
-\ {\isadigit{3}}\isanewline
-\ \ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{apply}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ bnullable{\isacharunderscore}{\kern0pt}Hdbmkeps{\isacharunderscore}{\kern0pt}Hd{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \isacommand{prefer}\isamarkupfalse%
-\ {\isadigit{2}}\isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}case{\isacharunderscore}{\kern0pt}tac\ {\isachardoublequoteopen}{\isasymexists}x{\isasymin}set\ x{\isadigit{5}}{\isadigit{2}}{\isachardot}{\kern0pt}\ bnullable\ x{\isachardoublequoteclose}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}case{\isacharunderscore}{\kern0pt}tac\ {\isachardoublequoteopen}list{\isachardoublequoteclose}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{apply}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ b{\isadigit{2}}\ fuse{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{4}}{\isacharparenright}{\kern0pt}\ q{\isadigit{3}}a\ r{\isadigit{2}}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}erule\ disjE{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}subst\ qq{\isadigit{1}}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}auto{\isacharparenright}{\kern0pt}{\isacharbrackleft}{\kern0pt}{\isadigit{1}}{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ bnullable{\isacharunderscore}{\kern0pt}correctness\ erase{\isacharunderscore}{\kern0pt}fuse{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ b{\isadigit{2}}\ fuse{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{4}}{\isacharparenright}{\kern0pt}\ q{\isadigit{3}}a\ r{\isadigit{2}}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}auto{\isacharparenright}{\kern0pt}{\isacharbrackleft}{\kern0pt}{\isadigit{1}}{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}subst\ qq{\isadigit{1}}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ bnullable{\isacharunderscore}{\kern0pt}correctness\ erase{\isacharunderscore}{\kern0pt}fuse\ image{\isacharunderscore}{\kern0pt}eqI\ set{\isacharunderscore}{\kern0pt}map{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ b{\isadigit{2}}\ fuse{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{4}}{\isacharparenright}{\kern0pt}\ q{\isadigit{3}}a\ r{\isadigit{2}}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}subst\ qq{\isadigit{1}}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ bnullable{\isacharunderscore}{\kern0pt}correctness\ erase{\isacharunderscore}{\kern0pt}fuse\ image{\isacharunderscore}{\kern0pt}eqI\ set{\isacharunderscore}{\kern0pt}map{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{apply}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ b{\isadigit{2}}\ fuse{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{4}}{\isacharparenright}{\kern0pt}\ q{\isadigit{3}}a\ r{\isadigit{2}}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}subst\ qq{\isadigit{2}}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ bnullable{\isacharunderscore}{\kern0pt}correctness\ erase{\isacharunderscore}{\kern0pt}fuse\ imageE\ set{\isacharunderscore}{\kern0pt}map{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{prefer}\isamarkupfalse%
-\ {\isadigit{2}}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}case{\isacharunderscore}{\kern0pt}tac\ {\isachardoublequoteopen}list{\isachardoublequoteclose}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ qq{\isadigit{4}}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}auto{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}case{\isacharunderscore}{\kern0pt}tac\ list{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ bnullable{\isacharunderscore}{\kern0pt}Hdbmkeps{\isacharunderscore}{\kern0pt}Hd{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}case{\isacharunderscore}{\kern0pt}tac\ {\isachardoublequoteopen}bnullable\ {\isacharparenleft}{\kern0pt}ASEQ\ x{\isadigit{4}}{\isadigit{1}}\ x{\isadigit{4}}{\isadigit{2}}\ x{\isadigit{4}}{\isadigit{3}}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}case{\isacharunderscore}{\kern0pt}tac\ list{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ bnullable{\isacharunderscore}{\kern0pt}Hdbmkeps{\isacharunderscore}{\kern0pt}Hd{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{using}\isamarkupfalse%
-\ qq{\isadigit{4}}\ r{\isadigit{1}}\ r{\isadigit{2}}\ \isacommand{by}\isamarkupfalse%
-\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isanewline
-\ \ \isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ bder{\isacharunderscore}{\kern0pt}fuse{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}bder\ c\ {\isacharparenleft}{\kern0pt}fuse\ bs\ a{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ fuse\ bs\ \ {\isacharparenleft}{\kern0pt}bder\ c\ a{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}induct\ a\ arbitrary{\isacharcolon}{\kern0pt}\ bs\ c{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp{\isacharunderscore}{\kern0pt}all{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{done}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{fun}\isamarkupfalse%
-\ flts{\isadigit{2}}\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}char\ {\isasymRightarrow}\ arexp\ list\ {\isasymRightarrow}\ arexp\ list{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{where}\ \isanewline
-\ \ {\isachardoublequoteopen}flts{\isadigit{2}}\ {\isacharunderscore}{\kern0pt}\ {\isacharbrackleft}{\kern0pt}{\isacharbrackright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharbrackleft}{\kern0pt}{\isacharbrackright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-{\isacharbar}{\kern0pt}\ {\isachardoublequoteopen}flts{\isadigit{2}}\ c\ {\isacharparenleft}{\kern0pt}AZERO\ {\isacharhash}{\kern0pt}\ rs{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ flts{\isadigit{2}}\ c\ rs{\isachardoublequoteclose}\isanewline
-{\isacharbar}{\kern0pt}\ {\isachardoublequoteopen}flts{\isadigit{2}}\ c\ {\isacharparenleft}{\kern0pt}AONE\ {\isacharunderscore}{\kern0pt}\ {\isacharhash}{\kern0pt}\ rs{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ flts{\isadigit{2}}\ c\ rs{\isachardoublequoteclose}\isanewline
-{\isacharbar}{\kern0pt}\ {\isachardoublequoteopen}flts{\isadigit{2}}\ c\ {\isacharparenleft}{\kern0pt}ACHAR\ bs\ d\ {\isacharhash}{\kern0pt}\ rs{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}if\ c\ {\isacharequal}{\kern0pt}\ d\ then\ {\isacharparenleft}{\kern0pt}ACHAR\ bs\ d\ {\isacharhash}{\kern0pt}\ flts{\isadigit{2}}\ c\ rs{\isacharparenright}{\kern0pt}\ else\ flts{\isadigit{2}}\ c\ rs{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-{\isacharbar}{\kern0pt}\ {\isachardoublequoteopen}flts{\isadigit{2}}\ c\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}AALTs\ bs\ rs{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isacharhash}{\kern0pt}\ rs{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}map\ {\isacharparenleft}{\kern0pt}fuse\ bs{\isacharparenright}{\kern0pt}\ rs{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isacharat}{\kern0pt}\ flts{\isadigit{2}}\ c\ rs{\isachardoublequoteclose}\isanewline
-{\isacharbar}{\kern0pt}\ {\isachardoublequoteopen}flts{\isadigit{2}}\ c\ {\isacharparenleft}{\kern0pt}ASEQ\ bs\ r{\isadigit{1}}\ r{\isadigit{2}}\ {\isacharhash}{\kern0pt}\ rs{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}if\ {\isacharparenleft}{\kern0pt}bnullable{\isacharparenleft}{\kern0pt}r{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isasymand}\ r{\isadigit{2}}\ {\isacharequal}{\kern0pt}\ AZERO{\isacharparenright}{\kern0pt}\ then\ \isanewline
-\ \ \ \ flts{\isadigit{2}}\ c\ rs\isanewline
-\ \ \ \ else\ ASEQ\ bs\ r{\isadigit{1}}\ r{\isadigit{2}}\ {\isacharhash}{\kern0pt}\ flts{\isadigit{2}}\ c\ rs{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-{\isacharbar}{\kern0pt}\ {\isachardoublequoteopen}flts{\isadigit{2}}\ c\ {\isacharparenleft}{\kern0pt}r{\isadigit{1}}\ {\isacharhash}{\kern0pt}\ rs{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ r{\isadigit{1}}\ {\isacharhash}{\kern0pt}\ flts{\isadigit{2}}\ c\ rs{\isachardoublequoteclose}\isanewline
-\isanewline
-\isanewline
-\isanewline
-\isanewline
-\isanewline
-\isanewline
-\isanewline
-\isanewline
-\ \isanewline
-\isanewline
-\isanewline
-\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ WQ{\isadigit{1}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}s\ {\isasymin}\ L\ {\isacharparenleft}{\kern0pt}der\ c\ r{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}s\ {\isasymin}\ der\ c\ r\ {\isasymrightarrow}\ mkeps\ {\isacharparenleft}{\kern0pt}ders\ s\ {\isacharparenleft}{\kern0pt}der\ c\ r{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ assms\isanewline
-\ \ \isacommand{oops}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ bder{\isacharunderscore}{\kern0pt}bsimp{\isacharunderscore}{\kern0pt}AALTs{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}bder\ c\ {\isacharparenleft}{\kern0pt}bsimp{\isacharunderscore}{\kern0pt}AALTs\ bs\ rs{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ bsimp{\isacharunderscore}{\kern0pt}AALTs\ bs\ {\isacharparenleft}{\kern0pt}map\ {\isacharparenleft}{\kern0pt}bder\ c{\isacharparenright}{\kern0pt}\ rs{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}induct\ bs\ rs\ rule{\isacharcolon}{\kern0pt}\ bsimp{\isacharunderscore}{\kern0pt}AALTs{\isachardot}{\kern0pt}induct{\isacharparenright}{\kern0pt}\ \ \isanewline
-\ \ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ bder{\isacharunderscore}{\kern0pt}fuse{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{done}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}asize\ {\isacharparenleft}{\kern0pt}bsimp\ a{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ asize\ a{\isachardoublequoteclose}\ \ {\isachardoublequoteopen}a\ {\isacharequal}{\kern0pt}\ AALTs\ bs\ {\isacharbrackleft}{\kern0pt}AALTs\ bs{\isadigit{2}}\ {\isacharbrackleft}{\kern0pt}{\isacharbrackright}{\kern0pt}{\isacharcomma}{\kern0pt}\ AZERO{\isacharcomma}{\kern0pt}\ AONE\ bs{\isadigit{3}}{\isacharbrackright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}bsimp\ a\ {\isacharequal}{\kern0pt}\ a{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ assms\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{oops}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isanewline
-\isanewline
-\isanewline
-\isanewline
-\isanewline
-\isanewline
-\isacommand{inductive}\isamarkupfalse%
-\ rrewrite{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}arexp\ {\isasymRightarrow}\ arexp\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\ {\isacharparenleft}{\kern0pt}{\isachardoublequoteopen}{\isacharunderscore}{\kern0pt}\ {\isasymleadsto}\ {\isacharunderscore}{\kern0pt}{\isachardoublequoteclose}\ {\isacharbrackleft}{\kern0pt}{\isadigit{9}}{\isadigit{9}}{\isacharcomma}{\kern0pt}\ {\isadigit{9}}{\isadigit{9}}{\isacharbrackright}{\kern0pt}\ {\isadigit{9}}{\isadigit{9}}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}ASEQ\ bs\ AZERO\ r{\isadigit{2}}\ {\isasymleadsto}\ AZERO{\isachardoublequoteclose}\isanewline
-{\isacharbar}{\kern0pt}\ {\isachardoublequoteopen}ASEQ\ bs\ r{\isadigit{1}}\ AZERO\ {\isasymleadsto}\ AZERO{\isachardoublequoteclose}\isanewline
-{\isacharbar}{\kern0pt}\ {\isachardoublequoteopen}ASEQ\ bs\ {\isacharparenleft}{\kern0pt}AONE\ bs{\isadigit{1}}{\isacharparenright}{\kern0pt}\ r\ {\isasymleadsto}\ fuse\ {\isacharparenleft}{\kern0pt}bs{\isacharat}{\kern0pt}bs{\isadigit{1}}{\isacharparenright}{\kern0pt}\ r{\isachardoublequoteclose}\isanewline
-{\isacharbar}{\kern0pt}\ {\isachardoublequoteopen}r{\isadigit{1}}\ {\isasymleadsto}\ r{\isadigit{2}}\ {\isasymLongrightarrow}\ ASEQ\ bs\ r{\isadigit{1}}\ r{\isadigit{3}}\ {\isasymleadsto}\ ASEQ\ bs\ r{\isadigit{2}}\ r{\isadigit{3}}{\isachardoublequoteclose}\isanewline
-{\isacharbar}{\kern0pt}\ {\isachardoublequoteopen}r{\isadigit{3}}\ {\isasymleadsto}\ r{\isadigit{4}}\ {\isasymLongrightarrow}\ ASEQ\ bs\ r{\isadigit{1}}\ r{\isadigit{3}}\ {\isasymleadsto}\ ASEQ\ bs\ r{\isadigit{1}}\ r{\isadigit{4}}{\isachardoublequoteclose}\isanewline
-{\isacharbar}{\kern0pt}\ {\isachardoublequoteopen}r\ {\isasymleadsto}\ r{\isacharprime}{\kern0pt}\ {\isasymLongrightarrow}\ {\isacharparenleft}{\kern0pt}AALTs\ bs\ {\isacharparenleft}{\kern0pt}rs{\isadigit{1}}\ {\isacharat}{\kern0pt}\ {\isacharbrackleft}{\kern0pt}r{\isacharbrackright}{\kern0pt}\ {\isacharat}{\kern0pt}\ rs{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymleadsto}\ {\isacharparenleft}{\kern0pt}AALTs\ bs\ {\isacharparenleft}{\kern0pt}rs{\isadigit{1}}\ {\isacharat}{\kern0pt}\ {\isacharbrackleft}{\kern0pt}r{\isacharprime}{\kern0pt}{\isacharbrackright}{\kern0pt}\ {\isacharat}{\kern0pt}\ rs{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-{\isacharbar}{\kern0pt}\ {\isachardoublequoteopen}AALTs\ bs\ {\isacharparenleft}{\kern0pt}rsa{\isacharat}{\kern0pt}AZERO\ {\isacharhash}{\kern0pt}\ rsb{\isacharparenright}{\kern0pt}\ {\isasymleadsto}\ AALTs\ bs\ {\isacharparenleft}{\kern0pt}rsa{\isacharat}{\kern0pt}rsb{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-{\isacharbar}{\kern0pt}\ {\isachardoublequoteopen}AALTs\ bs\ {\isacharparenleft}{\kern0pt}rsa{\isacharat}{\kern0pt}{\isacharparenleft}{\kern0pt}AALTs\ bs{\isadigit{1}}\ rs{\isadigit{1}}{\isacharparenright}{\kern0pt}{\isacharhash}{\kern0pt}\ rsb{\isacharparenright}{\kern0pt}\ {\isasymleadsto}\ AALTs\ bs\ {\isacharparenleft}{\kern0pt}rsa{\isacharat}{\kern0pt}{\isacharparenleft}{\kern0pt}map\ {\isacharparenleft}{\kern0pt}fuse\ bs{\isadigit{1}}{\isacharparenright}{\kern0pt}\ rs{\isadigit{1}}{\isacharparenright}{\kern0pt}{\isacharat}{\kern0pt}rsb{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-{\isacharbar}{\kern0pt}\ {\isachardoublequoteopen}AALTs\ bs\ {\isacharparenleft}{\kern0pt}map\ {\isacharparenleft}{\kern0pt}fuse\ bs{\isadigit{1}}{\isacharparenright}{\kern0pt}\ rs{\isacharparenright}{\kern0pt}\ {\isasymleadsto}\ AALTs\ {\isacharparenleft}{\kern0pt}bs{\isacharat}{\kern0pt}bs{\isadigit{1}}{\isacharparenright}{\kern0pt}\ rs{\isachardoublequoteclose}\isanewline
-\isanewline
-{\isacharbar}{\kern0pt}\ {\isachardoublequoteopen}AALTs\ {\isacharparenleft}{\kern0pt}bs{\isacharat}{\kern0pt}bs{\isadigit{1}}{\isacharparenright}{\kern0pt}\ rs\ {\isasymleadsto}\ AALTs\ bs\ {\isacharparenleft}{\kern0pt}map\ {\isacharparenleft}{\kern0pt}fuse\ bs{\isadigit{1}}{\isacharparenright}{\kern0pt}\ rs{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-{\isacharbar}{\kern0pt}\ {\isachardoublequoteopen}AALTs\ bs\ {\isacharbrackleft}{\kern0pt}{\isacharbrackright}{\kern0pt}\ {\isasymleadsto}\ AZERO{\isachardoublequoteclose}\isanewline
-{\isacharbar}{\kern0pt}\ {\isachardoublequoteopen}AALTs\ bs\ {\isacharbrackleft}{\kern0pt}r{\isacharbrackright}{\kern0pt}\ {\isasymleadsto}\ fuse\ bs\ r{\isachardoublequoteclose}\isanewline
-{\isacharbar}{\kern0pt}\ {\isachardoublequoteopen}erase\ a{\isadigit{1}}\ {\isacharequal}{\kern0pt}\ erase\ a{\isadigit{2}}\ {\isasymLongrightarrow}\ AALTs\ bs\ {\isacharparenleft}{\kern0pt}rsa{\isacharat}{\kern0pt}{\isacharbrackleft}{\kern0pt}a{\isadigit{1}}{\isacharbrackright}{\kern0pt}{\isacharat}{\kern0pt}rsb{\isacharat}{\kern0pt}{\isacharbrackleft}{\kern0pt}a{\isadigit{2}}{\isacharbrackright}{\kern0pt}{\isacharat}{\kern0pt}rsc{\isacharparenright}{\kern0pt}\ {\isasymleadsto}\ AALTs\ bs\ {\isacharparenleft}{\kern0pt}rsa{\isacharat}{\kern0pt}{\isacharbrackleft}{\kern0pt}a{\isadigit{1}}{\isacharbrackright}{\kern0pt}{\isacharat}{\kern0pt}rsb{\isacharat}{\kern0pt}rsc{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isanewline
-\isacommand{inductive}\isamarkupfalse%
-\ rrewrites{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}arexp\ {\isasymRightarrow}\ arexp\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\ {\isacharparenleft}{\kern0pt}{\isachardoublequoteopen}{\isacharunderscore}{\kern0pt}\ {\isasymleadsto}{\isacharasterisk}{\kern0pt}\ {\isacharunderscore}{\kern0pt}{\isachardoublequoteclose}\ {\isacharbrackleft}{\kern0pt}{\isadigit{1}}{\isadigit{0}}{\isadigit{0}}{\isacharcomma}{\kern0pt}\ {\isadigit{1}}{\isadigit{0}}{\isadigit{0}}{\isacharbrackright}{\kern0pt}\ {\isadigit{1}}{\isadigit{0}}{\isadigit{0}}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isakeyword{where}\ \isanewline
-rs{\isadigit{1}}{\isacharbrackleft}{\kern0pt}intro{\isacharcomma}{\kern0pt}\ simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}{\isachardoublequoteopen}r\ {\isasymleadsto}{\isacharasterisk}{\kern0pt}\ r{\isachardoublequoteclose}\isanewline
-{\isacharbar}{\kern0pt}\ rs{\isadigit{2}}{\isacharbrackleft}{\kern0pt}intro{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymlbrakk}r{\isadigit{1}}\ {\isasymleadsto}{\isacharasterisk}{\kern0pt}\ r{\isadigit{2}}{\isacharsemicolon}{\kern0pt}\ r{\isadigit{2}}\ {\isasymleadsto}\ r{\isadigit{3}}{\isasymrbrakk}\ {\isasymLongrightarrow}\ r{\isadigit{1}}\ {\isasymleadsto}{\isacharasterisk}{\kern0pt}\ r{\isadigit{3}}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{inductive}\isamarkupfalse%
-\ srewrites{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}arexp\ list\ {\isasymRightarrow}\ arexp\ list\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\ {\isacharparenleft}{\kern0pt}{\isachardoublequoteopen}\ {\isacharunderscore}{\kern0pt}\ s{\isasymleadsto}{\isacharasterisk}{\kern0pt}\ {\isacharunderscore}{\kern0pt}{\isachardoublequoteclose}\ {\isacharbrackleft}{\kern0pt}{\isadigit{1}}{\isadigit{0}}{\isadigit{0}}{\isacharcomma}{\kern0pt}\ {\isadigit{1}}{\isadigit{0}}{\isadigit{0}}{\isacharbrackright}{\kern0pt}\ {\isadigit{1}}{\isadigit{0}}{\isadigit{0}}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isakeyword{where}\isanewline
-ss{\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharbrackleft}{\kern0pt}{\isacharbrackright}{\kern0pt}\ s{\isasymleadsto}{\isacharasterisk}{\kern0pt}\ {\isacharbrackleft}{\kern0pt}{\isacharbrackright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-{\isacharbar}{\kern0pt}ss{\isadigit{2}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymlbrakk}r\ {\isasymleadsto}{\isacharasterisk}{\kern0pt}\ r{\isacharprime}{\kern0pt}{\isacharsemicolon}{\kern0pt}\ rs\ s{\isasymleadsto}{\isacharasterisk}{\kern0pt}\ rs{\isacharprime}{\kern0pt}{\isasymrbrakk}\ {\isasymLongrightarrow}\ {\isacharparenleft}{\kern0pt}r{\isacharhash}{\kern0pt}rs{\isacharparenright}{\kern0pt}\ s{\isasymleadsto}{\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}r{\isacharprime}{\kern0pt}{\isacharhash}{\kern0pt}rs{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isanewline
-\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ r{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}rstar\ {\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}r{\isadigit{1}}\ {\isasymleadsto}\ r{\isadigit{2}}\ {\isasymLongrightarrow}\ r{\isadigit{1}}\ {\isasymleadsto}{\isacharasterisk}{\kern0pt}\ r{\isadigit{2}}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ rrewrites{\isachardot}{\kern0pt}intros{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}\ rrewrites{\isachardot}{\kern0pt}intros{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}\ \isacommand{by}\isamarkupfalse%
-\ blast%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\ \isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ real{\isacharunderscore}{\kern0pt}trans{\isacharcolon}{\kern0pt}\ \isanewline
