diff --git a/thys/MDP-Algorithms/ROOT b/thys/MDP-Algorithms/ROOT
--- a/thys/MDP-Algorithms/ROOT
+++ b/thys/MDP-Algorithms/ROOT
@@ -1,15 +1,15 @@
 chapter AFP
 session "MDP-Algorithms" (AFP) = "MDP-Rewards" +
-  options [document = pdf, document_output = "output", timeout = 600]
+  options [timeout = 600]
   sessions
     Gauss_Jordan
   directories
     code
     examples
   theories
     Algorithms
     Code_Mod
     Examples
   document_files
     "root.bib"
     "root.tex"
diff --git a/thys/MDP-Algorithms/output/document.pdf b/thys/MDP-Algorithms/output/document.pdf
deleted file mode 100644
index f79aa81fe9ef01b8402802d0652cb0cc4209582b..e69de29bb2d1d6434b8b29ae775ad8c2e48c5391
GIT binary patch
literal 0
Hc$@<O00001

diff --git a/thys/MDP-Algorithms/output/document/Algorithms.tex b/thys/MDP-Algorithms/output/document/Algorithms.tex
deleted file mode 100644
--- a/thys/MDP-Algorithms/output/document/Algorithms.tex
+++ /dev/null
@@ -1,42 +0,0 @@
-%
-\begin{isabellebody}%
-\setisabellecontext{Algorithms}%
-%
-\isadelimtheory
-\isanewline
-\isanewline
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{theory}\isamarkupfalse%
-\ Algorithms\isanewline
-\ \ \isakeyword{imports}\isanewline
-\ \ \ \ Value{\isacharunderscore}{\kern0pt}Iteration\isanewline
-\ \ \ \ Policy{\isacharunderscore}{\kern0pt}Iteration\isanewline
-\ \ \ \ Modified{\isacharunderscore}{\kern0pt}Policy{\isacharunderscore}{\kern0pt}Iteration\isanewline
-\ \ \ \ Splitting{\isacharunderscore}{\kern0pt}Methods\isanewline
-\isakeyword{begin}\isanewline
-\isacommand{end}\isamarkupfalse%
-%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\end{isabellebody}%
-\endinput
-%:%file=Algorithms.tex%:%
-%:%6=1%:%
-%:%7=2%:%
-%:%12=3%:%
-%:%13=3%:%
-%:%14=4%:%
-%:%15=5%:%
-%:%16=6%:%
-%:%17=7%:%
-%:%18=8%:%
-%:%19=9%:%
-%:%20=10%:%
diff --git a/thys/MDP-Algorithms/output/document/Blinfun_Matrix.tex b/thys/MDP-Algorithms/output/document/Blinfun_Matrix.tex
deleted file mode 100644
--- a/thys/MDP-Algorithms/output/document/Blinfun_Matrix.tex
+++ /dev/null
@@ -1,1342 +0,0 @@
-%
-\begin{isabellebody}%
-\setisabellecontext{Blinfun{\isacharunderscore}{\kern0pt}Matrix}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{theory}\isamarkupfalse%
-\ Blinfun{\isacharunderscore}{\kern0pt}Matrix\isanewline
-\ \ \isakeyword{imports}\ \isanewline
-\ \ \ \ {\isachardoublequoteopen}MDP{\isacharminus}{\kern0pt}Rewards{\isachardot}{\kern0pt}Blinfun{\isacharunderscore}{\kern0pt}Util{\isachardoublequoteclose}\ \isanewline
-\ \ \ \ Matrix{\isacharunderscore}{\kern0pt}Util\isanewline
-\isakeyword{begin}%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsection{Bounded Linear Functions and Matrices%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{definition}\isamarkupfalse%
-\ {\isachardoublequoteopen}blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ {\isacharparenleft}{\kern0pt}f\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}b{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}finite\ {\isasymRightarrow}\isactrlsub b\ real{\isacharparenright}{\kern0pt}\ {\isasymRightarrow}\isactrlsub L\ {\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}c{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}finite\ {\isasymRightarrow}\isactrlsub b\ {\isacharunderscore}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ \isanewline
-\ \ matrix\ {\isacharparenleft}{\kern0pt}{\isasymlambda}v{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymchi}\ j{\isachardot}{\kern0pt}\ f\ {\isacharparenleft}{\kern0pt}Bfun\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}{\isachardollar}{\kern0pt}{\isacharparenright}{\kern0pt}\ v{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ j{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ {\isachardoublequoteopen}matrix{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}blinfun\ X\ {\isacharequal}{\kern0pt}\ Blinfun\ {\isacharparenleft}{\kern0pt}{\isasymlambda}v{\isachardot}{\kern0pt}\ Bfun\ {\isacharparenleft}{\kern0pt}{\isasymlambda}i{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}X\ {\isacharasterisk}{\kern0pt}v\ {\isacharparenleft}{\kern0pt}{\isasymchi}\ i{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}apply{\isacharunderscore}{\kern0pt}bfun\ v\ i{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isachardollar}{\kern0pt}\ i{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ plus{\isacharunderscore}{\kern0pt}vec{\isacharunderscore}{\kern0pt}eq{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isasymchi}\ i{\isachardot}{\kern0pt}\ f\ i\ \ {\isacharplus}{\kern0pt}\ g\ i{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymchi}\ i{\isachardot}{\kern0pt}\ f\ i{\isacharparenright}{\kern0pt}\ {\isacharplus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymchi}\ i{\isachardot}{\kern0pt}\ g\ i{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ Finite{\isacharunderscore}{\kern0pt}Cartesian{\isacharunderscore}{\kern0pt}Product{\isachardot}{\kern0pt}plus{\isacharunderscore}{\kern0pt}vec{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ matrix{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}blinfun{\isacharunderscore}{\kern0pt}mult{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}matrix{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}blinfun\ m\ {\isacharparenleft}{\kern0pt}v\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isacharprime}{\kern0pt}c{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}finite\ {\isasymRightarrow}\isactrlsub b\ real{\isacharparenright}{\kern0pt}\ i\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}m\ {\isacharasterisk}{\kern0pt}v\ {\isacharparenleft}{\kern0pt}{\isasymchi}\ i{\isachardot}{\kern0pt}\ v\ i{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isachardollar}{\kern0pt}\ i{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isasymchi}\ i{\isachardot}{\kern0pt}\ c\ {\isacharasterisk}{\kern0pt}\ x\ i{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ c\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isacharparenleft}{\kern0pt}{\isasymchi}\ i{\isachardot}{\kern0pt}\ x\ i{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{for}\ c\ x\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ vector{\isacharunderscore}{\kern0pt}scalar{\isacharunderscore}{\kern0pt}mult{\isacharunderscore}{\kern0pt}def\ scalar{\isacharunderscore}{\kern0pt}mult{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}scaleR{\isacharbrackleft}{\kern0pt}symmetric{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}bounded{\isacharunderscore}{\kern0pt}linear\ {\isacharparenleft}{\kern0pt}{\isasymlambda}v{\isachardot}{\kern0pt}\ bfun{\isachardot}{\kern0pt}Bfun\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}{\isachardollar}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}m\ {\isacharasterisk}{\kern0pt}v\ vec{\isacharunderscore}{\kern0pt}lambda\ {\isacharparenleft}{\kern0pt}apply{\isacharunderscore}{\kern0pt}bfun\ v{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}rule\ bounded{\isacharunderscore}{\kern0pt}linear{\isacharunderscore}{\kern0pt}compose{\isacharbrackleft}{\kern0pt}of\ {\isachardoublequoteopen}{\isasymlambda}x{\isachardot}{\kern0pt}\ bfun{\isachardot}{\kern0pt}Bfun\ {\isacharparenleft}{\kern0pt}{\isasymlambda}y{\isachardot}{\kern0pt}\ x\ {\isachardollar}{\kern0pt}\ y{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}{\isacharbrackright}{\kern0pt}{\isacharcomma}{\kern0pt}\ goal{\isacharunderscore}{\kern0pt}cases{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{case}\isamarkupfalse%
-\ {\isadigit{1}}\isanewline
-\ \ \ \ \isacommand{then}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}case\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ bounded{\isacharunderscore}{\kern0pt}linear{\isacharunderscore}{\kern0pt}bfun{\isacharunderscore}{\kern0pt}nth{\isacharbrackleft}{\kern0pt}of\ id{\isacharcomma}{\kern0pt}\ simplified{\isacharbrackright}{\kern0pt}\ bounded{\isacharunderscore}{\kern0pt}linear{\isacharunderscore}{\kern0pt}ident\ eq{\isacharunderscore}{\kern0pt}id{\isacharunderscore}{\kern0pt}iff\ \isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{case}\isamarkupfalse%
-\ {\isadigit{2}}\isanewline
-\ \ \ \ \isacommand{then}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}case\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ norm{\isacharunderscore}{\kern0pt}vec{\isacharunderscore}{\kern0pt}le{\isacharunderscore}{\kern0pt}norm{\isacharunderscore}{\kern0pt}bfun\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ matrix{\isacharunderscore}{\kern0pt}vector{\isacharunderscore}{\kern0pt}right{\isacharunderscore}{\kern0pt}distrib\ plus{\isacharunderscore}{\kern0pt}vec{\isacharunderscore}{\kern0pt}eq\isanewline
-\ \ \ \ \ \ \ \ \ \ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ bounded{\isacharunderscore}{\kern0pt}linear{\isacharunderscore}{\kern0pt}intro\ bounded{\isacharunderscore}{\kern0pt}linear{\isacharunderscore}{\kern0pt}compose{\isacharbrackleft}{\kern0pt}OF\ matrix{\isacharunderscore}{\kern0pt}vector{\isacharunderscore}{\kern0pt}mul{\isacharunderscore}{\kern0pt}bounded{\isacharunderscore}{\kern0pt}linear{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ Blinfun{\isacharunderscore}{\kern0pt}inverse\ matrix{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}blinfun{\isacharunderscore}{\kern0pt}def\ Bfun{\isacharunderscore}{\kern0pt}inverse{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}mult{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ f\ {\isacharasterisk}{\kern0pt}v\ {\isacharparenleft}{\kern0pt}{\isasymchi}\ i{\isachardot}{\kern0pt}\ apply{\isacharunderscore}{\kern0pt}bfun\ v\ i{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isachardollar}{\kern0pt}\ i\ {\isacharequal}{\kern0pt}\ f\ v\ i{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ f\ {\isacharasterisk}{\kern0pt}v\ {\isacharparenleft}{\kern0pt}{\isasymchi}\ i{\isachardot}{\kern0pt}\ v\ i{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isachardollar}{\kern0pt}\ i\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSum}j{\isasymin}UNIV{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}f\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}v\ j\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ bfun{\isachardot}{\kern0pt}Bfun\ {\isacharparenleft}{\kern0pt}{\isasymlambda}i{\isachardot}{\kern0pt}\ if\ i\ {\isacharequal}{\kern0pt}\ j\ then\ {\isadigit{1}}\ else\ {\isadigit{0}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ i{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}def\ matrix{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ matrix{\isacharunderscore}{\kern0pt}vector{\isacharunderscore}{\kern0pt}mult{\isacharunderscore}{\kern0pt}def\ mult{\isachardot}{\kern0pt}commute\ axis{\isacharunderscore}{\kern0pt}def\ blinfun{\isachardot}{\kern0pt}scaleR{\isacharunderscore}{\kern0pt}right\ vec{\isacharunderscore}{\kern0pt}lambda{\isacharunderscore}{\kern0pt}inverse{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSum}j{\isasymin}UNIV{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}f\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}v\ j\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ bfun{\isachardot}{\kern0pt}Bfun\ {\isacharparenleft}{\kern0pt}{\isasymlambda}i{\isachardot}{\kern0pt}\ if\ i\ {\isacharequal}{\kern0pt}\ j\ then\ {\isadigit{1}}\ else\ {\isadigit{0}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ i{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ intro{\isacharcolon}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}induct{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isacharequal}{\kern0pt}\ f\ {\isacharparenleft}{\kern0pt}{\isasymSum}j{\isasymin}UNIV{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}v\ j\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ bfun{\isachardot}{\kern0pt}Bfun\ {\isacharparenleft}{\kern0pt}{\isasymlambda}i{\isachardot}{\kern0pt}\ if\ i\ {\isacharequal}{\kern0pt}\ j\ then\ {\isadigit{1}}\ else\ {\isadigit{0}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ i{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ blinfun{\isachardot}{\kern0pt}sum{\isacharunderscore}{\kern0pt}right{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isacharequal}{\kern0pt}\ f\ v\ i{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isasymSum}j{\isasymin}UNIV{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}v\ j\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ bfun{\isachardot}{\kern0pt}Bfun\ {\isacharparenleft}{\kern0pt}{\isasymlambda}i{\isachardot}{\kern0pt}\ if\ i\ {\isacharequal}{\kern0pt}\ j\ then\ {\isadigit{1}}\ else\ {\isadigit{0}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ x\ {\isacharequal}{\kern0pt}\ v\ x{\isachardoublequoteclose}\ \isakeyword{for}\ x\isanewline
-\ \ \ \ \isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isasymSum}j{\isasymin}UNIV{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}v\ j\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ bfun{\isachardot}{\kern0pt}Bfun\ {\isacharparenleft}{\kern0pt}{\isasymlambda}i{\isachardot}{\kern0pt}\ if\ i\ {\isacharequal}{\kern0pt}\ j\ then\ {\isadigit{1}}\ else\ {\isadigit{0}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ x\ {\isacharequal}{\kern0pt}\ \isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymSum}j{\isasymin}UNIV{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}v\ j\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ bfun{\isachardot}{\kern0pt}Bfun\ {\isacharparenleft}{\kern0pt}{\isasymlambda}i{\isachardot}{\kern0pt}\ if\ i\ {\isacharequal}{\kern0pt}\ j\ then\ {\isadigit{1}}\ else\ {\isadigit{0}}{\isacharparenright}{\kern0pt}\ x{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ intro{\isacharcolon}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}induct{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSum}j{\isasymin}UNIV{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}v\ j\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isacharparenleft}{\kern0pt}{\isasymlambda}i{\isachardot}{\kern0pt}\ if\ i\ {\isacharequal}{\kern0pt}\ j\ then\ {\isadigit{1}}\ else\ {\isadigit{0}}{\isacharparenright}{\kern0pt}\ x{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}subst\ Bfun{\isacharunderscore}{\kern0pt}inverse{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}metis\ vec{\isacharunderscore}{\kern0pt}bfun\ vec{\isacharunderscore}{\kern0pt}lambda{\isacharunderscore}{\kern0pt}inverse{\isacharbrackleft}{\kern0pt}OF\ UNIV{\isacharunderscore}{\kern0pt}I{\isacharcomma}{\kern0pt}\ symmetric{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharplus}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSum}j{\isasymin}UNIV{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}if\ x\ {\isacharequal}{\kern0pt}\ j\ then\ v\ j\ {\isacharasterisk}{\kern0pt}\ {\isadigit{1}}\ else\ v\ j\ {\isacharasterisk}{\kern0pt}\ {\isadigit{0}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ if{\isacharunderscore}{\kern0pt}distrib\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ sum{\isachardot}{\kern0pt}cong{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSum}j{\isasymin}UNIV{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}if\ x\ {\isacharequal}{\kern0pt}\ j\ then\ v\ j\ else\ {\isadigit{0}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}meson\ more{\isacharunderscore}{\kern0pt}arith{\isacharunderscore}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{6}}{\isacharparenright}{\kern0pt}\ mult{\isacharunderscore}{\kern0pt}zero{\isacharunderscore}{\kern0pt}right{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isacharequal}{\kern0pt}\ v\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \ \ \isacommand{finally}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isacommand{{\isachardot}{\kern0pt}}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ bfun{\isacharunderscore}{\kern0pt}eqI\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ fastforce\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{finally}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isacommand{{\isachardot}{\kern0pt}}\isamarkupfalse%
-\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}mult{\isacharprime}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ f\ {\isacharasterisk}{\kern0pt}v\ v{\isacharparenright}{\kern0pt}\ {\isachardollar}{\kern0pt}\ i\ {\isacharequal}{\kern0pt}\ f\ {\isacharparenleft}{\kern0pt}Bfun\ {\isacharparenleft}{\kern0pt}{\isasymlambda}i{\isachardot}{\kern0pt}\ v\ {\isachardollar}{\kern0pt}\ i{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ i{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ bfun{\isachardot}{\kern0pt}Bfun{\isacharunderscore}{\kern0pt}inverse\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}mult\ vec{\isacharunderscore}{\kern0pt}bfun\ vec{\isacharunderscore}{\kern0pt}nth{\isacharunderscore}{\kern0pt}inverse{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}mult{\isacharprime}{\kern0pt}{\isacharprime}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ f\ {\isacharasterisk}{\kern0pt}v\ v{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymchi}\ i{\isachardot}{\kern0pt}\ f\ {\isacharparenleft}{\kern0pt}Bfun\ {\isacharparenleft}{\kern0pt}{\isasymlambda}i{\isachardot}{\kern0pt}\ v\ {\isachardollar}{\kern0pt}\ i{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ i{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}mult{\isacharprime}{\kern0pt}\ vec{\isacharunderscore}{\kern0pt}lambda{\isacharunderscore}{\kern0pt}unique{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ matrix{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}blinfun{\isacharunderscore}{\kern0pt}inv{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}matrix{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}blinfun\ {\isacharparenleft}{\kern0pt}blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ f{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ f{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ matrix{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}blinfun{\isacharunderscore}{\kern0pt}mult\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}mult\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ blinfun{\isacharunderscore}{\kern0pt}eqI{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}add{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ {\isacharparenleft}{\kern0pt}f\ {\isacharplus}{\kern0pt}\ g{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ f\ {\isacharplus}{\kern0pt}\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ g{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ matrix{\isacharunderscore}{\kern0pt}eq\ \ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}mult{\isacharprime}{\kern0pt}{\isacharprime}{\kern0pt}\ matrix{\isacharunderscore}{\kern0pt}vector{\isacharunderscore}{\kern0pt}mult{\isacharunderscore}{\kern0pt}add{\isacharunderscore}{\kern0pt}rdistrib\ blinfun{\isachardot}{\kern0pt}add{\isacharunderscore}{\kern0pt}left\ plus{\isacharunderscore}{\kern0pt}vec{\isacharunderscore}{\kern0pt}eq{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}diff{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ {\isacharparenleft}{\kern0pt}f\ {\isacharminus}{\kern0pt}\ g{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ f\ {\isacharminus}{\kern0pt}\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ g{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}add\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ add{\isacharunderscore}{\kern0pt}right{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}eq\ diff{\isacharunderscore}{\kern0pt}add{\isacharunderscore}{\kern0pt}cancel{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}scaleR{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ {\isacharparenleft}{\kern0pt}c\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ f{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ c\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ f{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ matrix{\isacharunderscore}{\kern0pt}eq\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}mult{\isacharprime}{\kern0pt}{\isacharprime}{\kern0pt}\ scaleR{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}vector{\isacharunderscore}{\kern0pt}assoc{\isacharbrackleft}{\kern0pt}symmetric{\isacharbrackright}{\kern0pt}\ \isanewline
-\ \ \ \ \ \ blinfun{\isachardot}{\kern0pt}scaleR{\isacharunderscore}{\kern0pt}left\ vector{\isacharunderscore}{\kern0pt}scalar{\isacharunderscore}{\kern0pt}mult{\isacharunderscore}{\kern0pt}def{\isacharbrackleft}{\kern0pt}of\ c{\isacharcomma}{\kern0pt}\ unfolded\ scalar{\isacharunderscore}{\kern0pt}mult{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}scaleR{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\ \isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ matrix{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}blinfun{\isacharunderscore}{\kern0pt}add{\isacharcolon}{\kern0pt}\ \isanewline
-\ \ {\isachardoublequoteopen}matrix{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}blinfun\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}f\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ real{\isacharcircum}{\kern0pt}{\isacharunderscore}{\kern0pt}{\isacharcircum}{\kern0pt}{\isacharunderscore}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharplus}{\kern0pt}\ g{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ matrix{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}blinfun\ f\ {\isacharplus}{\kern0pt}\ matrix{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}blinfun\ g{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ blinfun{\isacharunderscore}{\kern0pt}eqI\ simp{\isacharcolon}{\kern0pt}\ matrix{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}blinfun{\isacharunderscore}{\kern0pt}mult\ blinfun{\isachardot}{\kern0pt}add{\isacharunderscore}{\kern0pt}left\ matrix{\isacharunderscore}{\kern0pt}vector{\isacharunderscore}{\kern0pt}mult{\isacharunderscore}{\kern0pt}add{\isacharunderscore}{\kern0pt}rdistrib{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ matrix{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}blinfun{\isacharunderscore}{\kern0pt}diff{\isacharcolon}{\kern0pt}\ \isanewline
-\ \ {\isachardoublequoteopen}matrix{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}blinfun\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}f\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ real{\isacharcircum}{\kern0pt}{\isacharunderscore}{\kern0pt}{\isacharcircum}{\kern0pt}{\isacharunderscore}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharminus}{\kern0pt}\ g{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ matrix{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}blinfun\ f\ {\isacharminus}{\kern0pt}\ matrix{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}blinfun\ g{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ matrix{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}blinfun{\isacharunderscore}{\kern0pt}add\ diff{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}eq\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ metis%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ matrix{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}blinfun{\isacharunderscore}{\kern0pt}scaleR{\isacharcolon}{\kern0pt}\ \isanewline