-\ \ \isakeyword{assumes}\ a{\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}r{\isadigit{1}}\ {\isasymleadsto}{\isacharasterisk}{\kern0pt}\ r{\isadigit{2}}{\isachardoublequoteclose}\ \ \isakeyword{and}\ a{\isadigit{2}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}r{\isadigit{2}}\ {\isasymleadsto}{\isacharasterisk}{\kern0pt}\ r{\isadigit{3}}{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}r{\isadigit{1}}\ {\isasymleadsto}{\isacharasterisk}{\kern0pt}\ r{\isadigit{3}}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ a{\isadigit{2}}\ a{\isadigit{1}}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}induct\ r{\isadigit{2}}\ r{\isadigit{3}}\ arbitrary{\isacharcolon}{\kern0pt}\ r{\isadigit{1}}\ rule{\isacharcolon}{\kern0pt}\ rrewrites{\isachardot}{\kern0pt}induct{\isacharparenright}{\kern0pt}\ \isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}auto{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{done}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ \ many{\isacharunderscore}{\kern0pt}steps{\isacharunderscore}{\kern0pt}later{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymlbrakk}r{\isadigit{1}}\ {\isasymleadsto}\ r{\isadigit{2}}{\isacharsemicolon}{\kern0pt}\ r{\isadigit{2}}\ {\isasymleadsto}{\isacharasterisk}{\kern0pt}\ r{\isadigit{3}}\ {\isasymrbrakk}\ {\isasymLongrightarrow}\ r{\isadigit{1}}\ {\isasymleadsto}{\isacharasterisk}{\kern0pt}\ r{\isadigit{3}}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}meson\ r{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}rstar\ real{\isacharunderscore}{\kern0pt}trans{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ contextrewrites{\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}r\ {\isasymleadsto}{\isacharasterisk}{\kern0pt}\ r{\isacharprime}{\kern0pt}\ {\isasymLongrightarrow}\ {\isacharparenleft}{\kern0pt}AALTs\ bs\ {\isacharparenleft}{\kern0pt}r{\isacharhash}{\kern0pt}rs{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymleadsto}{\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}AALTs\ bs\ {\isacharparenleft}{\kern0pt}r{\isacharprime}{\kern0pt}{\isacharhash}{\kern0pt}rs{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}induct\ r\ r{\isacharprime}{\kern0pt}\ rule{\isacharcolon}{\kern0pt}\ rrewrites{\isachardot}{\kern0pt}induct{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ append{\isacharunderscore}{\kern0pt}Cons\ append{\isacharunderscore}{\kern0pt}Nil\ rrewrite{\isachardot}{\kern0pt}intros{\isacharparenleft}{\kern0pt}{\isadigit{6}}{\isacharparenright}{\kern0pt}\ rs{\isadigit{2}}{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ contextrewrites{\isadigit{2}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}r\ {\isasymleadsto}{\isacharasterisk}{\kern0pt}\ r{\isacharprime}{\kern0pt}\ {\isasymLongrightarrow}\ {\isacharparenleft}{\kern0pt}AALTs\ bs\ {\isacharparenleft}{\kern0pt}rs{\isadigit{1}}{\isacharat}{\kern0pt}{\isacharbrackleft}{\kern0pt}r{\isacharbrackright}{\kern0pt}{\isacharat}{\kern0pt}rs{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymleadsto}{\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}AALTs\ bs\ {\isacharparenleft}{\kern0pt}rs{\isadigit{1}}{\isacharat}{\kern0pt}{\isacharbrackleft}{\kern0pt}r{\isacharprime}{\kern0pt}{\isacharbrackright}{\kern0pt}{\isacharat}{\kern0pt}rs{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}induct\ r\ r{\isacharprime}{\kern0pt}\ rule{\isacharcolon}{\kern0pt}\ rrewrites{\isachardot}{\kern0pt}induct{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{using}\isamarkupfalse%
-\ rrewrite{\isachardot}{\kern0pt}intros{\isacharparenleft}{\kern0pt}{\isadigit{6}}{\isacharparenright}{\kern0pt}\ \isacommand{by}\isamarkupfalse%
-\ blast%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ srewrites{\isacharunderscore}{\kern0pt}alt{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}rs{\isadigit{1}}\ s{\isasymleadsto}{\isacharasterisk}{\kern0pt}\ rs{\isadigit{2}}\ {\isasymLongrightarrow}\ {\isacharparenleft}{\kern0pt}AALTs\ bs\ {\isacharparenleft}{\kern0pt}rs{\isacharat}{\kern0pt}rs{\isadigit{1}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymleadsto}{\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}AALTs\ bs\ {\isacharparenleft}{\kern0pt}rs{\isacharat}{\kern0pt}rs{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\isanewline
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}induct\ rs{\isadigit{1}}\ rs{\isadigit{2}}\ arbitrary{\isacharcolon}{\kern0pt}\ bs\ rs\ rule{\isacharcolon}{\kern0pt}\ srewrites{\isachardot}{\kern0pt}induct{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}rule\ rs{\isadigit{1}}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}drule{\isacharunderscore}{\kern0pt}tac\ x\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}bs{\isachardoublequoteclose}\ \isakeyword{in}\ meta{\isacharunderscore}{\kern0pt}spec{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}drule{\isacharunderscore}{\kern0pt}tac\ x\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}rsa{\isacharat}{\kern0pt}{\isacharbrackleft}{\kern0pt}r{\isacharprime}{\kern0pt}{\isacharbrackright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{in}\ meta{\isacharunderscore}{\kern0pt}spec{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}rule\ real{\isacharunderscore}{\kern0pt}trans{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \isacommand{prefer}\isamarkupfalse%
-\ {\isadigit{2}}\isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}assumption{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}drule\ contextrewrites{\isadigit{2}}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{done}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{corollary}\isamarkupfalse%
-\ srewrites{\isacharunderscore}{\kern0pt}alt{\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}rs{\isadigit{1}}\ s{\isasymleadsto}{\isacharasterisk}{\kern0pt}\ rs{\isadigit{2}}\ {\isasymLongrightarrow}\ AALTs\ bs\ rs{\isadigit{1}}\ {\isasymleadsto}{\isacharasterisk}{\kern0pt}\ AALTs\ bs\ rs{\isadigit{2}}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ append{\isachardot}{\kern0pt}left{\isacharunderscore}{\kern0pt}neutral\ srewrites{\isacharunderscore}{\kern0pt}alt{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ star{\isacharunderscore}{\kern0pt}seq{\isacharcolon}{\kern0pt}\ \ {\isachardoublequoteopen}r{\isadigit{1}}\ {\isasymleadsto}{\isacharasterisk}{\kern0pt}\ r{\isadigit{2}}\ {\isasymLongrightarrow}\ ASEQ\ bs\ r{\isadigit{1}}\ r{\isadigit{3}}\ {\isasymleadsto}{\isacharasterisk}{\kern0pt}\ ASEQ\ bs\ r{\isadigit{2}}\ r{\isadigit{3}}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}induct\ r{\isadigit{1}}\ r{\isadigit{2}}\ arbitrary{\isacharcolon}{\kern0pt}\ r{\isadigit{3}}\ rule{\isacharcolon}{\kern0pt}\ rrewrites{\isachardot}{\kern0pt}induct{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}rule\ rs{\isadigit{1}}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}erule\ rrewrites{\isachardot}{\kern0pt}cases{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}rule\ r{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}rstar{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}rule\ rrewrite{\isachardot}{\kern0pt}intros{\isacharparenleft}{\kern0pt}{\isadigit{4}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}rule\ rs{\isadigit{2}}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}assumption{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}rule\ rrewrite{\isachardot}{\kern0pt}intros{\isacharparenleft}{\kern0pt}{\isadigit{4}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ assumption%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ star{\isacharunderscore}{\kern0pt}seq{\isadigit{2}}{\isacharcolon}{\kern0pt}\ \ {\isachardoublequoteopen}r{\isadigit{3}}\ {\isasymleadsto}{\isacharasterisk}{\kern0pt}\ r{\isadigit{4}}\ {\isasymLongrightarrow}\ ASEQ\ bs\ r{\isadigit{1}}\ r{\isadigit{3}}\ {\isasymleadsto}{\isacharasterisk}{\kern0pt}\ ASEQ\ bs\ r{\isadigit{1}}\ r{\isadigit{4}}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}induct\ r{\isadigit{3}}\ r{\isadigit{4}}\ arbitrary{\isacharcolon}{\kern0pt}\ r{\isadigit{1}}\ rule{\isacharcolon}{\kern0pt}\ rrewrites{\isachardot}{\kern0pt}induct{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{using}\isamarkupfalse%
-\ rrewrite{\isachardot}{\kern0pt}intros{\isacharparenleft}{\kern0pt}{\isadigit{5}}{\isacharparenright}{\kern0pt}\ \isacommand{by}\isamarkupfalse%
-\ blast%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ continuous{\isacharunderscore}{\kern0pt}rewrite{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymlbrakk}r{\isadigit{1}}\ {\isasymleadsto}{\isacharasterisk}{\kern0pt}\ AZERO{\isasymrbrakk}\ {\isasymLongrightarrow}\ ASEQ\ bs{\isadigit{1}}\ r{\isadigit{1}}\ r{\isadigit{2}}\ {\isasymleadsto}{\isacharasterisk}{\kern0pt}\ AZERO{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}induction\ ra{\isasymequiv}{\isachardoublequoteopen}r{\isadigit{1}}{\isachardoublequoteclose}\ rb{\isasymequiv}{\isachardoublequoteopen}AZERO{\isachardoublequoteclose}\ arbitrary{\isacharcolon}{\kern0pt}\ bs{\isadigit{1}}\ r{\isadigit{1}}\ r{\isadigit{2}}\ rule{\isacharcolon}{\kern0pt}\ rrewrites{\isachardot}{\kern0pt}induct{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ r{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}rstar\ rrewrite{\isachardot}{\kern0pt}intros{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}meson\ rrewrite{\isachardot}{\kern0pt}intros{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}\ rrewrites{\isachardot}{\kern0pt}intros{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}\ star{\isacharunderscore}{\kern0pt}seq{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\ \ \isanewline
-\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ bsimp{\isacharunderscore}{\kern0pt}aalts{\isacharunderscore}{\kern0pt}simpcases{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}AONE\ bs\ {\isasymleadsto}{\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}bsimp\ {\isacharparenleft}{\kern0pt}AONE\ bs{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \ {\isachardoublequoteopen}AZERO\ {\isasymleadsto}{\isacharasterisk}{\kern0pt}\ bsimp\ AZERO{\isachardoublequoteclose}\ {\isachardoublequoteopen}ACHAR\ bs\ c\ {\isasymleadsto}{\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}bsimp\ {\isacharparenleft}{\kern0pt}ACHAR\ bs\ c{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{apply}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ rrewrites{\isachardot}{\kern0pt}intros{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ rrewrites{\isachardot}{\kern0pt}intros{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ rrewrites{\isachardot}{\kern0pt}intros{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ trivialbsimpsrewrites{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymlbrakk}{\isasymAnd}x{\isachardot}{\kern0pt}\ x\ {\isasymin}\ set\ rs\ {\isasymLongrightarrow}\ x\ {\isasymleadsto}{\isacharasterisk}{\kern0pt}\ f\ x\ {\isasymrbrakk}\ {\isasymLongrightarrow}\ rs\ s{\isasymleadsto}{\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}map\ f\ rs{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\isanewline
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}induction\ rs{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}rule\ ss{\isadigit{1}}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ insert{\isacharunderscore}{\kern0pt}iff\ list{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isadigit{5}}{\isacharparenright}{\kern0pt}\ list{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{9}}{\isacharparenright}{\kern0pt}\ srewrites{\isachardot}{\kern0pt}simps{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ bsimp{\isacharunderscore}{\kern0pt}AALTsrewrites{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}AALTs\ bs{\isadigit{1}}\ rs\ {\isasymleadsto}{\isacharasterisk}{\kern0pt}\ bsimp{\isacharunderscore}{\kern0pt}AALTs\ bs{\isadigit{1}}\ rs{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}induction\ rs{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}rule\ r{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}rstar{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ \ rrewrite{\isachardot}{\kern0pt}intros{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isadigit{1}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}case{\isacharunderscore}{\kern0pt}tac\ {\isachardoublequoteopen}rs\ {\isacharequal}{\kern0pt}\ Nil{\isachardoublequoteclose}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{using}\isamarkupfalse%
-\ rrewrite{\isachardot}{\kern0pt}intros{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isadigit{2}}{\isacharparenright}{\kern0pt}\ \isacommand{apply}\isamarkupfalse%
-\ auto{\isacharbrackleft}{\kern0pt}{\isadigit{1}}{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}subgoal{\isacharunderscore}{\kern0pt}tac\ {\isachardoublequoteopen}length\ {\isacharparenleft}{\kern0pt}a{\isacharhash}{\kern0pt}rs{\isacharparenright}{\kern0pt}\ {\isachargreater}{\kern0pt}\ {\isadigit{1}}{\isachardoublequoteclose}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ bsimp{\isacharunderscore}{\kern0pt}AALTs{\isacharunderscore}{\kern0pt}qq{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{done}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\ \isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{inductive}\isamarkupfalse%
-\ frewrites{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}arexp\ list\ {\isasymRightarrow}\ arexp\ list\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\ {\isacharparenleft}{\kern0pt}{\isachardoublequoteopen}\ {\isacharunderscore}{\kern0pt}\ f{\isasymleadsto}{\isacharasterisk}{\kern0pt}\ {\isacharunderscore}{\kern0pt}{\isachardoublequoteclose}\ {\isacharbrackleft}{\kern0pt}{\isadigit{1}}{\isadigit{0}}{\isadigit{0}}{\isacharcomma}{\kern0pt}\ {\isadigit{1}}{\isadigit{0}}{\isadigit{0}}{\isacharbrackright}{\kern0pt}\ {\isadigit{1}}{\isadigit{0}}{\isadigit{0}}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isakeyword{where}\isanewline
-fs{\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharbrackleft}{\kern0pt}{\isacharbrackright}{\kern0pt}\ f{\isasymleadsto}{\isacharasterisk}{\kern0pt}\ {\isacharbrackleft}{\kern0pt}{\isacharbrackright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-{\isacharbar}{\kern0pt}fs{\isadigit{2}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymlbrakk}rs\ f{\isasymleadsto}{\isacharasterisk}{\kern0pt}\ rs{\isacharprime}{\kern0pt}{\isasymrbrakk}\ {\isasymLongrightarrow}\ {\isacharparenleft}{\kern0pt}AZERO{\isacharhash}{\kern0pt}rs{\isacharparenright}{\kern0pt}\ f{\isasymleadsto}{\isacharasterisk}{\kern0pt}\ rs{\isacharprime}{\kern0pt}{\isachardoublequoteclose}\isanewline
-{\isacharbar}{\kern0pt}fs{\isadigit{3}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymlbrakk}rs\ f{\isasymleadsto}{\isacharasterisk}{\kern0pt}\ rs{\isacharprime}{\kern0pt}{\isasymrbrakk}\ {\isasymLongrightarrow}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}AALTs\ bs\ rs{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isacharhash}{\kern0pt}\ rs{\isacharparenright}{\kern0pt}\ f{\isasymleadsto}{\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}map\ {\isacharparenleft}{\kern0pt}fuse\ bs{\isacharparenright}{\kern0pt}\ rs{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isacharat}{\kern0pt}\ rs{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-{\isacharbar}{\kern0pt}fs{\isadigit{4}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymlbrakk}rs\ f{\isasymleadsto}{\isacharasterisk}{\kern0pt}\ rs{\isacharprime}{\kern0pt}{\isacharsemicolon}{\kern0pt}nonalt\ r{\isacharsemicolon}{\kern0pt}\ nonazero\ r{\isasymrbrakk}\ {\isasymLongrightarrow}\ {\isacharparenleft}{\kern0pt}r{\isacharhash}{\kern0pt}rs{\isacharparenright}{\kern0pt}\ f{\isasymleadsto}{\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}r{\isacharhash}{\kern0pt}rs{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isanewline
-\isanewline
-\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ flts{\isacharunderscore}{\kern0pt}prepend{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymlbrakk}nonalt\ a{\isacharsemicolon}{\kern0pt}\ nonazero\ a{\isasymrbrakk}\ {\isasymLongrightarrow}\ flts\ {\isacharparenleft}{\kern0pt}a{\isacharhash}{\kern0pt}rs{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ a\ {\isacharhash}{\kern0pt}\ {\isacharparenleft}{\kern0pt}flts\ rs{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ append{\isacharunderscore}{\kern0pt}Cons\ append{\isacharunderscore}{\kern0pt}Nil\ flts{\isacharunderscore}{\kern0pt}single{\isadigit{1}}\ k{\isadigit{0}}{\isadigit{0}}{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ fltsfrewrites{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}rs\ f{\isasymleadsto}{\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}flts\ rs{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}induction\ rs{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}rule\ fs{\isadigit{1}}{\isacharparenright}{\kern0pt}\isanewline
-\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}case{\isacharunderscore}{\kern0pt}tac\ {\isachardoublequoteopen}a\ {\isacharequal}{\kern0pt}\ AZERO{\isachardoublequoteclose}{\isacharparenright}{\kern0pt}\isanewline
-\isanewline
-\ \ \ \isanewline
-\ \ \isacommand{using}\isamarkupfalse%
-\ fs{\isadigit{2}}\ \isacommand{apply}\isamarkupfalse%
-\ auto{\isacharbrackleft}{\kern0pt}{\isadigit{1}}{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}case{\isacharunderscore}{\kern0pt}tac\ {\isachardoublequoteopen}{\isasymexists}bs\ rs{\isachardot}{\kern0pt}\ a\ {\isacharequal}{\kern0pt}\ AALTs\ bs\ rs{\isachardoublequoteclose}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}erule\ exE{\isacharparenright}{\kern0pt}{\isacharplus}{\kern0pt}\isanewline
-\ \ \ \isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ fs{\isadigit{3}}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}subst\ flts{\isacharunderscore}{\kern0pt}prepend{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}rule\ nonalt{\isachardot}{\kern0pt}elims{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{prefer}\isamarkupfalse%
-\ {\isadigit{2}}\isanewline
-\ \ \isacommand{thm}\isamarkupfalse%
-\ nonalt{\isachardot}{\kern0pt}elims\isanewline
-\ \ \ \isanewline
-\ \ \ \ \ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-\ blast\isanewline
-\ \ \ \isanewline
-\ \ \isacommand{using}\isamarkupfalse%
-\ bbbbs{\isadigit{1}}\ \isacommand{apply}\isamarkupfalse%
-\ blast\isanewline
-\ \ \ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ nonalt{\isachardot}{\kern0pt}simps{\isacharparenright}{\kern0pt}{\isacharplus}{\kern0pt}\isanewline
-\ \ \ \isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}meson\ nonazero{\isachardot}{\kern0pt}elims{\isacharparenleft}{\kern0pt}{\isadigit{3}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}meson\ fs{\isadigit{4}}\ nonalt{\isachardot}{\kern0pt}elims{\isacharparenleft}{\kern0pt}{\isadigit{3}}{\isacharparenright}{\kern0pt}\ nonazero{\isachardot}{\kern0pt}elims{\isacharparenleft}{\kern0pt}{\isadigit{3}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ rrewrite{\isadigit{0}}away{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}AALTs\ bs\ {\isacharparenleft}{\kern0pt}\ AZERO\ {\isacharhash}{\kern0pt}\ rsb{\isacharparenright}{\kern0pt}\ {\isasymleadsto}\ AALTs\ bs\ rsb{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ append{\isacharunderscore}{\kern0pt}Nil\ rrewrite{\isachardot}{\kern0pt}intros{\isacharparenleft}{\kern0pt}{\isadigit{7}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ frewritesaalts{\isacharcolon}{\kern0pt}{\isachardoublequoteopen}rs\ f{\isasymleadsto}{\isacharasterisk}{\kern0pt}\ rs{\isacharprime}{\kern0pt}\ {\isasymLongrightarrow}\ {\isacharparenleft}{\kern0pt}AALTs\ bs\ {\isacharparenleft}{\kern0pt}rs{\isadigit{1}}{\isacharat}{\kern0pt}rs{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymleadsto}{\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}AALTs\ bs\ {\isacharparenleft}{\kern0pt}rs{\isadigit{1}}{\isacharat}{\kern0pt}rs{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}induct\ rs\ rs{\isacharprime}{\kern0pt}\ arbitrary{\isacharcolon}{\kern0pt}\ bs\ rs{\isadigit{1}}\ rule{\isacharcolon}{\kern0pt}frewrites{\isachardot}{\kern0pt}induct{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}rule\ rs{\isadigit{1}}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}drule{\isacharunderscore}{\kern0pt}tac\ x\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}bs{\isachardoublequoteclose}\ \isakeyword{in}\ meta{\isacharunderscore}{\kern0pt}spec{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}drule{\isacharunderscore}{\kern0pt}tac\ x\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}rs{\isadigit{1}}\ {\isacharat}{\kern0pt}\ {\isacharbrackleft}{\kern0pt}AZERO{\isacharbrackright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{in}\ meta{\isacharunderscore}{\kern0pt}spec{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}rule\ real{\isacharunderscore}{\kern0pt}trans{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{using}\isamarkupfalse%
-\ r{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}rstar\ rrewrite{\isachardot}{\kern0pt}intros{\isacharparenleft}{\kern0pt}{\isadigit{7}}{\isacharparenright}{\kern0pt}\ \isacommand{apply}\isamarkupfalse%
-\ presburger\isanewline
-\ \ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}drule{\isacharunderscore}{\kern0pt}tac\ x\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}bsa{\isachardoublequoteclose}\ \isakeyword{in}\ meta{\isacharunderscore}{\kern0pt}spec{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}drule{\isacharunderscore}{\kern0pt}tac\ x\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}rs{\isadigit{1}}a\ {\isacharat}{\kern0pt}\ {\isacharbrackleft}{\kern0pt}AALTs\ bs\ rs{\isadigit{1}}{\isacharbrackright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{in}\ meta{\isacharunderscore}{\kern0pt}spec{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}rule\ real{\isacharunderscore}{\kern0pt}trans{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{apply}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{using}\isamarkupfalse%
-\ r{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}rstar\ rrewrite{\isachardot}{\kern0pt}intros{\isacharparenleft}{\kern0pt}{\isadigit{8}}{\isacharparenright}{\kern0pt}\ \isacommand{apply}\isamarkupfalse%
-\ presburger\isanewline
-\ \ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}drule{\isacharunderscore}{\kern0pt}tac\ x\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}bs{\isachardoublequoteclose}\ \isakeyword{in}\ meta{\isacharunderscore}{\kern0pt}spec{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}drule{\isacharunderscore}{\kern0pt}tac\ x\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}rs{\isadigit{1}}{\isacharat}{\kern0pt}{\isacharbrackleft}{\kern0pt}r{\isacharbrackright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{in}\ meta{\isacharunderscore}{\kern0pt}spec{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}rule\ real{\isacharunderscore}{\kern0pt}trans{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{done}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ fltsrewrites{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}\ \ AALTs\ bs{\isadigit{1}}\ rs\ {\isasymleadsto}{\isacharasterisk}{\kern0pt}\ AALTs\ bs{\isadigit{1}}\ {\isacharparenleft}{\kern0pt}flts\ rs{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}induction\ rs{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}case{\isacharunderscore}{\kern0pt}tac\ {\isachardoublequoteopen}a\ {\isacharequal}{\kern0pt}\ AZERO{\isachardoublequoteclose}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ append{\isacharunderscore}{\kern0pt}Nil\ flts{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}\ many{\isacharunderscore}{\kern0pt}steps{\isacharunderscore}{\kern0pt}later\ rrewrite{\isachardot}{\kern0pt}intros{\isacharparenleft}{\kern0pt}{\isadigit{7}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\isanewline
-\isanewline
-\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}case{\isacharunderscore}{\kern0pt}tac\ {\isachardoublequoteopen}{\isasymexists}bs{\isadigit{2}}\ rs{\isadigit{2}}{\isachardot}{\kern0pt}\ a\ {\isacharequal}{\kern0pt}\ AALTs\ bs{\isadigit{2}}\ rs{\isadigit{2}}{\isachardoublequoteclose}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}erule\ exE{\isacharparenright}{\kern0pt}{\isacharplus}{\kern0pt}\isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ flts{\isachardot}{\kern0pt}simps{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \isacommand{prefer}\isamarkupfalse%
-\ {\isadigit{2}}\isanewline
-\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}subst\ flts{\isacharunderscore}{\kern0pt}prepend{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \isanewline
-\ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}meson\ nonalt{\isachardot}{\kern0pt}elims{\isacharparenleft}{\kern0pt}{\isadigit{3}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \isanewline
-\ \ \ \ \isacommand{apply}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}meson\ nonazero{\isachardot}{\kern0pt}elims{\isacharparenleft}{\kern0pt}{\isadigit{3}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}subgoal{\isacharunderscore}{\kern0pt}tac\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}a{\isacharhash}{\kern0pt}rs{\isacharparenright}{\kern0pt}\ f{\isasymleadsto}{\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}a{\isacharhash}{\kern0pt}flts\ rs{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ append{\isacharunderscore}{\kern0pt}Nil\ frewritesaalts{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}meson\ fltsfrewrites\ fs{\isadigit{4}}\ nonalt{\isachardot}{\kern0pt}elims{\isacharparenleft}{\kern0pt}{\isadigit{3}}{\isacharparenright}{\kern0pt}\ nonazero{\isachardot}{\kern0pt}elims{\isacharparenleft}{\kern0pt}{\isadigit{3}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ append{\isacharunderscore}{\kern0pt}Cons\ append{\isacharunderscore}{\kern0pt}Nil\ fltsfrewrites\ frewritesaalts\ k{\isadigit{0}}{\isadigit{0}}\ k{\isadigit{0}}a{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ alts{\isacharunderscore}{\kern0pt}simpalts{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymAnd}bs{\isadigit{1}}\ rs{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymAnd}x{\isachardot}{\kern0pt}\ x\ {\isasymin}\ set\ rs\ {\isasymLongrightarrow}\ x\ {\isasymleadsto}{\isacharasterisk}{\kern0pt}\ bsimp\ x{\isacharparenright}{\kern0pt}\ {\isasymLongrightarrow}\ \isanewline
-AALTs\ bs{\isadigit{1}}\ rs\ {\isasymleadsto}{\isacharasterisk}{\kern0pt}\ AALTs\ bs{\isadigit{1}}\ {\isacharparenleft}{\kern0pt}map\ bsimp\ rs{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}subgoal{\isacharunderscore}{\kern0pt}tac\ {\isachardoublequoteopen}\ rs\ s{\isasymleadsto}{\isacharasterisk}{\kern0pt}\ \ {\isacharparenleft}{\kern0pt}map\ bsimp\ rs{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \isacommand{prefer}\isamarkupfalse%
-\ {\isadigit{2}}\isanewline
-\ \ \isacommand{using}\isamarkupfalse%
-\ trivialbsimpsrewrites\ \isacommand{apply}\isamarkupfalse%
-\ auto{\isacharbrackleft}{\kern0pt}{\isadigit{1}}{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \isacommand{using}\isamarkupfalse%
-\ srewrites{\isacharunderscore}{\kern0pt}alt{\isadigit{1}}\ \isacommand{by}\isamarkupfalse%
-\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ threelistsappend{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}rsa{\isacharat}{\kern0pt}a{\isacharhash}{\kern0pt}rsb\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}rsa{\isacharat}{\kern0pt}{\isacharbrackleft}{\kern0pt}a{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharat}{\kern0pt}rsb{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{apply}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{done}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{fun}\isamarkupfalse%
-\ distinctByAcc\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ list\ {\isasymRightarrow}\ {\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}a\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}b{\isacharparenright}{\kern0pt}\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}b\ set\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}b\ set{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}distinctByAcc\ {\isacharbrackleft}{\kern0pt}{\isacharbrackright}{\kern0pt}\ f\ acc\ {\isacharequal}{\kern0pt}\ acc{\isachardoublequoteclose}\isanewline
-{\isacharbar}{\kern0pt}\ {\isachardoublequoteopen}distinctByAcc\ {\isacharparenleft}{\kern0pt}x{\isacharhash}{\kern0pt}xs{\isacharparenright}{\kern0pt}\ f\ acc\ {\isacharequal}{\kern0pt}\ \isanewline
-\ \ \ \ \ {\isacharparenleft}{\kern0pt}if\ {\isacharparenleft}{\kern0pt}f\ x{\isacharparenright}{\kern0pt}\ {\isasymin}\ acc\ then\ distinctByAcc\ xs\ f\ acc\ \isanewline
-\ \ \ \ \ \ else\ \ {\isacharparenleft}{\kern0pt}distinctByAcc\ xs\ f\ {\isacharparenleft}{\kern0pt}{\isacharbraceleft}{\kern0pt}f\ x{\isacharbraceright}{\kern0pt}\ {\isasymunion}\ acc{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ dB{\isacharunderscore}{\kern0pt}single{\isacharunderscore}{\kern0pt}step{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}distinctBy\ {\isacharparenleft}{\kern0pt}a{\isacharhash}{\kern0pt}rs{\isacharparenright}{\kern0pt}\ f\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ a\ {\isacharhash}{\kern0pt}\ distinctBy\ rs\ f\ {\isacharbraceleft}{\kern0pt}f\ a{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{apply}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{done}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ somewhereInside{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}r\ {\isasymin}\ set\ rs\ {\isasymLongrightarrow}\ {\isasymexists}rs{\isadigit{1}}\ rs{\isadigit{2}}{\isachardot}{\kern0pt}\ rs\ {\isacharequal}{\kern0pt}\ rs{\isadigit{1}}{\isacharat}{\kern0pt}{\isacharbrackleft}{\kern0pt}r{\isacharbrackright}{\kern0pt}{\isacharat}{\kern0pt}rs{\isadigit{2}}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ split{\isacharunderscore}{\kern0pt}list\ \isacommand{by}\isamarkupfalse%
-\ fastforce%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ somewhereMapInside{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}f\ r\ {\isasymin}\ f\ {\isacharbackquote}{\kern0pt}\ set\ rs\ {\isasymLongrightarrow}\ {\isasymexists}rs{\isadigit{1}}\ rs{\isadigit{2}}\ a{\isachardot}{\kern0pt}\ rs\ {\isacharequal}{\kern0pt}\ rs{\isadigit{1}}{\isacharat}{\kern0pt}{\isacharbrackleft}{\kern0pt}a{\isacharbrackright}{\kern0pt}{\isacharat}{\kern0pt}rs{\isadigit{2}}\ {\isasymand}\ f\ a\ {\isacharequal}{\kern0pt}\ f\ r{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{apply}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ split{\isacharunderscore}{\kern0pt}list{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ alts{\isacharunderscore}{\kern0pt}dBrewrites{\isacharunderscore}{\kern0pt}withFront{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}\ AALTs\ bs\ {\isacharparenleft}{\kern0pt}rsa\ {\isacharat}{\kern0pt}\ rs{\isacharparenright}{\kern0pt}\ {\isasymleadsto}{\isacharasterisk}{\kern0pt}\ AALTs\ bs\ {\isacharparenleft}{\kern0pt}rsa\ {\isacharat}{\kern0pt}\ distinctBy\ rs\ erase\ {\isacharparenleft}{\kern0pt}erase\ {\isacharbackquote}{\kern0pt}\ set\ rsa{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}induction\ rs\ arbitrary{\isacharcolon}{\kern0pt}\ rsa{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}drule{\isacharunderscore}{\kern0pt}tac\ x\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}rsa{\isacharat}{\kern0pt}{\isacharbrackleft}{\kern0pt}a{\isacharbrackright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{in}\ meta{\isacharunderscore}{\kern0pt}spec{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}subst\ threelistsappend{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}rule\ real{\isacharunderscore}{\kern0pt}trans{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}case{\isacharunderscore}{\kern0pt}tac\ {\isachardoublequoteopen}a\ {\isasymin}\ set\ rsa{\isachardoublequoteclose}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}drule\ somewhereInside{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}erule\ exE{\isacharparenright}{\kern0pt}{\isacharplus}{\kern0pt}\isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}subgoal{\isacharunderscore}{\kern0pt}tac\ {\isachardoublequoteopen}\ AALTs\ bs\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}rs{\isadigit{1}}\ {\isacharat}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ a\ {\isacharhash}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ rs{\isadigit{2}}\ {\isacharat}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ a\ {\isacharhash}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ distinctBy\ rs\ erase\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}insert\ {\isacharparenleft}{\kern0pt}erase\ a{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}erase\ {\isacharbackquote}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}set\ rs{\isadigit{1}}\ {\isasymunion}\ set\ rs{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymleadsto}\ AALTs\ bs\ {\isacharparenleft}{\kern0pt}rs{\isadigit{1}}{\isacharat}{\kern0pt}\ a\ {\isacharhash}{\kern0pt}\ rs{\isadigit{2}}\ {\isacharat}{\kern0pt}\ \ distinctBy\ rs\ erase\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}insert\ {\isacharparenleft}{\kern0pt}erase\ a{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}erase\ {\isacharbackquote}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}set\ rs{\isadigit{1}}\ {\isasymunion}\ set\ rs{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isachardoublequoteclose}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{prefer}\isamarkupfalse%
-\ {\isadigit{2}}\isanewline
-\ \ \isacommand{using}\isamarkupfalse%
-\ rrewrite{\isachardot}{\kern0pt}intros{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isadigit{3}}{\isacharparenright}{\kern0pt}\ \isacommand{apply}\isamarkupfalse%
-\ force\isanewline
-\ \ \isacommand{using}\isamarkupfalse%
-\ r{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}rstar\ \isacommand{apply}\isamarkupfalse%
-\ force\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}subgoal{\isacharunderscore}{\kern0pt}tac\ {\isachardoublequoteopen}erase\ {\isacharbackquote}{\kern0pt}\ set\ {\isacharparenleft}{\kern0pt}rsa\ {\isacharat}{\kern0pt}\ {\isacharbrackleft}{\kern0pt}a{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ insert\ {\isacharparenleft}{\kern0pt}erase\ a{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}erase\ {\isacharbackquote}{\kern0pt}\ set\ rsa{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{prefer}\isamarkupfalse%
-\ {\isadigit{2}}\isanewline
-\ \ \ \ \isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-\ auto{\isacharbrackleft}{\kern0pt}{\isadigit{1}}{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}case{\isacharunderscore}{\kern0pt}tac\ {\isachardoublequoteopen}erase\ a\ {\isasymin}\ erase\ {\isacharbackquote}{\kern0pt}set\ rsa{\isachardoublequoteclose}{\isacharparenright}{\kern0pt}\isanewline
-\isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}subgoal{\isacharunderscore}{\kern0pt}tac\ {\isachardoublequoteopen}AALTs\ bs\ {\isacharparenleft}{\kern0pt}rsa\ {\isacharat}{\kern0pt}\ a\ {\isacharhash}{\kern0pt}\ distinctBy\ rs\ erase\ {\isacharparenleft}{\kern0pt}insert\ {\isacharparenleft}{\kern0pt}erase\ a{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}erase\ {\isacharbackquote}{\kern0pt}\ set\ rsa{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymleadsto}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ AALTs\ bs\ {\isacharparenleft}{\kern0pt}rsa\ {\isacharat}{\kern0pt}\ distinctBy\ rs\ erase\ {\isacharparenleft}{\kern0pt}insert\ {\isacharparenleft}{\kern0pt}erase\ a{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}erase\ {\isacharbackquote}{\kern0pt}\ set\ rsa{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-\ force\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}smt\ {\isacharparenleft}{\kern0pt}verit{\isacharcomma}{\kern0pt}\ ccfv{\isacharunderscore}{\kern0pt}threshold{\isacharparenright}{\kern0pt}\ append{\isacharunderscore}{\kern0pt}Cons\ append{\isacharunderscore}{\kern0pt}assoc\ append{\isacharunderscore}{\kern0pt}self{\isacharunderscore}{\kern0pt}conv{\isadigit{2}}\ r{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}rstar\ rrewrite{\isachardot}{\kern0pt}intros{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isadigit{3}}{\isacharparenright}{\kern0pt}\ same{\isacharunderscore}{\kern0pt}append{\isacharunderscore}{\kern0pt}eq\ somewhereMapInside{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ force%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\ \isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ alts{\isacharunderscore}{\kern0pt}dBrewrites{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}AALTs\ bs\ rs\ {\isasymleadsto}{\isacharasterisk}{\kern0pt}\ AALTs\ bs\ {\isacharparenleft}{\kern0pt}distinctBy\ rs\ erase\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}induction\ rs{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{using}\isamarkupfalse%
-\ alts{\isacharunderscore}{\kern0pt}dBrewrites{\isacharunderscore}{\kern0pt}withFront\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ append{\isacharunderscore}{\kern0pt}Nil\ dB{\isacharunderscore}{\kern0pt}single{\isacharunderscore}{\kern0pt}step\ empty{\isacharunderscore}{\kern0pt}set\ image{\isacharunderscore}{\kern0pt}empty{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isanewline
-\ \ \isanewline
-\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ bsimp{\isacharunderscore}{\kern0pt}rewrite{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}\ {\isacharparenleft}{\kern0pt}rrewrites\ r\ {\isacharparenleft}{\kern0pt}\ bsimp\ r{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}induction\ r\ rule{\isacharcolon}{\kern0pt}\ bsimp{\isachardot}{\kern0pt}induct{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}case{\isacharunderscore}{\kern0pt}tac\ {\isachardoublequoteopen}bsimp\ r{\isadigit{1}}\ {\isacharequal}{\kern0pt}\ AZERO{\isachardoublequoteclose}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{using}\isamarkupfalse%
-\ continuous{\isacharunderscore}{\kern0pt}rewrite\ \isacommand{apply}\isamarkupfalse%
-\ blast\isanewline
-\ \ \ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}case{\isacharunderscore}{\kern0pt}tac\ {\isachardoublequoteopen}{\isasymexists}bs{\isachardot}{\kern0pt}\ bsimp\ r{\isadigit{1}}\ {\isacharequal}{\kern0pt}\ AONE\ bs{\isachardoublequoteclose}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}erule\ exE{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}subst\ bsimp{\isacharunderscore}{\kern0pt}ASEQ{\isadigit{2}}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}meson\ real{\isacharunderscore}{\kern0pt}trans\ rrewrite{\isachardot}{\kern0pt}intros{\isacharparenleft}{\kern0pt}{\isadigit{3}}{\isacharparenright}{\kern0pt}\ rrewrites{\isachardot}{\kern0pt}intros{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}\ star{\isacharunderscore}{\kern0pt}seq\ star{\isacharunderscore}{\kern0pt}seq{\isadigit{2}}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}smt\ {\isacharparenleft}{\kern0pt}verit{\isacharcomma}{\kern0pt}\ best{\isacharparenright}{\kern0pt}\ bsimp{\isacharunderscore}{\kern0pt}ASEQ{\isadigit{0}}\ bsimp{\isacharunderscore}{\kern0pt}ASEQ{\isadigit{1}}\ real{\isacharunderscore}{\kern0pt}trans\ rrewrite{\isachardot}{\kern0pt}intros{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}\ rs{\isadigit{2}}\ star{\isacharunderscore}{\kern0pt}seq\ star{\isacharunderscore}{\kern0pt}seq{\isadigit{2}}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{defer}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{using}\isamarkupfalse%
-\ bsimp{\isacharunderscore}{\kern0pt}aalts{\isacharunderscore}{\kern0pt}simpcases{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}\ \isacommand{apply}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-\ simp\isanewline
-\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-\ auto\isanewline
-\isanewline
-\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}subgoal{\isacharunderscore}{\kern0pt}tac\ {\isachardoublequoteopen}AALTs\ bs{\isadigit{1}}\ rs\ {\isasymleadsto}{\isacharasterisk}{\kern0pt}\ AALTs\ bs{\isadigit{1}}\ {\isacharparenleft}{\kern0pt}map\ bsimp\ rs{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}subgoal{\isacharunderscore}{\kern0pt}tac\ {\isachardoublequoteopen}AALTs\ bs{\isadigit{1}}\ {\isacharparenleft}{\kern0pt}map\ bsimp\ rs{\isacharparenright}{\kern0pt}\ {\isasymleadsto}{\isacharasterisk}{\kern0pt}\ AALTs\ bs{\isadigit{1}}\ {\isacharparenleft}{\kern0pt}flts\ {\isacharparenleft}{\kern0pt}map\ bsimp\ rs{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}subgoal{\isacharunderscore}{\kern0pt}tac\ {\isachardoublequoteopen}AALTs\ bs{\isadigit{1}}\ {\isacharparenleft}{\kern0pt}flts\ {\isacharparenleft}{\kern0pt}map\ bsimp\ rs{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymleadsto}{\isacharasterisk}{\kern0pt}\ AALTs\ bs{\isadigit{1}}\ {\isacharparenleft}{\kern0pt}distinctBy\ {\isacharparenleft}{\kern0pt}flts\ {\isacharparenleft}{\kern0pt}map\ bsimp\ rs{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ erase\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}subgoal{\isacharunderscore}{\kern0pt}tac\ {\isachardoublequoteopen}AALTs\ bs{\isadigit{1}}\ {\isacharparenleft}{\kern0pt}distinctBy\ {\isacharparenleft}{\kern0pt}flts\ {\isacharparenleft}{\kern0pt}map\ bsimp\ rs{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ erase\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymleadsto}{\isacharasterisk}{\kern0pt}\ bsimp{\isacharunderscore}{\kern0pt}AALTs\ bs{\isadigit{1}}\ {\isacharparenleft}{\kern0pt}distinctBy\ {\isacharparenleft}{\kern0pt}flts\ {\isacharparenleft}{\kern0pt}map\ bsimp\ rs{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ erase\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharparenright}{\kern0pt}{\isachardoublequoteclose}{\isacharparenright}{\kern0pt}\isanewline
-\isanewline
-\ \ \isanewline
-\ \ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}meson\ real{\isacharunderscore}{\kern0pt}trans{\isacharparenright}{\kern0pt}\isanewline
-\isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}meson\ bsimp{\isacharunderscore}{\kern0pt}AALTsrewrites{\isacharparenright}{\kern0pt}\isanewline
-\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}meson\ alts{\isacharunderscore}{\kern0pt}dBrewrites{\isacharparenright}{\kern0pt}\isanewline
-\isanewline
-\ \ \isacommand{using}\isamarkupfalse%
-\ fltsrewrites\ \isacommand{apply}\isamarkupfalse%
-\ auto{\isacharbrackleft}{\kern0pt}{\isadigit{1}}{\isacharbrackright}{\kern0pt}\isanewline
-\isanewline
-\ \ \isacommand{using}\isamarkupfalse%
-\ alts{\isacharunderscore}{\kern0pt}simpalts\ \isacommand{by}\isamarkupfalse%
-\ force%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ rewritenullable{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymlbrakk}r{\isadigit{1}}\ {\isasymleadsto}\ r{\isadigit{2}}{\isacharsemicolon}{\kern0pt}\ bnullable\ r{\isadigit{1}}\ {\isasymrbrakk}\ {\isasymLongrightarrow}\ bnullable\ r{\isadigit{2}}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}induction\ r{\isadigit{1}}\ r{\isadigit{2}}\ rule{\isacharcolon}{\kern0pt}\ rrewrite{\isachardot}{\kern0pt}induct{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp{\isacharparenright}{\kern0pt}{\isacharplus}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ bnullable{\isacharunderscore}{\kern0pt}correctness\ erase{\isacharunderscore}{\kern0pt}fuse{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-\ auto{\isacharbrackleft}{\kern0pt}{\isadigit{1}}{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-\ auto{\isacharbrackleft}{\kern0pt}{\isadigit{1}}{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-\ auto{\isacharbrackleft}{\kern0pt}{\isadigit{4}}{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ UnCI\ bnullable{\isacharunderscore}{\kern0pt}correctness\ erase{\isacharunderscore}{\kern0pt}fuse\ imageI{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{apply}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ bnullable{\isacharunderscore}{\kern0pt}correctness\ erase{\isacharunderscore}{\kern0pt}fuse{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{apply}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ bnullable{\isacharunderscore}{\kern0pt}correctness\ erase{\isacharunderscore}{\kern0pt}fuse{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ bnullable{\isacharunderscore}{\kern0pt}correctness\ erase{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{5}}{\isacharparenright}{\kern0pt}\ erase{\isacharunderscore}{\kern0pt}fuse{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isanewline
-\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}smt\ {\isacharparenleft}{\kern0pt}z{\isadigit{3}}{\isacharparenright}{\kern0pt}\ Un{\isacharunderscore}{\kern0pt}iff\ bnullable{\isacharunderscore}{\kern0pt}correctness\ insert{\isacharunderscore}{\kern0pt}iff\ list{\isachardot}{\kern0pt}set{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}\ qq{\isadigit{3}}\ set{\isacharunderscore}{\kern0pt}append{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ rewrite{\isacharunderscore}{\kern0pt}non{\isacharunderscore}{\kern0pt}nullable{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymlbrakk}r{\isadigit{1}}\ {\isasymleadsto}\ r{\isadigit{2}}{\isacharsemicolon}{\kern0pt}\ {\isasymnot}bnullable\ r{\isadigit{1}}\ {\isasymrbrakk}\ {\isasymLongrightarrow}\ {\isasymnot}bnullable\ r{\isadigit{2}}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}induction\ r{\isadigit{1}}\ r{\isadigit{2}}\ rule{\isacharcolon}{\kern0pt}\ rrewrite{\isachardot}{\kern0pt}induct{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-\ auto\ \isanewline
-\ \ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ bnullable{\isacharunderscore}{\kern0pt}correctness\ erase{\isacharunderscore}{\kern0pt}fuse{\isacharparenright}{\kern0pt}{\isacharplus}{\kern0pt}\isanewline
-\ \ \isacommand{done}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ rewritesnullable{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymlbrakk}\ r{\isadigit{1}}\ {\isasymleadsto}{\isacharasterisk}{\kern0pt}\ r{\isadigit{2}}{\isacharsemicolon}{\kern0pt}\ bnullable\ r{\isadigit{1}}\ {\isasymrbrakk}\ {\isasymLongrightarrow}\ bnullable\ r{\isadigit{2}}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}induction\ r{\isadigit{1}}\ r{\isadigit{2}}\ rule{\isacharcolon}{\kern0pt}\ rrewrites{\isachardot}{\kern0pt}induct{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}rule\ rewritenullable{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{done}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ nonbnullable{\isacharunderscore}{\kern0pt}lists{\isacharunderscore}{\kern0pt}concat{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}\ {\isasymlbrakk}\ {\isasymnot}\ {\isacharparenleft}{\kern0pt}{\isasymexists}r{\isadigit{0}}{\isasymin}set\ rs{\isadigit{1}}{\isachardot}{\kern0pt}\ bnullable\ r{\isadigit{0}}{\isacharparenright}{\kern0pt}{\isacharsemicolon}{\kern0pt}\ {\isasymnot}\ bnullable\ r{\isacharsemicolon}{\kern0pt}\ {\isasymnot}\ {\isacharparenleft}{\kern0pt}{\isasymexists}r{\isadigit{0}}{\isasymin}set\ rs{\isadigit{2}}{\isachardot}{\kern0pt}\ bnullable\ r{\isadigit{0}}{\isacharparenright}{\kern0pt}{\isasymrbrakk}\ {\isasymLongrightarrow}\ \isanewline
-{\isasymnot}{\isacharparenleft}{\kern0pt}{\isasymexists}r{\isadigit{0}}\ {\isasymin}\ {\isacharparenleft}{\kern0pt}set\ {\isacharparenleft}{\kern0pt}rs{\isadigit{1}}{\isacharat}{\kern0pt}{\isacharbrackleft}{\kern0pt}r{\isacharbrackright}{\kern0pt}{\isacharat}{\kern0pt}rs{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardot}{\kern0pt}\ bnullable\ r{\isadigit{0}}\ {\isacharparenright}{\kern0pt}\ {\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{apply}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{done}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ nomember{\isacharunderscore}{\kern0pt}bnullable{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymlbrakk}\ {\isasymnot}\ {\isacharparenleft}{\kern0pt}{\isasymexists}r{\isadigit{0}}{\isasymin}set\ rs{\isadigit{1}}{\isachardot}{\kern0pt}\ bnullable\ r{\isadigit{0}}{\isacharparenright}{\kern0pt}{\isacharsemicolon}{\kern0pt}\ {\isasymnot}\ bnullable\ r{\isacharsemicolon}{\kern0pt}\ {\isasymnot}\ {\isacharparenleft}{\kern0pt}{\isasymexists}r{\isadigit{0}}{\isasymin}set\ rs{\isadigit{2}}{\isachardot}{\kern0pt}\ bnullable\ r{\isadigit{0}}{\isacharparenright}{\kern0pt}{\isasymrbrakk}\isanewline
-\ {\isasymLongrightarrow}\ {\isasymnot}bnullable\ {\isacharparenleft}{\kern0pt}AALTs\ bs\ {\isacharparenleft}{\kern0pt}rs{\isadigit{1}}\ {\isacharat}{\kern0pt}\ {\isacharbrackleft}{\kern0pt}r{\isacharbrackright}{\kern0pt}\ {\isacharat}{\kern0pt}\ rs{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ nonbnullable{\isacharunderscore}{\kern0pt}lists{\isacharunderscore}{\kern0pt}concat\ qq{\isadigit{3}}\ \isacommand{by}\isamarkupfalse%
-\ presburger%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ bnullable{\isacharunderscore}{\kern0pt}segment{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}\ bnullable\ {\isacharparenleft}{\kern0pt}AALTs\ bs\ {\isacharparenleft}{\kern0pt}rs{\isadigit{1}}{\isacharat}{\kern0pt}{\isacharbrackleft}{\kern0pt}r{\isacharbrackright}{\kern0pt}{\isacharat}{\kern0pt}rs{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymLongrightarrow}\ bnullable\ {\isacharparenleft}{\kern0pt}AALTs\ bs\ rs{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isasymor}\ bnullable\ {\isacharparenleft}{\kern0pt}AALTs\ bs\ rs{\isadigit{2}}{\isacharparenright}{\kern0pt}\ {\isasymor}\ bnullable\ r{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}case{\isacharunderscore}{\kern0pt}tac\ {\isachardoublequoteopen}{\isasymexists}r{\isadigit{0}}{\isasymin}set\ rs{\isadigit{1}}{\isachardot}{\kern0pt}\ \ bnullable\ r{\isadigit{0}}{\isachardoublequoteclose}{\isacharparenright}{\kern0pt}\isanewline
-\isanewline
-\ \ \isacommand{using}\isamarkupfalse%
-\ qq{\isadigit{3}}\ \isacommand{apply}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}case{\isacharunderscore}{\kern0pt}tac\ {\isachardoublequoteopen}bnullable\ r{\isachardoublequoteclose}{\isacharparenright}{\kern0pt}\isanewline
-\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}case{\isacharunderscore}{\kern0pt}tac\ {\isachardoublequoteopen}{\isasymexists}r{\isadigit{0}}{\isasymin}set\ rs{\isadigit{2}}{\isachardot}{\kern0pt}\ \ bnullable\ r{\isadigit{0}}{\isachardoublequoteclose}{\isacharparenright}{\kern0pt}\isanewline
-\isanewline
-\ \ \isacommand{using}\isamarkupfalse%
-\ bnullable{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{4}}{\isacharparenright}{\kern0pt}\ \isacommand{apply}\isamarkupfalse%
-\ presburger\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}subgoal{\isacharunderscore}{\kern0pt}tac\ {\isachardoublequoteopen}False{\isachardoublequoteclose}{\isacharparenright}{\kern0pt}\isanewline
-\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-\ blast\isanewline
-\isanewline
-\ \ \isacommand{using}\isamarkupfalse%
-\ nomember{\isacharunderscore}{\kern0pt}bnullable\ \isacommand{by}\isamarkupfalse%
-\ blast%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\ \ \isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ bnullablewhichbmkeps{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymlbrakk}bnullable\ \ {\isacharparenleft}{\kern0pt}AALTs\ bs\ {\isacharparenleft}{\kern0pt}rs{\isadigit{1}}{\isacharat}{\kern0pt}{\isacharbrackleft}{\kern0pt}r{\isacharbrackright}{\kern0pt}{\isacharat}{\kern0pt}rs{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharsemicolon}{\kern0pt}\ {\isasymnot}\ bnullable\ {\isacharparenleft}{\kern0pt}AALTs\ bs\ rs{\isadigit{1}}{\isacharparenright}{\kern0pt}{\isacharsemicolon}{\kern0pt}\ bnullable\ r\ {\isasymrbrakk}\isanewline
-\ {\isasymLongrightarrow}\ bmkeps\ {\isacharparenleft}{\kern0pt}AALTs\ bs\ {\isacharparenleft}{\kern0pt}rs{\isadigit{1}}{\isacharat}{\kern0pt}{\isacharbrackleft}{\kern0pt}r{\isacharbrackright}{\kern0pt}{\isacharat}{\kern0pt}rs{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ bs\ {\isacharat}{\kern0pt}\ {\isacharparenleft}{\kern0pt}bmkeps\ r{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ qq{\isadigit{2}}\ bnullable{\isacharunderscore}{\kern0pt}Hdbmkeps{\isacharunderscore}{\kern0pt}Hd\ \isacommand{by}\isamarkupfalse%
-\ force%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ rrewrite{\isacharunderscore}{\kern0pt}nbnullable{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymlbrakk}\ r{\isadigit{1}}\ {\isasymleadsto}\ r{\isadigit{2}}\ {\isacharsemicolon}{\kern0pt}\ {\isasymnot}\ bnullable\ r{\isadigit{1}}\ {\isasymrbrakk}\ {\isasymLongrightarrow}\ {\isasymnot}bnullable\ r{\isadigit{2}}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}induction\ rule{\isacharcolon}{\kern0pt}\ rrewrite{\isachardot}{\kern0pt}induct{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-\ auto{\isacharbrackleft}{\kern0pt}{\isadigit{1}}{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-\ auto{\isacharbrackleft}{\kern0pt}{\isadigit{1}}{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-\ auto{\isacharbrackleft}{\kern0pt}{\isadigit{1}}{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ bnullable{\isacharunderscore}{\kern0pt}correctness\ erase{\isacharunderscore}{\kern0pt}fuse{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-\ auto{\isacharbrackleft}{\kern0pt}{\isadigit{1}}{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-\ auto{\isacharbrackleft}{\kern0pt}{\isadigit{1}}{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-\ auto{\isacharbrackleft}{\kern0pt}{\isadigit{1}}{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-\ auto{\isacharbrackleft}{\kern0pt}{\isadigit{1}}{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-\ auto{\isacharbrackleft}{\kern0pt}{\isadigit{1}}{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ bnullable{\isacharunderscore}{\kern0pt}correctness\ erase{\isacharunderscore}{\kern0pt}fuse{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-\ auto{\isacharbrackleft}{\kern0pt}{\isadigit{1}}{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ bnullable{\isacharunderscore}{\kern0pt}correctness\ erase{\isacharunderscore}{\kern0pt}fuse{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{apply}\isamarkupfalse%
-\ auto{\isacharbrackleft}{\kern0pt}{\isadigit{1}}{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{apply}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ bnullable{\isacharunderscore}{\kern0pt}correctness\ erase{\isacharunderscore}{\kern0pt}fuse{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-\ auto{\isacharbrackleft}{\kern0pt}{\isadigit{1}}{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-\ auto{\isacharbrackleft}{\kern0pt}{\isadigit{1}}{\isacharbrackright}{\kern0pt}\isanewline
-\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ bnullable{\isacharunderscore}{\kern0pt}correctness\ erase{\isacharunderscore}{\kern0pt}fuse{\isacharparenright}{\kern0pt}\isanewline
-\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}meson\ rewrite{\isacharunderscore}{\kern0pt}non{\isacharunderscore}{\kern0pt}nullable\ rrewrite{\isachardot}{\kern0pt}intros{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isadigit{3}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ spillbmkepslistr{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}bnullable\ {\isacharparenleft}{\kern0pt}AALTs\ bs{\isadigit{1}}\ rs{\isadigit{1}}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ {\isasymLongrightarrow}\ bmkeps\ {\isacharparenleft}{\kern0pt}AALTs\ bs\ {\isacharparenleft}{\kern0pt}AALTs\ bs{\isadigit{1}}\ rs{\isadigit{1}}\ {\isacharhash}{\kern0pt}\ rsb{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ bmkeps\ {\isacharparenleft}{\kern0pt}AALTs\ bs\ {\isacharparenleft}{\kern0pt}\ map\ {\isacharparenleft}{\kern0pt}fuse\ bs{\isadigit{1}}{\isacharparenright}{\kern0pt}\ rs{\isadigit{1}}\ {\isacharat}{\kern0pt}\ rsb{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}subst\ bnullable{\isacharunderscore}{\kern0pt}Hdbmkeps{\isacharunderscore}{\kern0pt}Hd{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ bmkeps{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{3}}{\isacharparenright}{\kern0pt}\ k{\isadigit{0}}a\ list{\isachardot}{\kern0pt}set{\isacharunderscore}{\kern0pt}intros{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}\ qq{\isadigit{1}}\ qq{\isadigit{4}}\ qs{\isadigit{3}}{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ third{\isacharunderscore}{\kern0pt}segment{\isacharunderscore}{\kern0pt}bnullable{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymlbrakk}bnullable\ {\isacharparenleft}{\kern0pt}AALTs\ bs\ {\isacharparenleft}{\kern0pt}rs{\isadigit{1}}{\isacharat}{\kern0pt}rs{\isadigit{2}}{\isacharat}{\kern0pt}rs{\isadigit{3}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharsemicolon}{\kern0pt}\ {\isasymnot}bnullable\ {\isacharparenleft}{\kern0pt}AALTs\ bs\ rs{\isadigit{1}}{\isacharparenright}{\kern0pt}{\isacharsemicolon}{\kern0pt}\ {\isasymnot}bnullable\ {\isacharparenleft}{\kern0pt}AALTs\ bs\ rs{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isasymrbrakk}\ {\isasymLongrightarrow}\ \isanewline
-bnullable\ {\isacharparenleft}{\kern0pt}AALTs\ bs\ rs{\isadigit{3}}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ \isanewline
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ append{\isachardot}{\kern0pt}left{\isacharunderscore}{\kern0pt}neutral\ append{\isacharunderscore}{\kern0pt}Cons\ bnullable{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}\ bnullable{\isacharunderscore}{\kern0pt}segment\ rrewrite{\isachardot}{\kern0pt}intros{\isacharparenleft}{\kern0pt}{\isadigit{7}}{\isacharparenright}{\kern0pt}\ rrewrite{\isacharunderscore}{\kern0pt}nbnullable{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ third{\isacharunderscore}{\kern0pt}segment{\isacharunderscore}{\kern0pt}bmkeps{\isacharcolon}{\kern0pt}\ \ {\isachardoublequoteopen}{\isasymlbrakk}bnullable\ {\isacharparenleft}{\kern0pt}AALTs\ bs\ {\isacharparenleft}{\kern0pt}rs{\isadigit{1}}{\isacharat}{\kern0pt}rs{\isadigit{2}}{\isacharat}{\kern0pt}rs{\isadigit{3}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharsemicolon}{\kern0pt}\ {\isasymnot}bnullable\ {\isacharparenleft}{\kern0pt}AALTs\ bs\ rs{\isadigit{1}}{\isacharparenright}{\kern0pt}{\isacharsemicolon}{\kern0pt}\ {\isasymnot}bnullable\ {\isacharparenleft}{\kern0pt}AALTs\ bs\ rs{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isasymrbrakk}\ {\isasymLongrightarrow}\ \isanewline
-bmkeps\ {\isacharparenleft}{\kern0pt}AALTs\ bs\ {\isacharparenleft}{\kern0pt}rs{\isadigit{1}}{\isacharat}{\kern0pt}rs{\isadigit{2}}{\isacharat}{\kern0pt}rs{\isadigit{3}}{\isacharparenright}{\kern0pt}\ {\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ bmkeps\ {\isacharparenleft}{\kern0pt}AALTs\ bs\ rs{\isadigit{3}}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}subgoal{\isacharunderscore}{\kern0pt}tac\ {\isachardoublequoteopen}bnullable\ {\isacharparenleft}{\kern0pt}AALTs\ bs\ rs{\isadigit{3}}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}subgoal{\isacharunderscore}{\kern0pt}tac\ {\isachardoublequoteopen}{\isasymforall}r\ {\isasymin}\ set\ {\isacharparenleft}{\kern0pt}rs{\isadigit{1}}{\isacharat}{\kern0pt}rs{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isachardot}{\kern0pt}\ {\isasymnot}bnullable\ r{\isachardoublequoteclose}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}subgoal{\isacharunderscore}{\kern0pt}tac\ {\isachardoublequoteopen}bmkeps\ {\isacharparenleft}{\kern0pt}AALTs\ bs\ {\isacharparenleft}{\kern0pt}rs{\isadigit{1}}{\isacharat}{\kern0pt}rs{\isadigit{2}}{\isacharat}{\kern0pt}rs{\isadigit{3}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ bmkeps\ {\isacharparenleft}{\kern0pt}AALTs\ bs\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}rs{\isadigit{1}}{\isacharat}{\kern0pt}rs{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isacharat}{\kern0pt}rs{\isadigit{3}}{\isacharparenright}{\kern0pt}\ {\isacharparenright}{\kern0pt}{\isachardoublequoteclose}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ qq{\isadigit{2}}\ qq{\isadigit{3}}{\isacharparenright}{\kern0pt}\isanewline
-\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ append{\isachardot}{\kern0pt}assoc{\isacharparenright}{\kern0pt}\isanewline
-\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ append{\isachardot}{\kern0pt}assoc\ in{\isacharunderscore}{\kern0pt}set{\isacharunderscore}{\kern0pt}conv{\isacharunderscore}{\kern0pt}decomp\ r{\isadigit{2}}\ third{\isacharunderscore}{\kern0pt}segment{\isacharunderscore}{\kern0pt}bnullable{\isacharparenright}{\kern0pt}\isanewline
-\isanewline
-\ \ \isacommand{using}\isamarkupfalse%
-\ third{\isacharunderscore}{\kern0pt}segment{\isacharunderscore}{\kern0pt}bnullable\ \isacommand{by}\isamarkupfalse%
-\ blast%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ rewrite{\isacharunderscore}{\kern0pt}bmkepsalt{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}\ {\isasymlbrakk}bnullable\ {\isacharparenleft}{\kern0pt}AALTs\ bs\ {\isacharparenleft}{\kern0pt}rsa\ {\isacharat}{\kern0pt}\ AALTs\ bs{\isadigit{1}}\ rs{\isadigit{1}}\ {\isacharhash}{\kern0pt}\ rsb{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharsemicolon}{\kern0pt}\ bnullable\ {\isacharparenleft}{\kern0pt}AALTs\ bs\ {\isacharparenleft}{\kern0pt}rsa\ {\isacharat}{\kern0pt}\ map\ {\isacharparenleft}{\kern0pt}fuse\ bs{\isadigit{1}}{\isacharparenright}{\kern0pt}\ rs{\isadigit{1}}\ {\isacharat}{\kern0pt}\ rsb{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isasymrbrakk}\isanewline
-\ \ \ \ \ \ \ {\isasymLongrightarrow}\ bmkeps\ {\isacharparenleft}{\kern0pt}AALTs\ bs\ {\isacharparenleft}{\kern0pt}rsa\ {\isacharat}{\kern0pt}\ AALTs\ bs{\isadigit{1}}\ rs{\isadigit{1}}\ {\isacharhash}{\kern0pt}\ rsb{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ bmkeps\ {\isacharparenleft}{\kern0pt}AALTs\ bs\ {\isacharparenleft}{\kern0pt}rsa\ {\isacharat}{\kern0pt}\ map\ {\isacharparenleft}{\kern0pt}fuse\ bs{\isadigit{1}}{\isacharparenright}{\kern0pt}\ rs{\isadigit{1}}\ {\isacharat}{\kern0pt}\ rsb{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}case{\isacharunderscore}{\kern0pt}tac\ {\isachardoublequoteopen}bnullable\ {\isacharparenleft}{\kern0pt}AALTs\ bs\ rsa{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isanewline
-\ \ \isacommand{using}\isamarkupfalse%
-\ qq{\isadigit{1}}\ \isacommand{apply}\isamarkupfalse%
-\ force\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}case{\isacharunderscore}{\kern0pt}tac\ {\isachardoublequoteopen}bnullable\ {\isacharparenleft}{\kern0pt}AALTs\ bs{\isadigit{1}}\ rs{\isadigit{1}}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}subst\ qq{\isadigit{2}}{\isacharparenright}{\kern0pt}\isanewline
-\isanewline
-\ \ \isanewline
-\ \ \isacommand{using}\isamarkupfalse%
-\ r{\isadigit{2}}\ \isacommand{apply}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isanewline
-\ \ \ \ \isacommand{apply}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ list{\isachardot}{\kern0pt}set{\isacharunderscore}{\kern0pt}intros{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}smt\ {\isacharparenleft}{\kern0pt}verit{\isacharcomma}{\kern0pt}\ ccfv{\isacharunderscore}{\kern0pt}threshold{\isacharparenright}{\kern0pt}\ append{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}append{\isacharunderscore}{\kern0pt}conv{\isadigit{2}}\ list{\isachardot}{\kern0pt}set{\isacharunderscore}{\kern0pt}intros{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}\ qq{\isadigit{2}}\ qq{\isadigit{3}}\ rewritenullable\ rrewrite{\isachardot}{\kern0pt}intros{\isacharparenleft}{\kern0pt}{\isadigit{8}}{\isacharparenright}{\kern0pt}\ self{\isacharunderscore}{\kern0pt}append{\isacharunderscore}{\kern0pt}conv{\isadigit{2}}\ spillbmkepslistr{\isacharparenright}{\kern0pt}\isanewline
-\isanewline
-\isanewline
-\ \ \isacommand{thm}\isamarkupfalse%
-\ qq{\isadigit{1}}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}subgoal{\isacharunderscore}{\kern0pt}tac\ {\isachardoublequoteopen}bmkeps\ \ {\isacharparenleft}{\kern0pt}AALTs\ bs\ {\isacharparenleft}{\kern0pt}rsa\ {\isacharat}{\kern0pt}\ AALTs\ bs{\isadigit{1}}\ rs{\isadigit{1}}\ {\isacharhash}{\kern0pt}\ rsb{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ bmkeps\ {\isacharparenleft}{\kern0pt}AALTs\ bs\ rsb{\isacharparenright}{\kern0pt}\ {\isachardoublequoteclose}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \isacommand{prefer}\isamarkupfalse%
-\ {\isadigit{2}}\isanewline
-\ \ \isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ append{\isacharunderscore}{\kern0pt}Cons\ append{\isacharunderscore}{\kern0pt}Nil\ bnullable{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}\ bnullable{\isacharunderscore}{\kern0pt}segment\ rewritenullable\ rrewrite{\isachardot}{\kern0pt}intros{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isadigit{1}}{\isacharparenright}{\kern0pt}\ third{\isacharunderscore}{\kern0pt}segment{\isacharunderscore}{\kern0pt}bmkeps{\isacharparenright}{\kern0pt}\isanewline
-\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ bnullable{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{4}}{\isacharparenright}{\kern0pt}\ rewrite{\isacharunderscore}{\kern0pt}non{\isacharunderscore}{\kern0pt}nullable\ rrewrite{\isachardot}{\kern0pt}intros{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isadigit{0}}{\isacharparenright}{\kern0pt}\ third{\isacharunderscore}{\kern0pt}segment{\isacharunderscore}{\kern0pt}bmkeps{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ rewrite{\isacharunderscore}{\kern0pt}bmkeps{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymlbrakk}\ r{\isadigit{1}}\ {\isasymleadsto}\ r{\isadigit{2}}{\isacharsemicolon}{\kern0pt}\ {\isacharparenleft}{\kern0pt}bnullable\ r{\isadigit{1}}{\isacharparenright}{\kern0pt}{\isasymrbrakk}\ {\isasymLongrightarrow}\ bmkeps\ r{\isadigit{1}}\ {\isacharequal}{\kern0pt}\ bmkeps\ r{\isadigit{2}}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\isanewline
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}frule\ rewritenullable{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}induction\ r{\isadigit{1}}\ r{\isadigit{2}}\ rule{\isacharcolon}{\kern0pt}\ rrewrite{\isachardot}{\kern0pt}induct{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{using}\isamarkupfalse%
-\ bnullable{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}\ bnullable{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{5}}{\isacharparenright}{\kern0pt}\ \isacommand{apply}\isamarkupfalse%
-\ blast\isanewline
-\ \ \ \ \ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ b{\isadigit{2}}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}frule\ bnullable{\isacharunderscore}{\kern0pt}segment{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}case{\isacharunderscore}{\kern0pt}tac\ {\isachardoublequoteopen}bnullable\ {\isacharparenleft}{\kern0pt}AALTs\ bs\ rs{\isadigit{1}}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{using}\isamarkupfalse%
-\ qq{\isadigit{1}}\ \isacommand{apply}\isamarkupfalse%
-\ force\isanewline
-\ \ \ \ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}case{\isacharunderscore}{\kern0pt}tac\ {\isachardoublequoteopen}bnullable\ r{\isachardoublequoteclose}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{using}\isamarkupfalse%
-\ bnullablewhichbmkeps\ rewritenullable\ \isacommand{apply}\isamarkupfalse%
-\ presburger\isanewline
-\ \ \ \ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}subgoal{\isacharunderscore}{\kern0pt}tac\ {\isachardoublequoteopen}bnullable\ {\isacharparenleft}{\kern0pt}AALTs\ bs\ rs{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}subgoal{\isacharunderscore}{\kern0pt}tac\ {\isachardoublequoteopen}{\isasymnot}\ bnullable\ r{\isacharprime}{\kern0pt}{\isachardoublequoteclose}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ qq{\isadigit{2}}\ r{\isadigit{1}}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isanewline
-\ \ \isacommand{using}\isamarkupfalse%
-\ rrewrite{\isacharunderscore}{\kern0pt}nbnullable\ \isacommand{apply}\isamarkupfalse%
-\ blast\isanewline
-\isanewline
-\ \ \ \ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-\ blast\isanewline
-\ \ \ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ flts{\isacharunderscore}{\kern0pt}append\ qs{\isadigit{3}}{\isacharparenright}{\kern0pt}\isanewline
-\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}meson\ rewrite{\isacharunderscore}{\kern0pt}bmkepsalt{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isanewline
-\ \ \isacommand{using}\isamarkupfalse%
-\ bnullable{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{4}}{\isacharparenright}{\kern0pt}\ q{\isadigit{3}}a\ \isacommand{apply}\isamarkupfalse%
-\ blast\isanewline
-\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ q{\isadigit{3}}a{\isacharparenright}{\kern0pt}\isanewline
-\isanewline
-\ \ \isacommand{using}\isamarkupfalse%
-\ bnullable{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}\ \isacommand{apply}\isamarkupfalse%
-\ blast\isanewline
-\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ b{\isadigit{2}}{\isacharparenright}{\kern0pt}\isanewline
-\ \isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}smt\ {\isacharparenleft}{\kern0pt}z{\isadigit{3}}{\isacharparenright}{\kern0pt}\ Un{\isacharunderscore}{\kern0pt}iff\ bnullable{\isacharunderscore}{\kern0pt}correctness\ erase{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{5}}{\isacharparenright}{\kern0pt}\ qq{\isadigit{1}}\ qq{\isadigit{2}}\ qq{\isadigit{3}}\ set{\isacharunderscore}{\kern0pt}append{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ rewrites{\isacharunderscore}{\kern0pt}bmkeps{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymlbrakk}\ {\isacharparenleft}{\kern0pt}r{\isadigit{1}}\ {\isasymleadsto}{\isacharasterisk}{\kern0pt}\ r{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isacharsemicolon}{\kern0pt}\ {\isacharparenleft}{\kern0pt}bnullable\ r{\isadigit{1}}{\isacharparenright}{\kern0pt}{\isasymrbrakk}\ {\isasymLongrightarrow}\ bmkeps\ r{\isadigit{1}}\ {\isacharequal}{\kern0pt}\ bmkeps\ r{\isadigit{2}}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}induction\ r{\isadigit{1}}\ r{\isadigit{2}}\ rule{\isacharcolon}{\kern0pt}\ rrewrites{\isachardot}{\kern0pt}induct{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}subgoal{\isacharunderscore}{\kern0pt}tac\ {\isachardoublequoteopen}bnullable\ r{\isadigit{2}}{\isachardoublequoteclose}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{prefer}\isamarkupfalse%
-\ {\isadigit{2}}\isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}metis\ rewritesnullable{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}subgoal{\isacharunderscore}{\kern0pt}tac\ {\isachardoublequoteopen}bmkeps\ r{\isadigit{1}}\ {\isacharequal}{\kern0pt}\ bmkeps\ r{\isadigit{2}}{\isachardoublequoteclose}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \isacommand{prefer}\isamarkupfalse%
-\ {\isadigit{2}}\isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-\ fastforce\isanewline
-\ \ \isacommand{using}\isamarkupfalse%
-\ rewrite{\isacharunderscore}{\kern0pt}bmkeps\ \isacommand{by}\isamarkupfalse%
-\ presburger%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{thm}\isamarkupfalse%
-\ rrewrite{\isachardot}{\kern0pt}intros{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isadigit{2}}{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ alts{\isacharunderscore}{\kern0pt}rewrite{\isacharunderscore}{\kern0pt}front{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}r\ {\isasymleadsto}\ r{\isacharprime}{\kern0pt}\ {\isasymLongrightarrow}\ AALTs\ bs\ {\isacharparenleft}{\kern0pt}r\ {\isacharhash}{\kern0pt}\ rs{\isacharparenright}{\kern0pt}\ {\isasymleadsto}\ AALTs\ bs\ {\isacharparenleft}{\kern0pt}r{\isacharprime}{\kern0pt}\ {\isacharhash}{\kern0pt}\ rs{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ append{\isacharunderscore}{\kern0pt}Cons\ append{\isacharunderscore}{\kern0pt}Nil\ rrewrite{\isachardot}{\kern0pt}intros{\isacharparenleft}{\kern0pt}{\isadigit{6}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ alt{\isacharunderscore}{\kern0pt}rewrite{\isacharunderscore}{\kern0pt}front{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}r\ {\isasymleadsto}\ r{\isacharprime}{\kern0pt}\ {\isasymLongrightarrow}\ AALT\ bs\ r\ r{\isadigit{2}}\ {\isasymleadsto}\ AALT\ bs\ r{\isacharprime}{\kern0pt}\ r{\isadigit{2}}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ alts{\isacharunderscore}{\kern0pt}rewrite{\isacharunderscore}{\kern0pt}front\ \isacommand{by}\isamarkupfalse%
-\ blast%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ to{\isacharunderscore}{\kern0pt}zero{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alt{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}\ AALT\ bs\ {\isacharparenleft}{\kern0pt}ASEQ\ {\isacharbrackleft}{\kern0pt}{\isacharbrackright}{\kern0pt}\ AZERO\ r{\isacharparenright}{\kern0pt}\ r{\isadigit{2}}\ {\isasymleadsto}\ \ AALT\ bs\ AZERO\ r{\isadigit{2}}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ alts{\isacharunderscore}{\kern0pt}rewrite{\isacharunderscore}{\kern0pt}front\ rrewrite{\isachardot}{\kern0pt}intros{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ alt{\isacharunderscore}{\kern0pt}remove{\isadigit{0}}{\isacharunderscore}{\kern0pt}front{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}\ AALT\ bs\ AZERO\ r\ {\isasymleadsto}\ AALTs\ bs\ {\isacharbrackleft}{\kern0pt}r{\isacharbrackright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ rrewrite{\isadigit{0}}away{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ alt{\isacharunderscore}{\kern0pt}rewrites{\isacharunderscore}{\kern0pt}back{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}r{\isadigit{2}}\ {\isasymleadsto}{\isacharasterisk}{\kern0pt}\ r{\isadigit{2}}{\isacharprime}{\kern0pt}\ {\isasymLongrightarrow}AALT\ bs\ r{\isadigit{1}}\ r{\isadigit{2}}\ {\isasymleadsto}{\isacharasterisk}{\kern0pt}\ AALT\ bs\ r{\isadigit{1}}\ r{\isadigit{2}}{\isacharprime}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}induction\ r{\isadigit{2}}\ r{\isadigit{2}}{\isacharprime}{\kern0pt}\ arbitrary{\isacharcolon}{\kern0pt}\ bs\ rule{\isacharcolon}{\kern0pt}\ rrewrites{\isachardot}{\kern0pt}induct{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}meson\ rs{\isadigit{1}}\ rs{\isadigit{2}}\ srewrites{\isacharunderscore}{\kern0pt}alt{\isadigit{1}}\ ss{\isadigit{1}}\ ss{\isadigit{2}}{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ rewrite{\isacharunderscore}{\kern0pt}fuse{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}\ r{\isadigit{2}}\ {\isasymleadsto}\ r{\isadigit{3}}\ {\isasymLongrightarrow}\ fuse\ bs\ r{\isadigit{2}}\ {\isasymleadsto}{\isacharasterisk}{\kern0pt}\ fuse\ bs\ r{\isadigit{3}}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}induction\ r{\isadigit{2}}\ r{\isadigit{3}}\ arbitrary{\isacharcolon}{\kern0pt}\ bs\ rule{\isacharcolon}{\kern0pt}\ rrewrite{\isachardot}{\kern0pt}induct{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-\ auto\isanewline
-\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ continuous{\isacharunderscore}{\kern0pt}rewrite{\isacharparenright}{\kern0pt}\isanewline
-\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ r{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}rstar\ rrewrite{\isachardot}{\kern0pt}intros{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\isanewline
-\ \ \ \ \ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ fuse{\isacharunderscore}{\kern0pt}append\ r{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}rstar\ rrewrite{\isachardot}{\kern0pt}intros{\isacharparenleft}{\kern0pt}{\isadigit{3}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\isanewline
-\ \ \isacommand{using}\isamarkupfalse%
-\ r{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}rstar\ star{\isacharunderscore}{\kern0pt}seq\ \isacommand{apply}\isamarkupfalse%
-\ blast\isanewline
-\isanewline
-\ \ \isacommand{using}\isamarkupfalse%
-\ r{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}rstar\ star{\isacharunderscore}{\kern0pt}seq{\isadigit{2}}\ \isacommand{apply}\isamarkupfalse%
-\ blast\isanewline
-\isanewline
-\ \ \isacommand{using}\isamarkupfalse%
-\ contextrewrites{\isadigit{2}}\ r{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}rstar\ \isacommand{apply}\isamarkupfalse%
-\ auto{\isacharbrackleft}{\kern0pt}{\isadigit{1}}{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \isanewline
-\ \ \ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ r{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}rstar\ rrewrite{\isachardot}{\kern0pt}intros{\isacharparenleft}{\kern0pt}{\isadigit{7}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\isanewline
-\ \ \isacommand{using}\isamarkupfalse%
-\ rrewrite{\isachardot}{\kern0pt}intros{\isacharparenleft}{\kern0pt}{\isadigit{8}}{\isacharparenright}{\kern0pt}\ \isacommand{apply}\isamarkupfalse%
-\ auto{\isacharbrackleft}{\kern0pt}{\isadigit{1}}{\isacharbrackright}{\kern0pt}\isanewline
-\isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ append{\isacharunderscore}{\kern0pt}assoc\ r{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}rstar\ rrewrite{\isachardot}{\kern0pt}intros{\isacharparenleft}{\kern0pt}{\isadigit{9}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ append{\isacharunderscore}{\kern0pt}assoc\ r{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}rstar\ rrewrite{\isachardot}{\kern0pt}intros{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isadigit{0}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ r{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}rstar\ rrewrite{\isachardot}{\kern0pt}intros{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isadigit{1}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ fuse{\isacharunderscore}{\kern0pt}append\ r{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}rstar\ rrewrite{\isachardot}{\kern0pt}intros{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\isanewline
-\ \ \isacommand{using}\isamarkupfalse%
-\ rrewrite{\isachardot}{\kern0pt}intros{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isadigit{3}}{\isacharparenright}{\kern0pt}\ \isacommand{by}\isamarkupfalse%
-\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\ \ \isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ rewrites{\isacharunderscore}{\kern0pt}fuse{\isacharcolon}{\kern0pt}\ \ {\isachardoublequoteopen}r{\isadigit{2}}\ {\isasymleadsto}{\isacharasterisk}{\kern0pt}\ r{\isadigit{2}}{\isacharprime}{\kern0pt}\ {\isasymLongrightarrow}\ \ {\isacharparenleft}{\kern0pt}fuse\ bs{\isadigit{1}}\ r{\isadigit{2}}{\isacharparenright}{\kern0pt}\ {\isasymleadsto}{\isacharasterisk}{\kern0pt}\ \ {\isacharparenleft}{\kern0pt}fuse\ bs{\isadigit{1}}\ r{\isadigit{2}}{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}induction\ r{\isadigit{2}}\ r{\isadigit{2}}{\isacharprime}{\kern0pt}\ arbitrary{\isacharcolon}{\kern0pt}\ bs{\isadigit{1}}\ rule{\isacharcolon}{\kern0pt}\ rrewrites{\isachardot}{\kern0pt}induct{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}meson\ real{\isacharunderscore}{\kern0pt}trans\ rewrite{\isacharunderscore}{\kern0pt}fuse{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ \ bder{\isacharunderscore}{\kern0pt}fuse{\isacharunderscore}{\kern0pt}list{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}\ map\ {\isacharparenleft}{\kern0pt}bder\ c\ {\isasymcirc}\ fuse\ bs{\isadigit{1}}{\isacharparenright}{\kern0pt}\ rs{\isadigit{1}}\ {\isacharequal}{\kern0pt}\ map\ {\isacharparenleft}{\kern0pt}fuse\ bs{\isadigit{1}}\ {\isasymcirc}\ bder\ c{\isacharparenright}{\kern0pt}\ rs{\isadigit{1}}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}induction\ rs{\isadigit{1}}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ bder{\isacharunderscore}{\kern0pt}fuse{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ rewrite{\isacharunderscore}{\kern0pt}der{\isacharunderscore}{\kern0pt}altmiddle{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}bder\ c\ {\isacharparenleft}{\kern0pt}AALTs\ bs\ {\isacharparenleft}{\kern0pt}rsa\ {\isacharat}{\kern0pt}\ AALTs\ bs{\isadigit{1}}\ rs{\isadigit{1}}\ {\isacharhash}{\kern0pt}\ rsb{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymleadsto}{\isacharasterisk}{\kern0pt}\ bder\ c\ {\isacharparenleft}{\kern0pt}AALTs\ bs\ {\isacharparenleft}{\kern0pt}rsa\ {\isacharat}{\kern0pt}\ map\ {\isacharparenleft}{\kern0pt}fuse\ bs{\isadigit{1}}{\isacharparenright}{\kern0pt}\ rs{\isadigit{1}}\ {\isacharat}{\kern0pt}\ rsb{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{apply}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ bder{\isacharunderscore}{\kern0pt}fuse{\isacharunderscore}{\kern0pt}list{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}rule\ many{\isacharunderscore}{\kern0pt}steps{\isacharunderscore}{\kern0pt}later{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}subst\ rrewrite{\isachardot}{\kern0pt}intros{\isacharparenleft}{\kern0pt}{\isadigit{8}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-\ simp\isanewline
-\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ fastforce%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ lock{\isacharunderscore}{\kern0pt}step{\isacharunderscore}{\kern0pt}der{\isacharunderscore}{\kern0pt}removal{\isacharcolon}{\kern0pt}\ \isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}\ erase\ a{\isadigit{1}}\ {\isacharequal}{\kern0pt}\ erase\ a{\isadigit{2}}\ {\isasymLongrightarrow}\ \isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ bder\ c\ {\isacharparenleft}{\kern0pt}AALTs\ bs\ {\isacharparenleft}{\kern0pt}rsa\ {\isacharat}{\kern0pt}\ {\isacharbrackleft}{\kern0pt}a{\isadigit{1}}{\isacharbrackright}{\kern0pt}\ {\isacharat}{\kern0pt}\ rsb\ {\isacharat}{\kern0pt}\ {\isacharbrackleft}{\kern0pt}a{\isadigit{2}}{\isacharbrackright}{\kern0pt}\ {\isacharat}{\kern0pt}\ rsc{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymleadsto}{\isacharasterisk}{\kern0pt}\ \isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ bder\ c\ {\isacharparenleft}{\kern0pt}AALTs\ bs\ {\isacharparenleft}{\kern0pt}rsa\ {\isacharat}{\kern0pt}\ {\isacharbrackleft}{\kern0pt}a{\isadigit{1}}{\isacharbrackright}{\kern0pt}\ {\isacharat}{\kern0pt}\ rsb\ {\isacharat}{\kern0pt}\ rsc{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isanewline
-\ \ \isacommand{using}\isamarkupfalse%
-\ rrewrite{\isachardot}{\kern0pt}intros{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isadigit{3}}{\isacharparenright}{\kern0pt}\ \isacommand{by}\isamarkupfalse%
-\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ rewrite{\isacharunderscore}{\kern0pt}after{\isacharunderscore}{\kern0pt}der{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}r{\isadigit{1}}\ {\isasymleadsto}\ r{\isadigit{2}}\ {\isasymLongrightarrow}\ {\isacharparenleft}{\kern0pt}bder\ c\ r{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isasymleadsto}{\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}bder\ c\ r{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}induction\ r{\isadigit{1}}\ r{\isadigit{2}}\ arbitrary{\isacharcolon}{\kern0pt}\ c\ rule{\isacharcolon}{\kern0pt}\ rrewrite{\isachardot}{\kern0pt}induct{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ r{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}rstar\ rrewrite{\isachardot}{\kern0pt}intros{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}meson\ contextrewrites{\isadigit{1}}\ r{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}rstar\ rrewrite{\isachardot}{\kern0pt}intros{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isadigit{1}}{\isacharparenright}{\kern0pt}\ rrewrite{\isachardot}{\kern0pt}intros{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}\ rrewrite{\isadigit{0}}away\ rs{\isadigit{2}}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}rule\ many{\isacharunderscore}{\kern0pt}steps{\isacharunderscore}{\kern0pt}later{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}rule\ to{\isacharunderscore}{\kern0pt}zero{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alt{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}rule\ many{\isacharunderscore}{\kern0pt}steps{\isacharunderscore}{\kern0pt}later{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}rule\ alt{\isacharunderscore}{\kern0pt}remove{\isadigit{0}}{\isacharunderscore}{\kern0pt}front{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}rule\ many{\isacharunderscore}{\kern0pt}steps{\isacharunderscore}{\kern0pt}later{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}rule\ rrewrite{\isachardot}{\kern0pt}intros{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{using}\isamarkupfalse%
-\ bder{\isacharunderscore}{\kern0pt}fuse\ fuse{\isacharunderscore}{\kern0pt}append\ rs{\isadigit{1}}\ \isacommand{apply}\isamarkupfalse%
-\ presburger\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}case{\isacharunderscore}{\kern0pt}tac\ {\isachardoublequoteopen}bnullable\ r{\isadigit{1}}{\isachardoublequoteclose}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{prefer}\isamarkupfalse%
-\ {\isadigit{2}}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}subgoal{\isacharunderscore}{\kern0pt}tac\ {\isachardoublequoteopen}{\isasymnot}bnullable\ r{\isadigit{2}}{\isachardoublequoteclose}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \isacommand{prefer}\isamarkupfalse%
-\ {\isadigit{2}}\isanewline
-\ \ \isacommand{using}\isamarkupfalse%
-\ rewrite{\isacharunderscore}{\kern0pt}non{\isacharunderscore}{\kern0pt}nullable\ \isacommand{apply}\isamarkupfalse%
-\ presburger\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-\ simp{\isacharplus}{\kern0pt}\isanewline
-\ \ \isanewline
-\ \ \isacommand{using}\isamarkupfalse%
-\ star{\isacharunderscore}{\kern0pt}seq\ \isacommand{apply}\isamarkupfalse%
-\ auto{\isacharbrackleft}{\kern0pt}{\isadigit{1}}{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}subgoal{\isacharunderscore}{\kern0pt}tac\ {\isachardoublequoteopen}bnullable\ r{\isadigit{2}}{\isachardoublequoteclose}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-\ simp{\isacharplus}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}subgoal{\isacharunderscore}{\kern0pt}tac\ {\isachardoublequoteopen}bmkeps\ r{\isadigit{1}}\ {\isacharequal}{\kern0pt}\ bmkeps\ r{\isadigit{2}}{\isachardoublequoteclose}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{prefer}\isamarkupfalse%
-\ {\isadigit{2}}\isanewline
-\ \ \isacommand{using}\isamarkupfalse%
-\ rewrite{\isacharunderscore}{\kern0pt}bmkeps\ \isacommand{apply}\isamarkupfalse%
-\ auto{\isacharbrackleft}{\kern0pt}{\isadigit{1}}{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \isacommand{using}\isamarkupfalse%
-\ contextrewrites{\isadigit{1}}\ star{\isacharunderscore}{\kern0pt}seq\ \isacommand{apply}\isamarkupfalse%
-\ auto{\isacharbrackleft}{\kern0pt}{\isadigit{1}}{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \isacommand{using}\isamarkupfalse%
-\ rewritenullable\ \isacommand{apply}\isamarkupfalse%
-\ auto{\isacharbrackleft}{\kern0pt}{\isadigit{1}}{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}case{\isacharunderscore}{\kern0pt}tac\ {\isachardoublequoteopen}bnullable\ r{\isadigit{1}}{\isachardoublequoteclose}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}subgoal{\isacharunderscore}{\kern0pt}tac\ {\isachardoublequoteopen}ASEQ\ {\isacharbrackleft}{\kern0pt}{\isacharbrackright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}bder\ c\ r{\isadigit{1}}{\isacharparenright}{\kern0pt}\ r{\isadigit{3}}\ {\isasymleadsto}\ ASEQ\ {\isacharbrackleft}{\kern0pt}{\isacharbrackright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}bder\ c\ r{\isadigit{1}}{\isacharparenright}{\kern0pt}\ r{\isadigit{4}}{\isachardoublequoteclose}{\isacharparenright}{\kern0pt}\ \isanewline
-\ \ \ \ \ \ \ \ \ \ \ \isacommand{prefer}\isamarkupfalse%
-\ {\isadigit{2}}\isanewline
-\ \ \isacommand{using}\isamarkupfalse%
-\ rrewrite{\isachardot}{\kern0pt}intros{\isacharparenleft}{\kern0pt}{\isadigit{5}}{\isacharparenright}{\kern0pt}\ \isacommand{apply}\isamarkupfalse%
-\ blast\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}rule\ many{\isacharunderscore}{\kern0pt}steps{\isacharunderscore}{\kern0pt}later{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}rule\ alt{\isacharunderscore}{\kern0pt}rewrite{\isacharunderscore}{\kern0pt}front{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-\ assumption\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}meson\ alt{\isacharunderscore}{\kern0pt}rewrites{\isacharunderscore}{\kern0pt}back\ rewrites{\isacharunderscore}{\kern0pt}fuse{\isacharparenright}{\kern0pt}\ \isanewline
-\isanewline
-\ \ \ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ r{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}rstar\ rrewrite{\isachardot}{\kern0pt}intros{\isacharparenleft}{\kern0pt}{\isadigit{5}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\isanewline
-\ \ \isacommand{using}\isamarkupfalse%
-\ contextrewrites{\isadigit{2}}\ \isacommand{apply}\isamarkupfalse%
-\ force\isanewline
-\isanewline
-\ \ \isacommand{using}\isamarkupfalse%
-\ rrewrite{\isachardot}{\kern0pt}intros{\isacharparenleft}{\kern0pt}{\isadigit{7}}{\isacharparenright}{\kern0pt}\ \isacommand{apply}\isamarkupfalse%
-\ force\isanewline
-\ \ \isanewline
-\ \ \isacommand{using}\isamarkupfalse%
-\ rewrite{\isacharunderscore}{\kern0pt}der{\isacharunderscore}{\kern0pt}altmiddle\ \isacommand{apply}\isamarkupfalse%
-\ auto{\isacharbrackleft}{\kern0pt}{\isadigit{1}}{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ bder{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{4}}{\isacharparenright}{\kern0pt}\ bder{\isacharunderscore}{\kern0pt}fuse{\isacharunderscore}{\kern0pt}list\ map{\isacharunderscore}{\kern0pt}map\ r{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}rstar\ rrewrite{\isachardot}{\kern0pt}intros{\isacharparenleft}{\kern0pt}{\isadigit{9}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ List{\isachardot}{\kern0pt}map{\isachardot}{\kern0pt}compositionality\ bder{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{4}}{\isacharparenright}{\kern0pt}\ bder{\isacharunderscore}{\kern0pt}fuse{\isacharunderscore}{\kern0pt}list\ r{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}rstar\ rrewrite{\isachardot}{\kern0pt}intros{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isadigit{0}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ r{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}rstar\ rrewrite{\isachardot}{\kern0pt}intros{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isadigit{1}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ bder{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{4}}{\isacharparenright}{\kern0pt}\ bder{\isacharunderscore}{\kern0pt}bsimp{\isacharunderscore}{\kern0pt}AALTs\ bsimp{\isacharunderscore}{\kern0pt}AALTs{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}\ bsimp{\isacharunderscore}{\kern0pt}AALTsrewrites{\isacharparenright}{\kern0pt}\isanewline
-\isanewline
-\ \ \isanewline
-\ \ \isacommand{using}\isamarkupfalse%
-\ lock{\isacharunderscore}{\kern0pt}step{\isacharunderscore}{\kern0pt}der{\isacharunderscore}{\kern0pt}removal\ \isacommand{by}\isamarkupfalse%
-\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ rewrites{\isacharunderscore}{\kern0pt}after{\isacharunderscore}{\kern0pt}der{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}\ \ r{\isadigit{1}}\ {\isasymleadsto}{\isacharasterisk}{\kern0pt}\ r{\isadigit{2}}\ \ {\isasymLongrightarrow}\ \ {\isacharparenleft}{\kern0pt}bder\ c\ r{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isasymleadsto}{\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}bder\ c\ r{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}induction\ r{\isadigit{1}}\ r{\isadigit{2}}\ rule{\isacharcolon}{\kern0pt}\ rrewrites{\isachardot}{\kern0pt}induct{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}rule\ rs{\isadigit{1}}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}meson\ real{\isacharunderscore}{\kern0pt}trans\ rewrite{\isacharunderscore}{\kern0pt}after{\isacharunderscore}{\kern0pt}der{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\ \ \isanewline
-\isanewline
-\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ central{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}\ {\isacharparenleft}{\kern0pt}bders\ r\ s{\isacharparenright}{\kern0pt}\ {\isasymleadsto}{\isacharasterisk}{\kern0pt}\ \ {\isacharparenleft}{\kern0pt}bders{\isacharunderscore}{\kern0pt}simp\ r\ s{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}induct\ s\ arbitrary{\isacharcolon}{\kern0pt}\ r\ rule{\isacharcolon}{\kern0pt}\ rev{\isacharunderscore}{\kern0pt}induct{\isacharparenright}{\kern0pt}\isanewline
-\isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}subst\ bders{\isacharunderscore}{\kern0pt}append{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}subst\ bders{\isacharunderscore}{\kern0pt}simp{\isacharunderscore}{\kern0pt}append{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ bders{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}\ bders{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}\ bders{\isacharunderscore}{\kern0pt}simp{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}\ bders{\isacharunderscore}{\kern0pt}simp{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}\ bsimp{\isacharunderscore}{\kern0pt}rewrite\ real{\isacharunderscore}{\kern0pt}trans\ rewrites{\isacharunderscore}{\kern0pt}after{\isacharunderscore}{\kern0pt}der{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isanewline
-\isacommand{thm}\isamarkupfalse%
-\ arexp{\isachardot}{\kern0pt}induct\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ quasi{\isacharunderscore}{\kern0pt}main{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}bnullable\ {\isacharparenleft}{\kern0pt}bders\ r\ s{\isacharparenright}{\kern0pt}\ {\isasymLongrightarrow}\ bmkeps\ {\isacharparenleft}{\kern0pt}bders\ r\ s{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ bmkeps\ {\isacharparenleft}{\kern0pt}bders{\isacharunderscore}{\kern0pt}simp\ r\ s{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ central\ rewrites{\isacharunderscore}{\kern0pt}bmkeps\ \isacommand{by}\isamarkupfalse%
-\ blast%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{theorem}\isamarkupfalse%
-\ main{\isacharunderscore}{\kern0pt}main{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}blexer\ r\ s\ {\isacharequal}{\kern0pt}\ blexer{\isacharunderscore}{\kern0pt}simp\ r\ s{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ b{\isadigit{4}}\ blexer{\isacharunderscore}{\kern0pt}def\ blexer{\isacharunderscore}{\kern0pt}simp{\isacharunderscore}{\kern0pt}def\ quasi{\isacharunderscore}{\kern0pt}main{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{theorem}\isamarkupfalse%
-\ blexersimp{\isacharunderscore}{\kern0pt}correctness{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}blexer{\isacharunderscore}{\kern0pt}simp\ r\ s{\isacharequal}{\kern0pt}\ lexer\ r\ s{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ blexer{\isacharunderscore}{\kern0pt}correctness\ main{\isacharunderscore}{\kern0pt}main\ \isacommand{by}\isamarkupfalse%
-\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{unused{\isacharunderscore}{\kern0pt}thms}\isamarkupfalse%
-\isanewline
-\isanewline
-%
-\isadelimtheory
-\isanewline
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{end}\isamarkupfalse%
-%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\end{isabellebody}%
-\endinput
-%:%file=~/Dropbox/Workspace/journalpaper/lexing/thys2/SizeBound.thy%:%
-%:%6=1%:%
-%:%11=2%:%
-%:%12=2%:%
-%:%13=3%:%
-%:%14=4%:%
-%:%28=6%:%
-%:%38=8%:%
-%:%39=8%:%
-%:%40=9%:%
-%:%41=10%:%
-%:%42=10%:%
-%:%43=11%:%
-%:%44=12%:%
-%:%45=13%:%
-%:%46=14%:%
-%:%47=15%:%
-%:%48=16%:%
-%:%49=17%:%
-%:%50=18%:%
-%:%51=19%:%
-%:%52=20%:%
-%:%53=21%:%
-%:%54=21%:%
-%:%55=22%:%
-%:%56=23%:%
-%:%57=24%:%
-%:%58=25%:%
-%:%59=26%:%
-%:%60=26%:%
-%:%61=27%:%
-%:%62=28%:%
-%:%63=29%:%
-%:%64=30%:%
-%:%65=31%:%
-%:%66=32%:%
-%:%67=33%:%
-%:%68=34%:%
-%:%69=35%:%
-%:%70=36%:%
-%:%71=37%:%
-%:%72=38%:%
-%:%73=39%:%
-%:%75=41%:%
-%:%82=42%:%
-%:%83=42%:%
-%:%88=42%:%
-%:%91=43%:%
-%:%92=44%:%
-%:%93=44%:%
-%:%94=45%:%
-%:%95=46%:%
-%:%102=47%:%
-%:%103=47%:%
-%:%104=48%:%
-%:%105=48%:%
-%:%106=49%:%
-%:%107=49%:%
-%:%108=50%:%
-%:%109=50%:%
-%:%110=50%:%
-%:%111=51%:%
-%:%112=51%:%
-%:%117=51%:%
-%:%120=52%:%
-%:%121=53%:%
-%:%122=53%:%
-%:%129=54%:%
-%:%130=54%:%
-%:%131=55%:%
-%:%132=55%:%
-%:%133=56%:%
-%:%134=56%:%
-%:%139=56%:%
-%:%142=57%:%
-%:%143=58%:%
-%:%144=58%:%
-%:%145=59%:%
-%:%146=60%:%
-%:%147=61%:%
-%:%148=62%:%
-%:%149=63%:%
-%:%150=64%:%
-%:%151=64%:%
-%:%152=65%:%
-%:%153=66%:%
-%:%156=67%:%
-%:%160=67%:%
-%:%161=67%:%
-%:%162=68%:%
-%:%163=68%:%
-%:%164=69%:%
-%:%165=69%:%
-%:%166=70%:%
-%:%172=70%:%
-%:%175=71%:%
-%:%176=72%:%
-%:%177=72%:%
-%:%178=73%:%
-%:%179=74%:%
-%:%186=75%:%
-%:%187=75%:%
-%:%188=76%:%
-%:%189=76%:%
-%:%190=77%:%
-%:%191=77%:%
-%:%192=78%:%
-%:%193=78%:%
-%:%194=78%:%
-%:%199=78%:%
-%:%202=79%:%
-%:%203=80%:%
-%:%204=80%:%
-%:%205=81%:%
-%:%206=82%:%
-%:%209=83%:%
-%:%213=83%:%
-%:%214=83%:%
-%:%215=83%:%
-%:%216=84%:%
-%:%217=84%:%
-%:%231=87%:%
-%:%241=89%:%
-%:%242=89%:%
-%:%243=90%:%
-%:%244=91%:%
-%:%245=92%:%
-%:%246=93%:%
-%:%247=94%:%
-%:%248=95%:%
-%:%249=96%:%
-%:%250=97%:%
-%:%251=97%:%
-%:%252=98%:%
-%:%253=99%:%
-%:%254=100%:%
-%:%255=100%:%
-%:%256=101%:%
-%:%257=102%:%
-%:%258=103%:%
-%:%259=104%:%
-%:%260=105%:%
-%:%261=106%:%
-%:%262=107%:%
-%:%263=108%:%
-%:%264=108%:%
-%:%265=109%:%
-%:%266=110%:%
-%:%267=111%:%
-%:%268=112%:%
-%:%269=113%:%
-%:%270=114%:%
-%:%271=115%:%
-%:%272=116%:%
-%:%273=117%:%
-%:%274=118%:%
-%:%275=119%:%
-%:%276=120%:%
-%:%277=121%:%
-%:%278=122%:%
-%:%279=123%:%
-%:%280=123%:%
-%:%281=124%:%
-%:%282=125%:%
-%:%283=126%:%
-%:%284=127%:%
-%:%285=128%:%
-%:%286=129%:%
-%:%287=129%:%
-%:%288=130%:%
-%:%289=131%:%
-%:%290=132%:%
-%:%291=133%:%
-%:%292=134%:%
-%:%293=135%:%
-%:%294=136%:%
-%:%295=137%:%
-%:%296=138%:%
-%:%297=139%:%
-%:%298=140%:%
-%:%299=141%:%
-%:%300=142%:%
-%:%301=143%:%
-%:%302=144%:%
-%:%303=145%:%
-%:%304=145%:%
-%:%305=146%:%
-%:%306=147%:%
-%:%307=148%:%
-%:%308=149%:%
-%:%309=150%:%
-%:%310=151%:%
-%:%311=152%:%
-%:%312=153%:%
-%:%313=153%:%
-%:%314=154%:%
-%:%317=155%:%
-%:%321=155%:%
-%:%322=155%:%
-%:%323=156%:%
-%:%324=156%:%
-%:%325=157%:%
-%:%331=157%:%
-%:%334=158%:%
-%:%335=159%:%
-%:%336=160%:%
-%:%337=160%:%
-%:%338=161%:%
-%:%339=162%:%
-%:%340=163%:%
-%:%341=164%:%
-%:%342=165%:%
-%:%343=166%:%
-%:%344=167%:%
-%:%345=168%:%
-%:%346=169%:%
-%:%347=170%:%
-%:%348=170%:%
-%:%349=171%:%
-%:%350=172%:%
-%:%351=173%:%
-%:%352=174%:%
-%:%353=175%:%
-%:%354=176%:%
-%:%355=177%:%
-%:%356=178%:%
-%:%357=179%:%
-%:%358=180%:%
-%:%359=181%:%
-%:%360=182%:%
-%:%361=183%:%
-%:%362=183%:%
-%:%363=184%:%
-%:%364=185%:%
-%:%365=186%:%
-%:%366=187%:%
-%:%367=188%:%
-%:%368=189%:%
-%:%369=190%:%
-%:%370=191%:%
-%:%371=192%:%
-%:%372=193%:%
-%:%373=193%:%
-%:%374=194%:%
-%:%375=195%:%
-%:%376=196%:%
-%:%377=197%:%
-%:%378=198%:%
-%:%379=199%:%
-%:%380=200%:%
-%:%381=201%:%
-%:%382=202%:%
-%:%383=203%:%
-%:%384=203%:%
-%:%385=204%:%
-%:%386=205%:%
-%:%387=206%:%
-%:%388=207%:%
-%:%389=208%:%
-%:%390=209%:%
-%:%391=210%:%
-%:%394=213%:%
-%:%395=214%:%
-%:%396=215%:%
-%:%397=216%:%
-%:%398=217%:%
-%:%399=217%:%
-%:%400=218%:%
-%:%401=219%:%
-%:%402=220%:%
-%:%403=221%:%
-%:%404=222%:%
-%:%405=223%:%
-%:%406=223%:%
-%:%407=224%:%
-%:%410=225%:%
-%:%414=225%:%
-%:%415=225%:%
-%:%416=226%:%
-%:%417=226%:%
-%:%418=227%:%
-%:%424=227%:%
-%:%427=228%:%
-%:%428=229%:%
-%:%429=229%:%
-%:%430=230%:%
-%:%433=231%:%
-%:%437=231%:%
-%:%438=231%:%
-%:%439=232%:%
-%:%440=232%:%
-%:%441=233%:%
-%:%447=233%:%
-%:%450=234%:%
-%:%451=235%:%
-%:%452=235%:%
-%:%453=236%:%
-%:%456=237%:%
-%:%460=237%:%
-%:%461=237%:%
-%:%462=238%:%
-%:%463=238%:%
-%:%464=239%:%
-%:%470=239%:%
-%:%473=240%:%
-%:%474=241%:%
-%:%475=241%:%
-%:%476=242%:%
-%:%477=243%:%
-%:%478=243%:%
-%:%479=244%:%
-%:%482=245%:%
-%:%486=245%:%
-%:%487=245%:%
-%:%488=246%:%
-%:%489=246%:%
-%:%490=247%:%
-%:%496=247%:%
-%:%499=248%:%
-%:%500=249%:%
-%:%501=249%:%
-%:%502=250%:%
-%:%505=251%:%
-%:%509=251%:%
-%:%510=251%:%
-%:%511=252%:%
-%:%512=252%:%
-%:%513=253%:%
-%:%519=253%:%
-%:%522=254%:%
-%:%523=255%:%
-%:%524=255%:%
-%:%525=256%:%
-%:%528=257%:%
-%:%532=257%:%
-%:%533=257%:%
-%:%534=258%:%
-%:%535=258%:%
-%:%536=259%:%
-%:%542=259%:%
-%:%545=260%:%
-%:%546=261%:%
-%:%547=261%:%
-%:%548=262%:%
-%:%549=263%:%
-%:%552=264%:%
-%:%556=264%:%
-%:%557=264%:%
-%:%558=265%:%
-%:%559=265%:%
-%:%560=266%:%
-%:%561=266%:%
-%:%562=267%:%
-%:%568=267%:%
-%:%571=268%:%
-%:%572=269%:%
-%:%573=270%:%
-%:%574=270%:%
-%:%575=271%:%
-%:%576=272%:%
-%:%579=273%:%
-%:%583=273%:%
-%:%584=273%:%
-%:%585=274%:%
-%:%586=274%:%
-%:%587=275%:%
-%:%588=275%:%
-%:%589=276%:%
-%:%590=276%:%
-%:%591=277%:%
-%:%592=277%:%
-%:%593=278%:%
-%:%594=278%:%
-%:%595=279%:%
-%:%596=279%:%
-%:%597=280%:%
-%:%598=280%:%
-%:%599=281%:%
-%:%600=281%:%
-%:%601=282%:%
-%:%602=283%:%
-%:%603=283%:%
-%:%604=284%:%
-%:%605=284%:%
-%:%606=285%:%
-%:%607=285%:%
-%:%608=286%:%
-%:%609=286%:%
-%:%610=287%:%
-%:%611=287%:%
-%:%612=288%:%
-%:%613=288%:%
-%:%614=289%:%
-%:%615=289%:%
-%:%616=290%:%
-%:%617=290%:%
-%:%618=291%:%
-%:%619=291%:%
-%:%620=292%:%
-%:%626=292%:%
-%:%629=293%:%
-%:%630=294%:%
-%:%631=294%:%
-%:%632=295%:%
-%:%633=296%:%
-%:%636=297%:%
-%:%640=297%:%
-%:%641=297%:%
-%:%642=298%:%
-%:%643=298%:%
-%:%648=298%:%
-%:%651=299%:%
-%:%652=300%:%
-%:%653=301%:%
-%:%654=301%:%
-%:%655=302%:%
-%:%656=303%:%
-%:%659=304%:%
-%:%663=304%:%
-%:%664=304%:%
-%:%665=305%:%
-%:%666=305%:%
-%:%667=306%:%
-%:%668=306%:%
-%:%669=307%:%
-%:%675=307%:%
-%:%678=308%:%
-%:%679=309%:%
-%:%680=310%:%
-%:%681=310%:%
-%:%682=311%:%
-%:%683=312%:%
-%:%686=313%:%
-%:%690=313%:%
-%:%691=313%:%
-%:%692=314%:%
-%:%693=314%:%
-%:%698=314%:%
-%:%701=315%:%
-%:%702=316%:%
-%:%703=316%:%
-%:%704=317%:%
-%:%705=318%:%
-%:%708=319%:%
-%:%712=319%:%
-%:%713=319%:%
-%:%714=320%:%
-%:%715=320%:%
-%:%716=321%:%
-%:%717=321%:%
-%:%718=322%:%
-%:%724=322%:%
-%:%727=323%:%
-%:%728=324%:%
-%:%729=324%:%
-%:%730=325%:%
-%:%731=326%:%
-%:%734=327%:%
-%:%738=327%:%
-%:%739=327%:%
-%:%740=328%:%
-%:%741=328%:%
-%:%742=329%:%
-%:%743=329%:%
-%:%744=330%:%
-%:%750=330%:%
-%:%753=331%:%
-%:%754=332%:%
-%:%755=332%:%
-%:%756=333%:%
-%:%757=334%:%
-%:%758=335%:%
-%:%759=336%:%
-%:%762=337%:%
-%:%766=337%:%
-%:%767=337%:%
-%:%768=338%:%
-%:%769=338%:%
-%:%770=339%:%
-%:%771=339%:%
-%:%772=340%:%
-%:%773=340%:%
-%:%774=341%:%
-%:%775=341%:%
-%:%776=341%:%
-%:%777=342%:%
-%:%778=342%:%
-%:%779=343%:%
-%:%780=343%:%
-%:%781=344%:%
-%:%782=344%:%
-%:%783=345%:%
-%:%784=345%:%
-%:%785=346%:%
-%:%786=346%:%
-%:%787=347%:%
-%:%788=347%:%
-%:%789=348%:%
-%:%790=348%:%
-%:%791=349%:%
-%:%792=349%:%
-%:%793=350%:%
-%:%794=350%:%
-%:%795=351%:%
-%:%796=351%:%
-%:%797=352%:%
-%:%798=352%:%
-%:%799=353%:%
-%:%800=353%:%
-%:%801=354%:%
-%:%802=354%:%
-%:%803=355%:%
-%:%804=355%:%
-%:%805=356%:%
-%:%806=356%:%
-%:%807=357%:%
-%:%808=357%:%
-%:%809=358%:%
-%:%810=358%:%
-%:%811=359%:%
-%:%812=359%:%
-%:%813=360%:%
-%:%814=360%:%
-%:%815=361%:%
-%:%816=361%:%
-%:%817=362%:%
-%:%818=362%:%
-%:%819=363%:%
-%:%825=363%:%
-%:%828=364%:%
-%:%829=365%:%
-%:%830=366%:%
-%:%831=366%:%
-%:%832=367%:%
-%:%833=368%:%
-%:%834=369%:%
-%:%837=370%:%
-%:%841=370%:%
-%:%842=370%:%
-%:%843=371%:%
-%:%844=371%:%
-%:%845=372%:%
-%:%846=372%:%
-%:%847=373%:%
-%:%848=373%:%
-%:%849=374%:%
-%:%850=374%:%
-%:%851=375%:%
-%:%852=375%:%
-%:%853=376%:%
-%:%854=376%:%
-%:%855=377%:%
-%:%856=377%:%
-%:%857=378%:%
-%:%858=378%:%
-%:%859=379%:%
-%:%860=379%:%
-%:%861=380%:%
-%:%862=380%:%
-%:%863=381%:%
-%:%864=381%:%
-%:%865=382%:%
-%:%866=382%:%
-%:%867=383%:%
-%:%868=383%:%
-%:%869=384%:%
-%:%870=384%:%
-%:%871=385%:%
-%:%872=385%:%
-%:%873=386%:%
-%:%874=386%:%
-%:%875=387%:%
-%:%876=387%:%
-%:%877=388%:%
-%:%878=388%:%
-%:%879=389%:%
-%:%880=389%:%
-%:%881=390%:%
-%:%882=390%:%
-%:%883=390%:%
-%:%884=391%:%
-%:%885=391%:%
-%:%886=392%:%
-%:%887=392%:%
-%:%888=393%:%
-%:%889=393%:%
-%:%890=394%:%
-%:%891=394%:%
-%:%892=394%:%
-%:%893=395%:%
-%:%894=395%:%
-%:%895=396%:%
-%:%896=396%:%
-%:%897=397%:%
-%:%898=397%:%
-%:%899=398%:%
-%:%900=398%:%
-%:%901=399%:%
-%:%902=399%:%
-%:%903=399%:%
-%:%904=400%:%
-%:%905=400%:%
-%:%906=401%:%
-%:%907=401%:%
-%:%908=401%:%
-%:%913=401%:%
-%:%916=402%:%
-%:%917=403%:%
-%:%918=403%:%
-%:%919=404%:%
-%:%920=405%:%
-%:%923=406%:%
-%:%927=406%:%
-%:%928=406%:%
-%:%929=407%:%
-%:%930=407%:%
-%:%931=408%:%
-%:%932=408%:%
-%:%933=409%:%
-%:%934=409%:%
-%:%935=410%:%
-%:%936=410%:%
-%:%937=411%:%
-%:%938=411%:%
-%:%939=412%:%
-%:%940=412%:%
-%:%941=413%:%
-%:%942=413%:%
-%:%943=414%:%
-%:%944=414%:%
-%:%945=415%:%
-%:%946=415%:%
-%:%947=416%:%
-%:%953=416%:%
-%:%956=417%:%
-%:%957=418%:%
-%:%958=418%:%
-%:%959=419%:%
-%:%960=420%:%
-%:%963=421%:%
-%:%967=421%:%
-%:%968=421%:%
-%:%969=422%:%
-%:%970=422%:%
-%:%971=423%:%
-%:%972=423%:%
-%:%973=424%:%
-%:%974=424%:%
-%:%975=425%:%
-%:%976=425%:%
-%:%977=426%:%
-%:%978=426%:%
-%:%979=427%:%
-%:%980=427%:%
-%:%981=428%:%
-%:%982=428%:%
-%:%983=429%:%
-%:%984=429%:%
-%:%985=430%:%
-%:%986=430%:%
-%:%987=431%:%
-%:%988=431%:%
-%:%989=432%:%
-%:%990=432%:%
-%:%991=433%:%
-%:%992=433%:%
-%:%993=434%:%
-%:%994=434%:%
-%:%995=435%:%
-%:%996=435%:%
-%:%997=436%:%
-%:%998=436%:%
-%:%999=437%:%
-%:%1000=437%:%
-%:%1001=438%:%
-%:%1002=438%:%
-%:%1003=439%:%
-%:%1004=439%:%
-%:%1005=440%:%
-%:%1006=440%:%
-%:%1007=441%:%
-%:%1008=441%:%
-%:%1009=442%:%
-%:%1010=442%:%
-%:%1011=443%:%
-%:%1012=443%:%
-%:%1013=444%:%
-%:%1014=444%:%
-%:%1015=445%:%
-%:%1016=445%:%
-%:%1017=446%:%
-%:%1018=446%:%
-%:%1019=447%:%
-%:%1020=447%:%
-%:%1021=448%:%
-%:%1022=448%:%
-%:%1023=449%:%
-%:%1024=449%:%
-%:%1025=450%:%
-%:%1026=450%:%
-%:%1027=451%:%
-%:%1028=451%:%
-%:%1029=452%:%
-%:%1030=452%:%
-%:%1031=452%:%
-%:%1032=453%:%
-%:%1033=453%:%
-%:%1034=454%:%
-%:%1035=454%:%
-%:%1036=455%:%
-%:%1037=455%:%
-%:%1038=456%:%
-%:%1039=456%:%
-%:%1040=457%:%
-%:%1041=457%:%
-%:%1042=458%:%
-%:%1043=458%:%
-%:%1044=458%:%
-%:%1045=459%:%
-%:%1046=459%:%
-%:%1047=460%:%
-%:%1048=460%:%
-%:%1049=461%:%
-%:%1050=461%:%
-%:%1051=462%:%
-%:%1052=462%:%
-%:%1053=463%:%
-%:%1054=463%:%
-%:%1055=464%:%
-%:%1056=464%:%
-%:%1057=465%:%
-%:%1058=465%:%
-%:%1059=465%:%
-%:%1060=466%:%
-%:%1061=466%:%
-%:%1062=467%:%
-%:%1063=467%:%
-%:%1064=468%:%
-%:%1065=468%:%
-%:%1066=469%:%
-%:%1067=469%:%
-%:%1068=470%:%
-%:%1069=470%:%
-%:%1070=471%:%
-%:%1071=471%:%
-%:%1072=472%:%
-%:%1073=472%:%
-%:%1074=473%:%
-%:%1075=473%:%
-%:%1076=474%:%
-%:%1077=474%:%
-%:%1078=475%:%
-%:%1079=475%:%
-%:%1080=476%:%
-%:%1081=476%:%
-%:%1082=477%:%
-%:%1083=477%:%
-%:%1084=478%:%
-%:%1090=478%:%
-%:%1093=479%:%
-%:%1094=480%:%
-%:%1095=481%:%
-%:%1096=482%:%
-%:%1097=482%:%
-%:%1098=483%:%
-%:%1099=484%:%
-%:%1102=485%:%
-%:%1106=485%:%
-%:%1107=485%:%
-%:%1108=486%:%
-%:%1109=486%:%
-%:%1110=487%:%
-%:%1111=487%:%
-%:%1112=488%:%
-%:%1113=488%:%
-%:%1114=488%:%
-%:%1115=489%:%
-%:%1116=489%:%
-%:%1117=489%:%
-%:%1118=489%:%
-%:%1119=490%:%
-%:%1120=490%:%
-%:%1121=490%:%
-%:%1122=491%:%
-%:%1123=491%:%
-%:%1124=491%:%
-%:%1125=492%:%
-%:%1126=492%:%
-%:%1127=492%:%
-%:%1128=493%:%
-%:%1129=493%:%
-%:%1130=494%:%
-%:%1131=494%:%
-%:%1132=495%:%
-%:%1133=495%:%
-%:%1134=496%:%
-%:%1135=496%:%
-%:%1136=497%:%
-%:%1137=497%:%
-%:%1138=498%:%
-%:%1139=498%:%
-%:%1140=498%:%
-%:%1141=499%:%
-%:%1142=499%:%
-%:%1143=499%:%
-%:%1144=500%:%
-%:%1145=500%:%
-%:%1146=501%:%
-%:%1147=501%:%
-%:%1148=502%:%
-%:%1149=502%:%
-%:%1150=503%:%
-%:%1151=503%:%
-%:%1152=503%:%
-%:%1153=504%:%
-%:%1154=504%:%
-%:%1155=504%:%
-%:%1156=505%:%
-%:%1157=505%:%
-%:%1158=505%:%
-%:%1159=506%:%
-%:%1160=506%:%
-%:%1161=506%:%
-%:%1162=507%:%
-%:%1163=507%:%
-%:%1164=507%:%
-%:%1165=508%:%
-%:%1166=508%:%
-%:%1167=509%:%
-%:%1173=509%:%
-%:%1176=510%:%
-%:%1177=511%:%
-%:%1178=512%:%
-%:%1179=512%:%
-%:%1180=513%:%
-%:%1181=514%:%
-%:%1182=515%:%
-%:%1183=516%:%
-%:%1184=517%:%
-%:%1185=517%:%
-%:%1186=518%:%
-%:%1187=519%:%
-%:%1188=520%:%
-%:%1189=521%:%
-%:%1190=521%:%
-%:%1191=522%:%
-%:%1198=523%:%
-%:%1199=523%:%
-%:%1200=524%:%
-%:%1201=524%:%
-%:%1202=524%:%
-%:%1203=525%:%
-%:%1204=525%:%
-%:%1205=526%:%
-%:%1206=526%:%
-%:%1207=527%:%
-%:%1208=527%:%
-%:%1209=528%:%
-%:%1210=528%:%
-%:%1211=528%:%
-%:%1212=529%:%
-%:%1213=529%:%
-%:%1214=530%:%
-%:%1215=530%:%
-%:%1216=530%:%
-%:%1217=531%:%
-%:%1218=531%:%
-%:%1219=532%:%
-%:%1220=532%:%
-%:%1221=533%:%
-%:%1222=533%:%
-%:%1223=533%:%
-%:%1224=534%:%
-%:%1225=534%:%
-%:%1226=534%:%
-%:%1227=535%:%
-%:%1228=535%:%
-%:%1229=535%:%
-%:%1230=536%:%
-%:%1231=536%:%
-%:%1232=536%:%
-%:%1233=537%:%
-%:%1234=537%:%
-%:%1235=537%:%
-%:%1236=538%:%
-%:%1237=538%:%
-%:%1238=539%:%
-%:%1239=539%:%
-%:%1240=539%:%
-%:%1241=540%:%
-%:%1242=540%:%
-%:%1243=541%:%
-%:%1244=541%:%
-%:%1245=542%:%
-%:%1246=542%:%
-%:%1247=543%:%
-%:%1248=543%:%
-%:%1249=544%:%
-%:%1255=544%:%
-%:%1258=545%:%
-%:%1259=546%:%
-%:%1260=547%:%
-%:%1261=547%:%
-%:%1262=548%:%
-%:%1263=549%:%
-%:%1264=550%:%
-%:%1266=552%:%
-%:%1267=553%:%
-%:%1268=554%:%
-%:%1269=555%:%
-%:%1270=556%:%
-%:%1271=557%:%
-%:%1272=557%:%
-%:%1273=558%:%
-%:%1274=559%:%
-%:%1275=560%:%
-%:%1276=561%:%
-%:%1277=562%:%
-%:%1278=563%:%
-%:%1279=564%:%
-%:%1280=565%:%
-%:%1281=566%:%
-%:%1282=567%:%
-%:%1283=567%:%
-%:%1284=568%:%
-%:%1285=569%:%
-%:%1286=570%:%
-%:%1287=571%:%
-%:%1288=572%:%
-%:%1289=573%:%
-%:%1290=574%:%
-%:%1291=575%:%
-%:%1292=576%:%
-%:%1293=576%:%
-%:%1294=577%:%
-%:%1295=578%:%
-%:%1296=579%:%
-%:%1297=580%:%
-%:%1298=581%:%
-%:%1299=582%:%
-%:%1300=583%:%
-%:%1301=584%:%
-%:%1302=584%:%
-%:%1303=585%:%
-%:%1304=586%:%
-%:%1305=587%:%
-%:%1306=588%:%
-%:%1307=589%:%
-%:%1308=590%:%
-%:%1309=591%:%
-%:%1310=591%:%
-%:%1311=592%:%
-%:%1312=593%:%
-%:%1313=594%:%
-%:%1314=595%:%
-%:%1315=596%:%
-%:%1316=597%:%
-%:%1317=598%:%
-%:%1318=599%:%
-%:%1319=600%:%
-%:%1320=600%:%
-%:%1321=601%:%
-%:%1322=602%:%
-%:%1323=603%:%
-%:%1324=604%:%
-%:%1325=605%:%
-%:%1326=606%:%
-%:%1327=606%:%
-%:%1328=607%:%
-%:%1329=608%:%
-%:%1330=609%:%
-%:%1331=610%:%
-%:%1332=610%:%
-%:%1333=611%:%
-%:%1334=612%:%
-%:%1335=612%:%
-%:%1336=613%:%
-%:%1339=614%:%
-%:%1343=614%:%
-%:%1344=614%:%
-%:%1345=615%:%
-%:%1346=615%:%
-%:%1347=616%:%
-%:%1348=616%:%
-%:%1349=617%:%
-%:%1355=617%:%
-%:%1358=618%:%
-%:%1359=619%:%
-%:%1360=620%:%
-%:%1361=621%:%
-%:%1362=622%:%
-%:%1363=623%:%
-%:%1364=624%:%
-%:%1365=625%:%
-%:%1366=625%:%
-%:%1367=626%:%
-%:%1370=627%:%
-%:%1374=627%:%
-%:%1375=627%:%
-%:%1376=628%:%
-%:%1377=628%:%
-%:%1378=629%:%
-%:%1379=629%:%
-%:%1384=629%:%
-%:%1387=630%:%
-%:%1388=631%:%
-%:%1389=631%:%
-%:%1390=632%:%
-%:%1393=633%:%
-%:%1397=633%:%
-%:%1398=633%:%
-%:%1399=634%:%
-%:%1400=634%:%
-%:%1401=635%:%
-%:%1407=635%:%
-%:%1410=636%:%
-%:%1411=637%:%
-%:%1412=637%:%
-%:%1413=638%:%
-%:%1416=639%:%
-%:%1420=639%:%
-%:%1421=639%:%
-%:%1422=640%:%
-%:%1423=640%:%
-%:%1424=641%:%
-%:%1425=641%:%
-%:%1426=642%:%
-%:%1427=642%:%
-%:%1428=643%:%
-%:%1429=643%:%
-%:%1430=644%:%
-%:%1431=644%:%
-%:%1432=645%:%
-%:%1438=645%:%
-%:%1441=646%:%
-%:%1442=647%:%
-%:%1443=647%:%
-%:%1444=648%:%
-%:%1447=649%:%
-%:%1451=649%:%
-%:%1452=649%:%
-%:%1453=650%:%
-%:%1454=650%:%
-%:%1455=651%:%
-%:%1456=651%:%
-%:%1457=652%:%
-%:%1458=652%:%
-%:%1459=652%:%
-%:%1460=653%:%
-%:%1461=653%:%
-%:%1466=653%:%
-%:%1469=654%:%
-%:%1470=655%:%
-%:%1471=655%:%
-%:%1472=656%:%
-%:%1475=657%:%
-%:%1479=657%:%
-%:%1480=657%:%
-%:%1481=658%:%
-%:%1482=658%:%
-%:%1483=659%:%
-%:%1484=659%:%
-%:%1485=660%:%
-%:%1486=660%:%
-%:%1491=660%:%
-%:%1494=661%:%
-%:%1495=662%:%
-%:%1496=662%:%
-%:%1497=663%:%
-%:%1500=664%:%
-%:%1504=664%:%
-%:%1505=664%:%
-%:%1510=664%:%
-%:%1513=665%:%
-%:%1514=666%:%
-%:%1515=666%:%
-%:%1516=667%:%
-%:%1519=668%:%
-%:%1523=668%:%
-%:%1524=668%:%
-%:%1525=669%:%
-%:%1526=669%:%
-%:%1527=670%:%
-%:%1528=670%:%
-%:%1529=671%:%
-%:%1530=671%:%
-%:%1531=672%:%
-%:%1532=672%:%
-%:%1533=673%:%
-%:%1534=673%:%
-%:%1535=674%:%
-%:%1536=674%:%
-%:%1537=675%:%
-%:%1538=675%:%
-%:%1539=676%:%
-%:%1540=676%:%
-%:%1541=677%:%
-%:%1542=677%:%
-%:%1543=678%:%
-%:%1544=678%:%
-%:%1545=679%:%
-%:%1546=679%:%
-%:%1547=680%:%
-%:%1548=680%:%
-%:%1549=681%:%
-%:%1550=681%:%
-%:%1551=682%:%
-%:%1552=682%:%
-%:%1553=683%:%
-%:%1554=683%:%
-%:%1555=684%:%
-%:%1556=684%:%
-%:%1557=685%:%
-%:%1558=685%:%
-%:%1559=686%:%
-%:%1560=686%:%
-%:%1561=686%:%
-%:%1566=686%:%
-%:%1569=687%:%
-%:%1570=688%:%
-%:%1571=688%:%
-%:%1572=689%:%
-%:%1575=690%:%
-%:%1579=690%:%
-%:%1580=690%:%
-%:%1581=691%:%
-%:%1582=691%:%
-%:%1583=692%:%
-%:%1589=692%:%
-%:%1592=693%:%
-%:%1593=694%:%
-%:%1594=695%:%
-%:%1595=696%:%
-%:%1596=696%:%
-%:%1597=697%:%
-%:%1598=698%:%
-%:%1601=699%:%
-%:%1605=699%:%
-%:%1606=699%:%
-%:%1607=700%:%
-%:%1608=700%:%
-%:%1609=701%:%
-%:%1610=701%:%
-%:%1611=702%:%
-%:%1617=702%:%
-%:%1620=703%:%
-%:%1621=704%:%
-%:%1622=704%:%
-%:%1623=705%:%
-%:%1626=706%:%
-%:%1630=706%:%
-%:%1631=706%:%
-%:%1632=707%:%
-%:%1633=707%:%
-%:%1634=708%:%
-%:%1640=708%:%
-%:%1643=709%:%
-%:%1644=710%:%
-%:%1645=711%:%
-%:%1646=711%:%
-%:%1647=712%:%
-%:%1650=713%:%
-%:%1654=713%:%
-%:%1655=713%:%
-%:%1656=714%:%
-%:%1657=714%:%
-%:%1658=715%:%
-%:%1659=715%:%
-%:%1660=716%:%
-%:%1661=716%:%
-%:%1662=717%:%
-%:%1663=717%:%
-%:%1664=718%:%
-%:%1665=718%:%
-%:%1666=719%:%
-%:%1667=719%:%
-%:%1672=719%:%
-%:%1675=720%:%
-%:%1676=721%:%
-%:%1677=722%:%
-%:%1678=722%:%
-%:%1679=723%:%
-%:%1680=724%:%
-%:%1683=725%:%
-%:%1687=725%:%
-%:%1688=725%:%
-%:%1693=725%:%
-%:%1696=726%:%
-%:%1697=727%:%
-%:%1698=728%:%
-%:%1699=728%:%
-%:%1700=729%:%
-%:%1703=730%:%
-%:%1707=730%:%
-%:%1708=730%:%
-%:%1713=730%:%
-%:%1716=731%:%
-%:%1717=732%:%
-%:%1718=733%:%
-%:%1719=733%:%
-%:%1720=734%:%
-%:%1721=735%:%
-%:%1724=736%:%
-%:%1728=736%:%
-%:%1729=736%:%
-%:%1730=737%:%
-%:%1731=737%:%
-%:%1732=738%:%
-%:%1733=738%:%
-%:%1734=739%:%
-%:%1735=739%:%
-%:%1736=740%:%
-%:%1737=740%:%
-%:%1742=740%:%
-%:%1745=741%:%
-%:%1746=742%:%
-%:%1747=742%:%
-%:%1748=743%:%
-%:%1749=744%:%
-%:%1752=745%:%
-%:%1756=745%:%
-%:%1757=745%:%
-%:%1758=746%:%
-%:%1759=746%:%
-%:%1760=747%:%
-%:%1761=747%:%
-%:%1762=748%:%
-%:%1763=748%:%
-%:%1764=749%:%
-%:%1765=749%:%
-%:%1770=749%:%
-%:%1773=750%:%
-%:%1774=751%:%
-%:%1775=751%:%
-%:%1776=752%:%
-%:%1779=753%:%
-%:%1783=753%:%
-%:%1784=753%:%
-%:%1785=754%:%
-%:%1786=754%:%
-%:%1787=755%:%
-%:%1788=755%:%
-%:%1789=756%:%
-%:%1795=756%:%
-%:%1798=757%:%
-%:%1799=758%:%
-%:%1800=759%:%
-%:%1801=760%:%
-%:%1802=761%:%
-%:%1803=762%:%
-%:%1804=762%:%
-%:%1805=763%:%
-%:%1806=764%:%
-%:%1807=765%:%
-%:%1808=766%:%
-%:%1809=767%:%
-%:%1810=768%:%
-%:%1811=769%:%
-%:%1812=770%:%
-%:%1813=770%:%
-%:%1814=771%:%
-%:%1817=772%:%
-%:%1821=772%:%
-%:%1822=772%:%
-%:%1823=773%:%
-%:%1824=773%:%
-%:%1825=774%:%
-%:%1831=774%:%
-%:%1834=775%:%
-%:%1835=776%:%
-%:%1836=776%:%
-%:%1837=777%:%
-%:%1840=778%:%
-%:%1844=778%:%
-%:%1845=778%:%
-%:%1846=779%:%
-%:%1847=779%:%
-%:%1848=780%:%
-%:%1849=780%:%
-%:%1854=780%:%
-%:%1857=781%:%
-%:%1858=782%:%
-%:%1859=782%:%
-%:%1860=783%:%
-%:%1863=784%:%
-%:%1867=784%:%
-%:%1868=784%:%
-%:%1869=785%:%
-%:%1875=785%:%
-%:%1878=786%:%
-%:%1879=787%:%
-%:%1880=788%:%
-%:%1881=789%:%
-%:%1882=790%:%
-%:%1883=791%:%
-%:%1884=792%:%
-%:%1885=793%:%
-%:%1886=794%:%
-%:%1887=794%:%
-%:%1888=795%:%
-%:%1889=796%:%
-%:%1892=797%:%
-%:%1896=797%:%
-%:%1897=797%:%
-%:%1898=798%:%
-%:%1899=798%:%
-%:%1900=799%:%
-%:%1901=799%:%
-%:%1902=800%:%
-%:%1903=800%:%
-%:%1904=801%:%
-%:%1905=801%:%
-%:%1906=802%:%
-%:%1912=802%:%
-%:%1915=803%:%
-%:%1916=804%:%
-%:%1917=805%:%
-%:%1918=806%:%
-%:%1919=806%:%
-%:%1920=807%:%
-%:%1923=808%:%
-%:%1927=808%:%
-%:%1928=808%:%
-%:%1929=808%:%
-%:%1934=808%:%
-%:%1937=809%:%
-%:%1938=810%:%
-%:%1939=811%:%
-%:%1940=812%:%
-%:%1941=813%:%
-%:%1942=814%:%
-%:%1943=814%:%
-%:%1944=815%:%
-%:%1947=816%:%
-%:%1951=816%:%
-%:%1952=816%:%
-%:%1953=817%:%
-%:%1954=817%:%
-%:%1955=818%:%
-%:%1956=818%:%
-%:%1957=819%:%
-%:%1958=819%:%
-%:%1959=820%:%
-%:%1960=820%:%
-%:%1961=821%:%
-%:%1962=821%:%
-%:%1963=822%:%
-%:%1964=822%:%
-%:%1965=823%:%
-%:%1966=823%:%
-%:%1967=824%:%
-%:%1973=824%:%
-%:%1976=825%:%
-%:%1977=826%:%
-%:%1978=826%:%
-%:%1979=827%:%
-%:%1980=828%:%
-%:%1981=829%:%
-%:%1982=830%:%
-%:%1983=831%:%
-%:%1984=832%:%
-%:%1985=832%:%
-%:%1986=833%:%
-%:%1987=834%:%
-%:%1990=835%:%
-%:%1994=835%:%
-%:%1995=835%:%
-%:%1996=836%:%
-%:%1997=836%:%
-%:%1998=837%:%
-%:%1999=837%:%
-%:%2000=838%:%
-%:%2006=838%:%
-%:%2009=839%:%
-%:%2010=840%:%
-%:%2011=841%:%
-%:%2012=842%:%
-%:%2013=842%:%
-%:%2014=843%:%
-%:%2015=844%:%
-%:%2018=845%:%
-%:%2022=845%:%
-%:%2023=845%:%
-%:%2024=846%:%
-%:%2025=846%:%
-%:%2026=847%:%
-%:%2027=847%:%
-%:%2028=848%:%
-%:%2029=848%:%
-%:%2030=849%:%
-%:%2031=849%:%
-%:%2032=850%:%
-%:%2033=850%:%
-%:%2034=851%:%
-%:%2035=851%:%
-%:%2036=852%:%
-%:%2037=852%:%
-%:%2038=853%:%
-%:%2039=853%:%
-%:%2040=854%:%
-%:%2041=854%:%
-%:%2042=855%:%
-%:%2043=855%:%
-%:%2044=856%:%
-%:%2045=856%:%
-%:%2046=857%:%
-%:%2047=857%:%
-%:%2048=858%:%
-%:%2054=858%:%
-%:%2057=859%:%
-%:%2058=860%:%
-%:%2059=860%:%
-%:%2060=861%:%
-%:%2061=862%:%
-%:%2064=863%:%
-%:%2068=863%:%
-%:%2069=863%:%
-%:%2070=864%:%
-%:%2071=864%:%
-%:%2072=865%:%
-%:%2073=865%:%
-%:%2074=866%:%
-%:%2075=866%:%
-%:%2080=866%:%
-%:%2083=867%:%
-%:%2084=868%:%
-%:%2085=869%:%
-%:%2086=869%:%
-%:%2087=870%:%
-%:%2088=871%:%
-%:%2091=872%:%
-%:%2095=872%:%
-%:%2096=872%:%
-%:%2097=873%:%
-%:%2098=873%:%
-%:%2099=874%:%
-%:%2100=874%:%
-%:%2101=875%:%
-%:%2102=875%:%
-%:%2103=876%:%
-%:%2104=876%:%
-%:%2105=877%:%
-%:%2106=877%:%
-%:%2107=878%:%
-%:%2108=878%:%
-%:%2109=879%:%
-%:%2110=879%:%
-%:%2111=880%:%
-%:%2112=880%:%
-%:%2113=881%:%
-%:%2114=881%:%
-%:%2115=882%:%
-%:%2116=882%:%
-%:%2117=883%:%
-%:%2118=883%:%
-%:%2119=884%:%
-%:%2120=884%:%
-%:%2121=885%:%
-%:%2122=885%:%
-%:%2123=886%:%
-%:%2124=886%:%
-%:%2125=887%:%
-%:%2126=887%:%
-%:%2127=888%:%
-%:%2128=888%:%
-%:%2129=889%:%
-%:%2130=889%:%
-%:%2131=890%:%
-%:%2132=890%:%
-%:%2133=891%:%
-%:%2134=891%:%
-%:%2135=892%:%
-%:%2136=892%:%
-%:%2137=893%:%
-%:%2138=893%:%
-%:%2139=894%:%
-%:%2140=894%:%
-%:%2141=895%:%
-%:%2142=895%:%
-%:%2143=896%:%
-%:%2144=896%:%
-%:%2145=897%:%
-%:%2146=897%:%
-%:%2147=898%:%
-%:%2148=898%:%
-%:%2149=899%:%
-%:%2150=899%:%
-%:%2151=900%:%
-%:%2152=900%:%
-%:%2153=901%:%
-%:%2154=901%:%
-%:%2155=902%:%
-%:%2156=902%:%
-%:%2157=903%:%
-%:%2158=903%:%
-%:%2159=904%:%
-%:%2160=904%:%
-%:%2161=905%:%
-%:%2162=905%:%
-%:%2163=906%:%
-%:%2164=906%:%
-%:%2165=907%:%
-%:%2166=907%:%
-%:%2167=908%:%
-%:%2168=908%:%
-%:%2169=909%:%
-%:%2170=909%:%
-%:%2171=910%:%
-%:%2172=910%:%
-%:%2173=911%:%
-%:%2174=911%:%
-%:%2175=912%:%
-%:%2176=912%:%
-%:%2177=913%:%
-%:%2178=913%:%
-%:%2179=914%:%
-%:%2180=914%:%
-%:%2181=915%:%
-%:%2182=915%:%
-%:%2183=916%:%
-%:%2184=916%:%
-%:%2185=917%:%
-%:%2186=917%:%
-%:%2187=918%:%
-%:%2188=918%:%
-%:%2189=919%:%
-%:%2190=919%:%
-%:%2191=920%:%
-%:%2192=920%:%
-%:%2193=921%:%
-%:%2194=921%:%
-%:%2195=922%:%
-%:%2196=922%:%
-%:%2197=923%:%
-%:%2198=923%:%
-%:%2199=924%:%
-%:%2200=924%:%
-%:%2201=925%:%
-%:%2202=925%:%
-%:%2203=926%:%
-%:%2204=926%:%
-%:%2205=927%:%
-%:%2206=927%:%
-%:%2207=928%:%
-%:%2208=928%:%
-%:%2209=929%:%
-%:%2210=929%:%
-%:%2211=930%:%
-%:%2212=930%:%
-%:%2213=931%:%
-%:%2214=931%:%
-%:%2215=931%:%
-%:%2220=931%:%
-%:%2223=932%:%
-%:%2224=933%:%
-%:%2225=934%:%
-%:%2226=935%:%
-%:%2227=936%:%
-%:%2228=936%:%
-%:%2229=937%:%
-%:%2232=938%:%
-%:%2236=938%:%
-%:%2237=938%:%
-%:%2238=939%:%
-%:%2239=939%:%
-%:%2240=940%:%
-%:%2246=940%:%
-%:%2249=941%:%
-%:%2250=942%:%
-%:%2251=943%:%
-%:%2252=943%:%
-%:%2253=944%:%
-%:%2254=945%:%
-%:%2255=946%:%
-%:%2256=947%:%
-%:%2257=948%:%
-%:%2258=949%:%
-%:%2259=950%:%
-%:%2261=952%:%
-%:%2262=953%:%
-%:%2263=954%:%
-%:%2264=955%:%
-%:%2265=956%:%
-%:%2266=957%:%
-%:%2267=958%:%
-%:%2268=959%:%
-%:%2269=960%:%
-%:%2270=961%:%
-%:%2271=962%:%
-%:%2272=963%:%
-%:%2273=964%:%
-%:%2274=965%:%
-%:%2275=966%:%
-%:%2276=967%:%
-%:%2277=967%:%
-%:%2278=968%:%
-%:%2279=969%:%
-%:%2282=970%:%
-%:%2286=970%:%
-%:%2287=970%:%
-%:%2288=971%:%
-%:%2294=971%:%
-%:%2297=972%:%
-%:%2298=973%:%
-%:%2299=974%:%
-%:%2300=975%:%
-%:%2301=975%:%
-%:%2302=976%:%
-%:%2305=977%:%
-%:%2309=977%:%
-%:%2310=977%:%
-%:%2311=978%:%
-%:%2312=978%:%
-%:%2313=979%:%
-%:%2314=979%:%
-%:%2315=980%:%
-%:%2316=980%:%
-%:%2317=981%:%
-%:%2318=981%:%
-%:%2319=982%:%
-%:%2325=982%:%
-%:%2328=983%:%
-%:%2329=984%:%
-%:%2330=985%:%
-%:%2331=986%:%
-%:%2332=986%:%
-%:%2333=987%:%
-%:%2334=988%:%
-%:%2337=989%:%
-%:%2341=989%:%
-%:%2342=989%:%
-%:%2343=990%:%
-%:%2344=990%:%
-%:%2345=991%:%
-%:%2351=991%:%
-%:%2354=992%:%
-%:%2355=993%:%
-%:%2356=994%:%
-%:%2357=995%:%
-%:%2358=996%:%
-%:%2359=997%:%
-%:%2360=998%:%
-%:%2361=999%:%
-%:%2362=1000%:%
-%:%2363=1000%:%
-%:%2364=1001%:%
-%:%2365=1002%:%
-%:%2366=1003%:%
-%:%2367=1004%:%
-%:%2368=1005%:%
-%:%2369=1006%:%
-%:%2370=1007%:%
-%:%2371=1008%:%
-%:%2372=1009%:%
-%:%2373=1010%:%
-%:%2374=1011%:%
-%:%2375=1012%:%
-%:%2376=1014%:%
-%:%2377=1015%:%
-%:%2378=1016%:%
-%:%2379=1017%:%
-%:%2380=1018%:%
-%:%2381=1019%:%
-%:%2382=1020%:%
-%:%2383=1021%:%
-%:%2384=1021%:%
-%:%2385=1022%:%
-%:%2386=1023%:%
-%:%2387=1024%:%
-%:%2388=1025%:%
-%:%2389=1026%:%
-%:%2390=1026%:%
-%:%2391=1027%:%
-%:%2392=1028%:%
-%:%2393=1029%:%
-%:%2394=1032%:%
-%:%2395=1033%:%
-%:%2396=1034%:%
-%:%2397=1035%:%
-%:%2398=1036%:%
-%:%2399=1036%:%
-%:%2402=1037%:%
-%:%2406=1037%:%
-%:%2407=1037%:%
-%:%2408=1037%:%
-%:%2413=1037%:%
-%:%2416=1038%:%
-%:%2417=1039%:%
-%:%2418=1039%:%
-%:%2419=1040%:%
-%:%2420=1041%:%
-%:%2423=1042%:%
-%:%2427=1042%:%
-%:%2428=1042%:%
-%:%2429=1043%:%
-%:%2430=1043%:%
-%:%2431=1044%:%
-%:%2432=1044%:%
-%:%2433=1045%:%
-%:%2439=1045%:%
-%:%2442=1046%:%
-%:%2443=1047%:%
-%:%2444=1048%:%
-%:%2445=1048%:%
-%:%2448=1049%:%
-%:%2452=1049%:%
-%:%2453=1049%:%
-%:%2458=1049%:%
-%:%2461=1050%:%
-%:%2462=1051%:%
-%:%2463=1052%:%
-%:%2464=1052%:%
-%:%2467=1053%:%
-%:%2471=1053%:%
-%:%2472=1053%:%
-%:%2473=1054%:%
-%:%2474=1054%:%
-%:%2475=1055%:%
-%:%2476=1055%:%
-%:%2481=1055%:%
-%:%2484=1056%:%
-%:%2485=1057%:%
-%:%2486=1058%:%
-%:%2487=1058%:%
-%:%2490=1059%:%
-%:%2494=1059%:%
-%:%2495=1059%:%
-%:%2496=1060%:%
-%:%2497=1060%:%
-%:%2498=1061%:%
-%:%2499=1061%:%
-%:%2500=1061%:%
-%:%2505=1061%:%
-%:%2508=1062%:%
-%:%2509=1063%:%
-%:%2510=1064%:%
-%:%2511=1065%:%
-%:%2512=1065%:%
-%:%2515=1066%:%
-%:%2516=1067%:%
-%:%2520=1067%:%
-%:%2521=1067%:%
-%:%2522=1068%:%
-%:%2523=1068%:%
-%:%2524=1069%:%
-%:%2525=1069%:%
-%:%2526=1070%:%
-%:%2527=1070%:%
-%:%2528=1071%:%
-%:%2529=1071%:%
-%:%2530=1072%:%
-%:%2531=1072%:%
-%:%2532=1073%:%
-%:%2533=1073%:%
-%:%2534=1074%:%
-%:%2535=1074%:%
-%:%2536=1075%:%
-%:%2537=1075%:%
-%:%2538=1076%:%
-%:%2539=1076%:%
-%:%2540=1077%:%
-%:%2546=1077%:%
-%:%2549=1078%:%
-%:%2550=1079%:%
-%:%2551=1080%:%
-%:%2552=1080%:%
-%:%2555=1081%:%
-%:%2559=1081%:%
-%:%2560=1081%:%
-%:%2565=1081%:%
-%:%2568=1082%:%
-%:%2569=1083%:%
-%:%2570=1084%:%
-%:%2571=1084%:%
-%:%2574=1085%:%
-%:%2578=1085%:%
-%:%2579=1085%:%
-%:%2580=1086%:%
-%:%2581=1086%:%
-%:%2582=1087%:%
-%:%2583=1087%:%
-%:%2584=1088%:%
-%:%2585=1088%:%
-%:%2586=1089%:%
-%:%2587=1089%:%
-%:%2588=1090%:%
-%:%2589=1090%:%
-%:%2590=1091%:%
-%:%2591=1091%:%
-%:%2592=1092%:%
-%:%2593=1092%:%
-%:%2594=1093%:%
-%:%2595=1093%:%
-%:%2596=1094%:%
-%:%2597=1094%:%
-%:%2598=1095%:%
-%:%2599=1095%:%
-%:%2604=1095%:%
-%:%2607=1096%:%
-%:%2608=1097%:%
-%:%2609=1097%:%
-%:%2612=1098%:%
-%:%2616=1098%:%
-%:%2617=1098%:%
-%:%2618=1099%:%
-%:%2619=1099%:%
-%:%2620=1100%:%
-%:%2621=1100%:%
-%:%2622=1100%:%
-%:%2627=1100%:%
-%:%2630=1101%:%
-%:%2631=1102%:%
-%:%2632=1103%:%
-%:%2633=1103%:%
-%:%2636=1104%:%
-%:%2640=1104%:%
-%:%2641=1104%:%
-%:%2642=1105%:%
-%:%2643=1105%:%
-%:%2644=1106%:%
-%:%2645=1107%:%
-%:%2646=1107%:%
-%:%2651=1107%:%
-%:%2654=1108%:%
-%:%2655=1109%:%
-%:%2656=1110%:%
-%:%2657=1111%:%
-%:%2658=1111%:%
-%:%2661=1112%:%
-%:%2665=1112%:%
-%:%2666=1112%:%
-%:%2667=1113%:%
-%:%2668=1113%:%
-%:%2669=1114%:%
-%:%2670=1114%:%
-%:%2675=1114%:%
-%:%2678=1115%:%
-%:%2679=1116%:%
-%:%2680=1116%:%
-%:%2683=1117%:%
-%:%2684=1118%:%
-%:%2688=1118%:%
-%:%2689=1118%:%
-%:%2690=1119%:%
-%:%2691=1119%:%
-%:%2692=1120%:%
-%:%2693=1120%:%
-%:%2694=1121%:%
-%:%2695=1121%:%
-%:%2700=1121%:%
-%:%2703=1122%:%
-%:%2704=1123%:%
-%:%2705=1124%:%
-%:%2706=1124%:%
-%:%2709=1125%:%
-%:%2713=1125%:%
-%:%2714=1125%:%
-%:%2715=1126%:%
-%:%2716=1126%:%
-%:%2717=1127%:%
-%:%2718=1127%:%
-%:%2719=1128%:%
-%:%2720=1128%:%
-%:%2721=1129%:%
-%:%2722=1129%:%
-%:%2723=1130%:%
-%:%2724=1130%:%
-%:%2725=1131%:%
-%:%2726=1131%:%
-%:%2727=1131%:%
-%:%2728=1132%:%
-%:%2729=1132%:%
-%:%2730=1133%:%
-%:%2731=1133%:%
-%:%2732=1134%:%
-%:%2733=1134%:%
-%:%2734=1135%:%
-%:%2740=1135%:%
-%:%2743=1136%:%
-%:%2744=1137%:%
-%:%2745=1137%:%
-%:%2746=1138%:%
-%:%2747=1139%:%
-%:%2748=1140%:%
-%:%2749=1141%:%
-%:%2750=1142%:%
-%:%2751=1143%:%
-%:%2752=1144%:%
-%:%2753=1145%:%
-%:%2754=1146%:%
-%:%2755=1147%:%
-%:%2756=1148%:%
-%:%2757=1148%:%
-%:%2760=1149%:%
-%:%2764=1149%:%
-%:%2765=1149%:%
-%:%2770=1149%:%
-%:%2773=1150%:%
-%:%2774=1151%:%
-%:%2775=1151%:%
-%:%2778=1152%:%
-%:%2782=1152%:%
-%:%2783=1152%:%
-%:%2784=1153%:%
-%:%2785=1153%:%
-%:%2786=1154%:%
-%:%2787=1154%:%
-%:%2788=1155%:%
-%:%2789=1156%:%
-%:%2790=1156%:%
-%:%2791=1157%:%
-%:%2792=1158%:%
-%:%2793=1159%:%
-%:%2794=1159%:%
-%:%2795=1159%:%
-%:%2796=1160%:%
-%:%2797=1160%:%
-%:%2798=1161%:%
-%:%2799=1161%:%
-%:%2800=1162%:%
-%:%2801=1163%:%
-%:%2802=1163%:%
-%:%2803=1164%:%
-%:%2804=1164%:%
-%:%2805=1165%:%
-%:%2806=1165%:%
-%:%2807=1166%:%
-%:%2808=1166%:%
-%:%2809=1167%:%
-%:%2810=1167%:%
-%:%2811=1168%:%
-%:%2812=1169%:%
-%:%2813=1169%:%
-%:%2814=1170%:%
-%:%2815=1171%:%
-%:%2816=1171%:%
-%:%2817=1171%:%
-%:%2818=1172%:%
-%:%2819=1172%:%
-%:%2820=1173%:%
-%:%2821=1174%:%
-%:%2822=1174%:%
-%:%2823=1175%:%
-%:%2824=1176%:%
-%:%2825=1176%:%
-%:%2830=1176%:%
-%:%2833=1177%:%
-%:%2834=1178%:%
-%:%2835=1179%:%
-%:%2836=1179%:%
-%:%2839=1180%:%
-%:%2843=1180%:%
-%:%2844=1180%:%
-%:%2849=1180%:%
-%:%2852=1181%:%
-%:%2853=1182%:%
-%:%2854=1183%:%
-%:%2855=1183%:%
-%:%2858=1184%:%
-%:%2862=1184%:%
-%:%2863=1184%:%
-%:%2864=1185%:%
-%:%2865=1185%:%
-%:%2866=1186%:%
-%:%2867=1186%:%
-%:%2868=1187%:%
-%:%2869=1187%:%
-%:%2870=1188%:%
-%:%2871=1188%:%
-%:%2872=1189%:%
-%:%2873=1189%:%
-%:%2874=1190%:%
-%:%2875=1190%:%
-%:%2876=1190%:%
-%:%2877=1191%:%
-%:%2878=1191%:%
-%:%2879=1192%:%
-%:%2880=1192%:%
-%:%2881=1193%:%
-%:%2882=1193%:%
-%:%2883=1194%:%
-%:%2884=1194%:%
-%:%2885=1195%:%
-%:%2886=1195%:%
-%:%2887=1195%:%
-%:%2888=1196%:%
-%:%2889=1196%:%
-%:%2890=1197%:%
-%:%2891=1197%:%
-%:%2892=1198%:%
-%:%2893=1198%:%
-%:%2894=1199%:%
-%:%2895=1199%:%
-%:%2896=1200%:%
-%:%2897=1200%:%
-%:%2898=1201%:%
-%:%2904=1201%:%
-%:%2907=1202%:%
-%:%2908=1203%:%
-%:%2909=1203%:%
-%:%2912=1204%:%
-%:%2916=1204%:%
-%:%2917=1204%:%
-%:%2918=1205%:%
-%:%2919=1205%:%
-%:%2920=1206%:%
-%:%2921=1206%:%
-%:%2922=1207%:%
-%:%2923=1207%:%
-%:%2924=1208%:%
-%:%2925=1209%:%
-%:%2926=1210%:%
-%:%2927=1211%:%
-%:%2928=1211%:%
-%:%2929=1212%:%
-%:%2930=1212%:%
-%:%2931=1213%:%
-%:%2932=1213%:%
-%:%2933=1214%:%
-%:%2934=1214%:%
-%:%2935=1215%:%
-%:%2936=1216%:%
-%:%2937=1216%:%
-%:%2938=1217%:%
-%:%2939=1218%:%
-%:%2940=1218%:%
-%:%2941=1219%:%
-%:%2942=1220%:%
-%:%2943=1220%:%
-%:%2944=1221%:%
-%:%2945=1221%:%
-%:%2946=1222%:%
-%:%2947=1222%:%
-%:%2948=1223%:%
-%:%2949=1223%:%
-%:%2950=1224%:%
-%:%2951=1224%:%
-%:%2956=1224%:%
-%:%2959=1225%:%
-%:%2960=1226%:%
-%:%2961=1226%:%
-%:%2962=1227%:%
-%:%2965=1228%:%
-%:%2969=1228%:%
-%:%2970=1228%:%
-%:%2971=1229%:%
-%:%2972=1229%:%
-%:%2973=1230%:%
-%:%2974=1230%:%
-%:%2975=1230%:%
-%:%2976=1231%:%
-%:%2977=1231%:%
-%:%2978=1231%:%
-%:%2983=1231%:%
-%:%2986=1232%:%
-%:%2987=1233%:%
-%:%2988=1234%:%
-%:%2989=1234%:%
-%:%2992=1235%:%
-%:%2996=1235%:%
-%:%2997=1235%:%
-%:%2998=1236%:%
-%:%3004=1236%:%
-%:%3007=1237%:%
-%:%3008=1238%:%
-%:%3009=1238%:%
-%:%3010=1239%:%
-%:%3011=1240%:%
-%:%3012=1241%:%
-%:%3014=1243%:%
-%:%3015=1244%:%
-%:%3016=1245%:%
-%:%3017=1245%:%
-%:%3020=1246%:%
-%:%3024=1246%:%
-%:%3025=1246%:%
-%:%3026=1247%:%
-%:%3032=1247%:%
-%:%3035=1248%:%
-%:%3036=1249%:%
-%:%3037=1249%:%
-%:%3040=1250%:%
-%:%3044=1250%:%
-%:%3045=1250%:%
-%:%3046=1250%:%
-%:%3051=1250%:%
-%:%3054=1251%:%
-%:%3055=1252%:%
-%:%3056=1252%:%
-%:%3059=1253%:%
-%:%3063=1253%:%
-%:%3064=1253%:%
-%:%3065=1254%:%
-%:%3066=1254%:%
-%:%3071=1254%:%
-%:%3074=1255%:%
-%:%3075=1256%:%
-%:%3076=1256%:%
-%:%3079=1257%:%
-%:%3083=1257%:%
-%:%3084=1257%:%
-%:%3085=1258%:%
-%:%3086=1258%:%
-%:%3087=1259%:%
-%:%3088=1259%:%
-%:%3089=1260%:%
-%:%3090=1260%:%
-%:%3091=1261%:%
-%:%3092=1261%:%
-%:%3093=1262%:%
-%:%3094=1262%:%
-%:%3095=1263%:%
-%:%3096=1263%:%
-%:%3097=1264%:%
-%:%3098=1264%:%
-%:%3099=1265%:%
-%:%3100=1265%:%
-%:%3101=1266%:%
-%:%3102=1266%:%
-%:%3103=1267%:%
-%:%3104=1267%:%
-%:%3105=1268%:%
-%:%3106=1268%:%
-%:%3117=1279%:%
-%:%3118=1280%:%
-%:%3119=1280%:%
-%:%3120=1281%:%
-%:%3121=1281%:%
-%:%3122=1281%:%
-%:%3123=1282%:%
-%:%3124=1282%:%
-%:%3125=1282%:%
-%:%3126=1283%:%
-%:%3127=1283%:%
-%:%3128=1284%:%
-%:%3129=1284%:%
-%:%3130=1285%:%
-%:%3131=1286%:%
-%:%3132=1286%:%
-%:%3133=1287%:%
-%:%3134=1287%:%
-%:%3135=1288%:%
-%:%3136=1289%:%
-%:%3137=1289%:%
-%:%3138=1290%:%
-%:%3139=1290%:%
-%:%3140=1291%:%
-%:%3141=1292%:%
-%:%3142=1292%:%
-%:%3143=1293%:%
-%:%3144=1293%:%
-%:%3145=1294%:%
-%:%3146=1294%:%
-%:%3151=1294%:%
-%:%3154=1295%:%
-%:%3155=1296%:%
-%:%3156=1297%:%
-%:%3157=1298%:%
-%:%3158=1298%:%
-%:%3161=1299%:%
-%:%3165=1299%:%
-%:%3166=1299%:%
-%:%3167=1300%:%
-%:%3168=1300%:%
-%:%3169=1301%:%
-%:%3170=1301%:%
-%:%3171=1302%:%
-%:%3172=1302%:%
-%:%3173=1303%:%
-%:%3174=1303%:%
-%:%3179=1303%:%
-%:%3182=1304%:%
-%:%3183=1305%:%
-%:%3184=1306%:%
-%:%3185=1307%:%
-%:%3186=1308%:%
-%:%3187=1309%:%
-%:%3188=1310%:%
-%:%3189=1310%:%
-%:%3192=1311%:%
-%:%3196=1311%:%
-%:%3197=1311%:%
-%:%3198=1312%:%
-%:%3199=1312%:%
-%:%3200=1313%:%
-%:%3201=1313%:%
-%:%3202=1314%:%
-%:%3203=1314%:%
-%:%3204=1315%:%
-%:%3205=1315%:%
-%:%3206=1315%:%
-%:%3207=1316%:%
-%:%3208=1316%:%
-%:%3209=1317%:%
-%:%3210=1317%:%
-%:%3211=1318%:%
-%:%3212=1318%:%
-%:%3213=1319%:%
-%:%3214=1319%:%
-%:%3215=1320%:%
-%:%3216=1320%:%
-%:%3217=1321%:%
-%:%3218=1321%:%
-%:%3219=1322%:%
-%:%3220=1322%:%
-%:%3221=1323%:%
-%:%3222=1323%:%
-%:%3223=1323%:%
-%:%3224=1324%:%
-%:%3225=1324%:%
-%:%3226=1325%:%
-%:%3227=1325%:%
-%:%3228=1326%:%
-%:%3229=1326%:%
-%:%3230=1327%:%
-%:%3231=1328%:%
-%:%3232=1328%:%
-%:%3233=1329%:%
-%:%3234=1330%:%
-%:%3235=1331%:%
-%:%3236=1331%:%
-%:%3237=1332%:%
-%:%3238=1332%:%
-%:%3239=1333%:%
-%:%3240=1333%:%
-%:%3241=1334%:%
-%:%3242=1334%:%
-%:%3243=1335%:%
-%:%3244=1336%:%
-%:%3245=1337%:%
-%:%3246=1337%:%
-%:%3247=1338%:%
-%:%3248=1339%:%
-%:%3249=1339%:%
-%:%3250=1340%:%
-%:%3251=1341%:%
-%:%3252=1341%:%
-%:%3253=1342%:%
-%:%3254=1343%:%
-%:%3255=1343%:%
-%:%3256=1343%:%
-%:%3257=1344%:%
-%:%3258=1345%:%
-%:%3259=1345%:%
-%:%3260=1345%:%
-%:%3265=1345%:%
-%:%3268=1346%:%
-%:%3269=1347%:%
-%:%3270=1348%:%
-%:%3271=1348%:%
-%:%3274=1349%:%
-%:%3278=1349%:%
-%:%3279=1349%:%
-%:%3280=1350%:%
-%:%3281=1350%:%
-%:%3282=1351%:%
-%:%3283=1351%:%
-%:%3284=1352%:%
-%:%3285=1352%:%
-%:%3286=1353%:%
-%:%3287=1353%:%
-%:%3288=1354%:%
-%:%3289=1354%:%
-%:%3290=1355%:%
-%:%3291=1355%:%
-%:%3292=1356%:%
-%:%3293=1356%:%
-%:%3294=1357%:%
-%:%3295=1357%:%
-%:%3296=1358%:%
-%:%3297=1358%:%
-%:%3298=1359%:%
-%:%3299=1359%:%
-%:%3300=1360%:%
-%:%3301=1361%:%
-%:%3302=1361%:%
-%:%3303=1362%:%
-%:%3304=1363%:%
-%:%3305=1364%:%
-%:%3306=1364%:%
-%:%3311=1364%:%
-%:%3314=1365%:%
-%:%3315=1366%:%
-%:%3316=1366%:%
-%:%3319=1367%:%
-%:%3323=1367%:%
-%:%3324=1367%:%
-%:%3325=1368%:%
-%:%3326=1368%:%
-%:%3327=1369%:%
-%:%3328=1369%:%
-%:%3329=1370%:%
-%:%3335=1370%:%
-%:%3338=1371%:%
-%:%3339=1372%:%
-%:%3340=1373%:%
-%:%3341=1373%:%
-%:%3344=1374%:%
-%:%3348=1374%:%
-%:%3349=1374%:%
-%:%3350=1375%:%
-%:%3351=1375%:%
-%:%3352=1376%:%
-%:%3353=1376%:%
-%:%3354=1377%:%
-%:%3355=1377%:%
-%:%3356=1378%:%
-%:%3357=1378%:%
-%:%3358=1379%:%
-%:%3364=1379%:%
-%:%3367=1380%:%
-%:%3368=1381%:%
-%:%3369=1381%:%
-%:%3370=1382%:%
-%:%3373=1383%:%
-%:%3377=1383%:%
-%:%3378=1383%:%
-%:%3379=1384%:%
-%:%3380=1384%:%
-%:%3381=1385%:%
-%:%3387=1385%:%
-%:%3390=1386%:%
-%:%3391=1387%:%
-%:%3392=1388%:%
-%:%3393=1389%:%
-%:%3394=1389%:%
-%:%3395=1390%:%
-%:%3398=1391%:%
-%:%3402=1391%:%
-%:%3403=1391%:%
-%:%3404=1391%:%
-%:%3409=1391%:%
-%:%3412=1392%:%
-%:%3413=1393%:%
-%:%3414=1393%:%
-%:%3417=1394%:%
-%:%3421=1394%:%
-%:%3422=1394%:%
-%:%3423=1395%:%
-%:%3424=1396%:%
-%:%3425=1396%:%
-%:%3426=1396%:%
-%:%3427=1397%:%
-%:%3428=1397%:%
-%:%3429=1398%:%
-%:%3430=1399%:%
-%:%3431=1399%:%
-%:%3432=1400%:%
-%:%3433=1400%:%
-%:%3434=1401%:%
-%:%3435=1402%:%
-%:%3436=1402%:%
-%:%3437=1402%:%
-%:%3438=1403%:%
-%:%3439=1403%:%
-%:%3440=1404%:%
-%:%3441=1405%:%
-%:%3442=1405%:%
-%:%3443=1406%:%
-%:%3444=1407%:%
-%:%3445=1407%:%
-%:%3446=1407%:%
-%:%3451=1407%:%
-%:%3454=1408%:%
-%:%3455=1409%:%
-%:%3456=1410%:%
-%:%3457=1411%:%
-%:%3458=1411%:%
-%:%3459=1412%:%
-%:%3462=1413%:%
-%:%3466=1413%:%
-%:%3467=1413%:%
-%:%3468=1413%:%
-%:%3473=1413%:%
-%:%3476=1414%:%
-%:%3477=1415%:%
-%:%3478=1415%:%
-%:%3481=1416%:%
-%:%3485=1416%:%
-%:%3486=1416%:%
-%:%3487=1417%:%
-%:%3488=1417%:%
-%:%3489=1418%:%
-%:%3490=1418%:%
-%:%3491=1419%:%
-%:%3492=1419%:%
-%:%3493=1420%:%
-%:%3494=1420%:%
-%:%3495=1421%:%
-%:%3496=1421%:%
-%:%3497=1422%:%
-%:%3498=1422%:%
-%:%3499=1423%:%
-%:%3500=1423%:%
-%:%3501=1424%:%
-%:%3502=1424%:%
-%:%3503=1425%:%
-%:%3504=1425%:%
-%:%3505=1426%:%
-%:%3506=1426%:%
-%:%3507=1427%:%
-%:%3508=1427%:%
-%:%3509=1428%:%
-%:%3510=1428%:%
-%:%3511=1429%:%
-%:%3512=1429%:%
-%:%3513=1430%:%
-%:%3514=1430%:%
-%:%3515=1431%:%
-%:%3516=1431%:%
-%:%3517=1432%:%
-%:%3518=1432%:%
-%:%3519=1433%:%
-%:%3520=1434%:%
-%:%3521=1434%:%
-%:%3522=1435%:%
-%:%3523=1436%:%
-%:%3524=1436%:%
-%:%3529=1436%:%
-%:%3532=1437%:%
-%:%3533=1438%:%
-%:%3534=1439%:%
-%:%3535=1440%:%
-%:%3536=1441%:%
-%:%3537=1441%:%
-%:%3538=1442%:%
-%:%3541=1443%:%
-%:%3545=1443%:%
-%:%3546=1443%:%
-%:%3547=1444%:%
-%:%3548=1445%:%
-%:%3549=1445%:%
-%:%3550=1446%:%
-%:%3551=1446%:%
-%:%3556=1446%:%
-%:%3559=1447%:%
-%:%3560=1448%:%
-%:%3561=1448%:%
-%:%3562=1449%:%
-%:%3565=1450%:%
-%:%3566=1451%:%
-%:%3570=1451%:%
-%:%3571=1451%:%
-%:%3576=1451%:%
-%:%3579=1452%:%
-%:%3580=1453%:%
-%:%3581=1454%:%
-%:%3582=1454%:%
-%:%3583=1455%:%
-%:%3586=1456%:%
-%:%3590=1456%:%
-%:%3591=1456%:%
-%:%3592=1457%:%
-%:%3593=1457%:%
-%:%3594=1458%:%
-%:%3595=1458%:%
-%:%3596=1459%:%
-%:%3597=1459%:%
-%:%3598=1460%:%
-%:%3599=1461%:%
-%:%3600=1461%:%
-%:%3601=1462%:%
-%:%3602=1463%:%
-%:%3603=1463%:%
-%:%3604=1464%:%
-%:%3605=1465%:%
-%:%3606=1465%:%
-%:%3607=1465%:%
-%:%3612=1465%:%
-%:%3615=1466%:%
-%:%3616=1467%:%
-%:%3617=1468%:%
-%:%3618=1468%:%
-%:%3619=1469%:%
-%:%3622=1470%:%
-%:%3626=1470%:%
-%:%3627=1470%:%
-%:%3628=1471%:%
-%:%3629=1472%:%
-%:%3630=1472%:%
-%:%3631=1472%:%
-%:%3632=1473%:%
-%:%3633=1473%:%
-%:%3634=1474%:%
-%:%3635=1474%:%
-%:%3636=1475%:%
-%:%3637=1476%:%
-%:%3638=1477%:%
-%:%3639=1477%:%
-%:%3640=1477%:%
-%:%3641=1478%:%
-%:%3642=1479%:%
-%:%3643=1479%:%
-%:%3644=1480%:%
-%:%3645=1480%:%
-%:%3646=1481%:%
-%:%3647=1482%:%
-%:%3648=1483%:%
-%:%3649=1483%:%
-%:%3650=1484%:%
-%:%3651=1484%:%
-%:%3652=1485%:%
-%:%3653=1485%:%
-%:%3654=1486%:%
-%:%3655=1487%:%
-%:%3656=1487%:%
-%:%3657=1488%:%
-%:%3658=1489%:%
-%:%3659=1489%:%
-%:%3664=1489%:%
-%:%3667=1490%:%
-%:%3668=1491%:%
-%:%3669=1492%:%
-%:%3670=1493%:%
-%:%3671=1493%:%
-%:%3674=1494%:%
-%:%3675=1495%:%
-%:%3679=1495%:%
-%:%3680=1495%:%
-%:%3681=1496%:%
-%:%3682=1496%:%
-%:%3683=1497%:%
-%:%3684=1497%:%
-%:%3685=1498%:%
-%:%3686=1498%:%
-%:%3687=1499%:%
-%:%3688=1499%:%
-%:%3689=1499%:%
-%:%3690=1500%:%
-%:%3691=1500%:%
-%:%3692=1501%:%
-%:%3693=1501%:%
-%:%3694=1502%:%
-%:%3695=1502%:%
-%:%3696=1503%:%
-%:%3697=1503%:%
-%:%3698=1504%:%
-%:%3699=1504%:%
-%:%3700=1505%:%
-%:%3701=1505%:%
-%:%3702=1505%:%
-%:%3703=1506%:%
-%:%3704=1506%:%
-%:%3705=1507%:%
-%:%3706=1507%:%
-%:%3707=1507%:%
-%:%3708=1508%:%
-%:%3709=1508%:%
-%:%3710=1509%:%
-%:%3711=1509%:%
-%:%3712=1510%:%
-%:%3713=1510%:%
-%:%3714=1511%:%
-%:%3715=1512%:%
-%:%3716=1512%:%
-%:%3717=1512%:%
-%:%3718=1513%:%
-%:%3719=1514%:%
-%:%3720=1514%:%
-%:%3721=1515%:%
-%:%3722=1515%:%
-%:%3723=1516%:%
-%:%3724=1517%:%
-%:%3725=1517%:%
-%:%3726=1518%:%
-%:%3727=1519%:%
-%:%3728=1519%:%
-%:%3729=1519%:%
-%:%3730=1520%:%
-%:%3731=1521%:%
-%:%3732=1521%:%
-%:%3733=1522%:%
-%:%3734=1523%:%
-%:%3735=1523%:%
-%:%3736=1523%:%
-%:%3737=1524%:%
-%:%3738=1525%:%
-%:%3739=1525%:%
-%:%3740=1526%:%
-%:%3741=1527%:%
-%:%3742=1527%:%
-%:%3747=1527%:%
-%:%3750=1528%:%
-%:%3751=1529%:%
-%:%3752=1530%:%
-%:%3753=1531%:%
-%:%3754=1531%:%
-%:%3757=1532%:%
-%:%3761=1532%:%
-%:%3762=1532%:%
-%:%3763=1533%:%
-%:%3764=1533%:%
-%:%3765=1534%:%
-%:%3766=1534%:%
-%:%3767=1535%:%
-%:%3768=1535%:%
-%:%3769=1536%:%
-%:%3770=1536%:%
-%:%3771=1537%:%
-%:%3772=1537%:%
-%:%3773=1538%:%
-%:%3774=1538%:%
-%:%3775=1539%:%
-%:%3776=1539%:%
-%:%3777=1540%:%
-%:%3778=1540%:%
-%:%3779=1540%:%
-%:%3784=1540%:%
-%:%3787=1541%:%
-%:%3788=1542%:%
-%:%3789=1543%:%
-%:%3790=1543%:%
-%:%3791=1544%:%
-%:%3792=1544%:%
-%:%3795=1545%:%
-%:%3799=1545%:%
-%:%3800=1545%:%
-%:%3805=1545%:%
-%:%3808=1546%:%
-%:%3809=1547%:%
-%:%3810=1547%:%
-%:%3813=1548%:%
-%:%3817=1548%:%
-%:%3818=1548%:%
-%:%3819=1548%:%
-%:%3824=1548%:%
-%:%3827=1549%:%
-%:%3828=1550%:%
-%:%3829=1550%:%
-%:%3832=1551%:%
-%:%3836=1551%:%
-%:%3837=1551%:%
-%:%3842=1551%:%
-%:%3845=1552%:%
-%:%3846=1553%:%
-%:%3847=1553%:%
-%:%3850=1554%:%
-%:%3854=1554%:%
-%:%3855=1554%:%
-%:%3860=1554%:%
-%:%3863=1555%:%
-%:%3864=1556%:%
-%:%3865=1556%:%
-%:%3868=1557%:%
-%:%3872=1557%:%
-%:%3873=1557%:%
-%:%3874=1558%:%
-%:%3875=1558%:%
-%:%3876=1559%:%
-%:%3877=1559%:%
-%:%3882=1559%:%
-%:%3885=1560%:%
-%:%3886=1561%:%
-%:%3887=1561%:%
-%:%3890=1562%:%
-%:%3894=1562%:%
-%:%3895=1562%:%
-%:%3896=1563%:%
-%:%3897=1563%:%
-%:%3898=1564%:%
-%:%3899=1565%:%
-%:%3900=1565%:%
-%:%3901=1566%:%
-%:%3902=1567%:%
-%:%3903=1567%:%
-%:%3904=1568%:%
-%:%3905=1569%:%
-%:%3906=1569%:%
-%:%3907=1570%:%
-%:%3908=1571%:%
-%:%3909=1571%:%
-%:%3910=1571%:%
-%:%3911=1572%:%
-%:%3912=1573%:%
-%:%3913=1573%:%
-%:%3914=1573%:%
-%:%3915=1574%:%
-%:%3916=1575%:%
-%:%3917=1575%:%
-%:%3918=1575%:%
-%:%3919=1576%:%
-%:%3920=1577%:%
-%:%3921=1577%:%
-%:%3922=1578%:%
-%:%3923=1579%:%
-%:%3924=1579%:%
-%:%3925=1579%:%
-%:%3926=1580%:%
-%:%3927=1581%:%
-%:%3928=1581%:%
-%:%3929=1582%:%
-%:%3930=1583%:%
-%:%3931=1583%:%
-%:%3932=1584%:%
-%:%3933=1585%:%
-%:%3934=1585%:%
-%:%3935=1586%:%
-%:%3936=1587%:%
-%:%3937=1587%:%
-%:%3938=1588%:%
-%:%3939=1589%:%
-%:%3940=1589%:%
-%:%3941=1589%:%
-%:%3946=1589%:%
-%:%3949=1590%:%
-%:%3950=1591%:%
-%:%3951=1592%:%
-%:%3952=1593%:%
-%:%3953=1593%:%
-%:%3956=1594%:%
-%:%3960=1594%:%
-%:%3961=1594%:%
-%:%3962=1595%:%
-%:%3963=1595%:%
-%:%3964=1596%:%
-%:%3965=1596%:%
-%:%3970=1596%:%
-%:%3973=1597%:%
-%:%3974=1598%:%
-%:%3975=1598%:%
-%:%3978=1599%:%
-%:%3982=1599%:%
-%:%3983=1599%:%
-%:%3984=1600%:%
-%:%3985=1600%:%
-%:%3986=1601%:%
-%:%3987=1601%:%
-%:%3992=1601%:%
-%:%3995=1602%:%
-%:%3996=1603%:%
-%:%3997=1604%:%
-%:%3998=1605%:%
-%:%3999=1605%:%
-%:%4002=1606%:%
-%:%4006=1606%:%
-%:%4007=1606%:%
-%:%4008=1607%:%
-%:%4009=1607%:%
-%:%4010=1608%:%
-%:%4011=1608%:%
-%:%4012=1609%:%
-%:%4013=1609%:%
-%:%4014=1610%:%
-%:%4015=1610%:%
-%:%4016=1611%:%
-%:%4017=1612%:%
-%:%4018=1612%:%
-%:%4023=1612%:%
-%:%4026=1613%:%
-%:%4027=1614%:%
-%:%4028=1614%:%
-%:%4029=1615%:%
-%:%4031=1617%:%
-%:%4034=1618%:%
-%:%4038=1618%:%
-%:%4039=1618%:%
-%:%4040=1619%:%
-%:%4041=1620%:%
-%:%4042=1620%:%
-%:%4043=1620%:%
-%:%4048=1620%:%
-%:%4051=1621%:%
-%:%4052=1622%:%
-%:%4053=1622%:%
-%:%4056=1623%:%
-%:%4060=1623%:%
-%:%4061=1623%:%
-%:%4062=1624%:%
-%:%4063=1625%:%
-%:%4064=1625%:%
-%:%4065=1626%:%
-%:%4066=1626%:%
-%:%4067=1627%:%
-%:%4068=1628%:%
-%:%4069=1628%:%
-%:%4070=1629%:%
-%:%4071=1629%:%
-%:%4072=1630%:%
-%:%4073=1630%:%
-%:%4074=1631%:%
-%:%4075=1631%:%
-%:%4076=1632%:%
-%:%4077=1632%:%
-%:%4078=1633%:%
-%:%4079=1633%:%
-%:%4080=1634%:%
-%:%4081=1634%:%
-%:%4082=1635%:%
-%:%4083=1635%:%
-%:%4084=1636%:%
-%:%4085=1636%:%
-%:%4086=1636%:%
-%:%4087=1637%:%
-%:%4088=1637%:%
-%:%4089=1638%:%
-%:%4090=1638%:%
-%:%4091=1639%:%
-%:%4092=1639%:%
-%:%4093=1640%:%
-%:%4094=1640%:%
-%:%4095=1641%:%
-%:%4096=1641%:%
-%:%4097=1641%:%
-%:%4098=1642%:%
-%:%4099=1642%:%
-%:%4100=1643%:%
-%:%4101=1644%:%
-%:%4102=1644%:%
-%:%4103=1644%:%
-%:%4104=1645%:%
-%:%4105=1645%:%
-%:%4106=1646%:%
-%:%4107=1646%:%
-%:%4108=1647%:%
-%:%4109=1647%:%
-%:%4110=1648%:%
-%:%4111=1648%:%
-%:%4112=1649%:%
-%:%4113=1649%:%
-%:%4114=1649%:%
-%:%4115=1650%:%
-%:%4116=1650%:%
-%:%4117=1650%:%
-%:%4118=1651%:%
-%:%4119=1651%:%
-%:%4120=1651%:%
-%:%4121=1652%:%
-%:%4122=1652%:%
-%:%4123=1653%:%
-%:%4124=1653%:%
-%:%4125=1654%:%
-%:%4126=1654%:%
-%:%4127=1655%:%
-%:%4128=1655%:%
-%:%4129=1656%:%
-%:%4130=1656%:%
-%:%4131=1656%:%
-%:%4132=1657%:%
-%:%4133=1657%:%
-%:%4134=1658%:%
-%:%4135=1658%:%
-%:%4136=1659%:%
-%:%4137=1659%:%
-%:%4138=1660%:%
-%:%4139=1660%:%
-%:%4140=1661%:%
-%:%4141=1662%:%
-%:%4142=1662%:%
-%:%4143=1663%:%
-%:%4144=1664%:%
-%:%4145=1664%:%
-%:%4146=1664%:%
-%:%4147=1665%:%
-%:%4148=1666%:%
-%:%4149=1666%:%
-%:%4150=1666%:%
-%:%4151=1667%:%
-%:%4152=1668%:%
-%:%4153=1668%:%
-%:%4154=1668%:%
-%:%4155=1669%:%
-%:%4156=1670%:%
-%:%4157=1670%:%
-%:%4158=1671%:%
-%:%4159=1672%:%
-%:%4160=1672%:%
-%:%4161=1673%:%
-%:%4162=1674%:%
-%:%4163=1674%:%
-%:%4164=1675%:%
-%:%4165=1676%:%
-%:%4166=1676%:%
-%:%4167=1677%:%
-%:%4168=1678%:%
-%:%4169=1679%:%
-%:%4170=1679%:%
-%:%4171=1679%:%
-%:%4176=1679%:%
-%:%4179=1680%:%
-%:%4180=1681%:%
-%:%4181=1682%:%
-%:%4182=1683%:%
-%:%4183=1683%:%
-%:%4186=1684%:%
-%:%4190=1684%:%
-%:%4191=1684%:%
-%:%4192=1685%:%
-%:%4193=1685%:%
-%:%4194=1686%:%
-%:%4195=1686%:%
-%:%4200=1686%:%
-%:%4203=1687%:%
-%:%4204=1688%:%
-%:%4205=1689%:%
-%:%4206=1690%:%
-%:%4207=1691%:%
-%:%4208=1691%:%
-%:%4211=1692%:%
-%:%4215=1692%:%
-%:%4216=1692%:%
-%:%4217=1693%:%
-%:%4218=1694%:%
-%:%4219=1694%:%
-%:%4220=1695%:%
-%:%4221=1695%:%
-%:%4222=1696%:%
-%:%4223=1696%:%
-%:%4224=1697%:%
-%:%4225=1697%:%
-%:%4230=1697%:%
-%:%4233=1698%:%
-%:%4234=1699%:%
-%:%4235=1700%:%
-%:%4236=1701%:%
-%:%4237=1701%:%
-%:%4238=1702%:%
-%:%4239=1703%:%
-%:%4240=1703%:%
-%:%4243=1704%:%
-%:%4247=1704%:%
-%:%4248=1704%:%
-%:%4249=1704%:%
-%:%4254=1704%:%
-%:%4257=1705%:%
-%:%4258=1706%:%
-%:%4259=1706%:%
-%:%4262=1707%:%
-%:%4266=1707%:%
-%:%4267=1707%:%
-%:%4272=1707%:%
-%:%4275=1708%:%
-%:%4276=1709%:%
-%:%4277=1710%:%
-%:%4278=1710%:%
-%:%4281=1711%:%
-%:%4285=1711%:%
-%:%4286=1711%:%
-%:%4287=1711%:%
-%:%4292=1711%:%
-%:%4295=1712%:%
-%:%4296=1713%:%
-%:%4297=1714%:%
-%:%4298=1714%:%
-%:%4299=1715%:%
-%:%4302=1716%:%
-%:%4307=1717%:%
\ No newline at end of file