-\ \ {\isachardoublequoteopen}matrix{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}blinfun\ {\isacharparenleft}{\kern0pt}c\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isacharparenleft}{\kern0pt}f\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ real{\isacharcircum}{\kern0pt}{\isacharunderscore}{\kern0pt}{\isacharcircum}{\kern0pt}{\isacharunderscore}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ c\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ matrix{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}blinfun\ f{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ blinfun{\isacharunderscore}{\kern0pt}eqI\ simp{\isacharcolon}{\kern0pt}\ matrix{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}blinfun{\isacharunderscore}{\kern0pt}mult\ blinfun{\isachardot}{\kern0pt}scaleR{\isacharunderscore}{\kern0pt}left\ \isanewline
-\ \ \ \ \ \ matrix{\isacharunderscore}{\kern0pt}vector{\isacharunderscore}{\kern0pt}mult{\isacharunderscore}{\kern0pt}add{\isacharunderscore}{\kern0pt}rdistrib\ scaleR{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}vector{\isacharunderscore}{\kern0pt}assoc{\isacharbrackleft}{\kern0pt}symmetric{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ matrix{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}blinfun{\isacharunderscore}{\kern0pt}comp{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}matrix{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}blinfun\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}m{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ real{\isacharcircum}{\kern0pt}{\isacharunderscore}{\kern0pt}{\isacharcircum}{\kern0pt}{\isacharunderscore}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharasterisk}{\kern0pt}{\isacharasterisk}{\kern0pt}\ n{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}blinfun\ m{\isacharparenright}{\kern0pt}\ o\isactrlsub L\ {\isacharparenleft}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}blinfun\ n{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ blinfun{\isacharunderscore}{\kern0pt}eqI\ simp{\isacharcolon}{\kern0pt}\ matrix{\isacharunderscore}{\kern0pt}vector{\isacharunderscore}{\kern0pt}mul{\isacharunderscore}{\kern0pt}assoc{\isacharbrackleft}{\kern0pt}symmetric{\isacharbrackright}{\kern0pt}\ matrix{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}blinfun{\isacharunderscore}{\kern0pt}mult{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}comp{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ {\isacharparenleft}{\kern0pt}f\ o\isactrlsub L\ g{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ f{\isacharparenright}{\kern0pt}\ {\isacharasterisk}{\kern0pt}{\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ g{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ matrix{\isacharunderscore}{\kern0pt}eq\ apply{\isacharunderscore}{\kern0pt}bfun{\isacharunderscore}{\kern0pt}inverse\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}mult{\isacharprime}{\kern0pt}{\isacharprime}{\kern0pt}\ matrix{\isacharunderscore}{\kern0pt}vector{\isacharunderscore}{\kern0pt}mul{\isacharunderscore}{\kern0pt}assoc{\isacharbrackleft}{\kern0pt}symmetric{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}id{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ id{\isacharunderscore}{\kern0pt}blinfun\ {\isacharequal}{\kern0pt}\ mat\ {\isadigit{1}}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ Bfun{\isacharunderscore}{\kern0pt}inverse\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}mult{\isacharprime}{\kern0pt}{\isacharprime}{\kern0pt}\ matrix{\isacharunderscore}{\kern0pt}eq{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ matrix{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}blinfun{\isacharunderscore}{\kern0pt}id{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}matrix{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}blinfun\ {\isacharparenleft}{\kern0pt}mat\ {\isadigit{1}}\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isacharparenleft}{\kern0pt}real\ {\isacharcircum}{\kern0pt}{\isacharunderscore}{\kern0pt}{\isacharcircum}{\kern0pt}{\isacharunderscore}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ id{\isacharunderscore}{\kern0pt}blinfun{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ blinfun{\isacharunderscore}{\kern0pt}eqI\ simp{\isacharcolon}{\kern0pt}\ matrix{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}blinfun{\isacharunderscore}{\kern0pt}mult{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ matrix{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}blinfun{\isacharunderscore}{\kern0pt}inv\isactrlsub L{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}invertible\ m{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}matrix{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}blinfun\ {\isacharparenleft}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}inv\ {\isacharparenleft}{\kern0pt}m\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ real{\isacharcircum}{\kern0pt}{\isacharunderscore}{\kern0pt}{\isacharcircum}{\kern0pt}{\isacharunderscore}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ inv\isactrlsub L\ {\isacharparenleft}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}blinfun\ m{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ {\isachardoublequoteopen}invertible\isactrlsub L\ {\isacharparenleft}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}blinfun\ m{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}m\ {\isacharasterisk}{\kern0pt}{\isacharasterisk}{\kern0pt}\ matrix{\isacharunderscore}{\kern0pt}inv\ m\ {\isacharequal}{\kern0pt}\ mat\ {\isadigit{1}}{\isachardoublequoteclose}\ {\isachardoublequoteopen}matrix{\isacharunderscore}{\kern0pt}inv\ m\ {\isacharasterisk}{\kern0pt}{\isacharasterisk}{\kern0pt}\ m\ {\isacharequal}{\kern0pt}\ mat\ {\isadigit{1}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assms\ \isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ matrix{\isacharunderscore}{\kern0pt}inv{\isacharunderscore}{\kern0pt}right\ matrix{\isacharunderscore}{\kern0pt}inv{\isacharunderscore}{\kern0pt}left{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}matrix{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}blinfun\ {\isacharparenleft}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}inv\ m{\isacharparenright}{\kern0pt}\ o\isactrlsub L\ matrix{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}blinfun\ m\ {\isacharequal}{\kern0pt}\ id{\isacharunderscore}{\kern0pt}blinfun{\isachardoublequoteclose}\ \isanewline
-\ \ \ \ {\isachardoublequoteopen}matrix{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}blinfun\ m\ o\isactrlsub L\ matrix{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}blinfun\ {\isacharparenleft}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}inv\ m{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ id{\isacharunderscore}{\kern0pt}blinfun{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ matrix{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}blinfun{\isacharunderscore}{\kern0pt}id\ matrix{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}blinfun{\isacharunderscore}{\kern0pt}comp{\isacharbrackleft}{\kern0pt}symmetric{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}matrix{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}blinfun\ {\isacharparenleft}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}inv\ m{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ inv\isactrlsub L\ {\isacharparenleft}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}blinfun\ m{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ {\isachardoublequoteopen}invertible\isactrlsub L\ {\isacharparenleft}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}blinfun\ m{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ intro{\isacharcolon}{\kern0pt}\ inv\isactrlsub L{\isacharunderscore}{\kern0pt}I{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}inverse{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}invertible\isactrlsub L\ X{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}invertible\ {\isacharparenleft}{\kern0pt}blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ {\isacharparenleft}{\kern0pt}X\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}b{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}finite\ {\isasymRightarrow}\isactrlsub b\ real{\isacharparenright}{\kern0pt}\ {\isasymRightarrow}\isactrlsub L\ {\isacharprime}{\kern0pt}c{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}finite\ {\isasymRightarrow}\isactrlsub b\ real{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isanewline
-\ \ \ \ {\isachardoublequoteopen}blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ {\isacharparenleft}{\kern0pt}inv\isactrlsub L\ X{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ matrix{\isacharunderscore}{\kern0pt}inv\ {\isacharparenleft}{\kern0pt}blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ X{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}X\ o\isactrlsub L\ inv\isactrlsub L\ X\ {\isacharequal}{\kern0pt}\ id{\isacharunderscore}{\kern0pt}blinfun{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ assms{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ X\ {\isacharasterisk}{\kern0pt}{\isacharasterisk}{\kern0pt}\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ {\isacharparenleft}{\kern0pt}inv\isactrlsub L\ X{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ mat\ {\isadigit{1}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}comp\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}id{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}inv\isactrlsub L\ X\ o\isactrlsub L\ X\ {\isacharequal}{\kern0pt}\ id{\isacharunderscore}{\kern0pt}blinfun{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ assms{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isadigit{2}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ {\isacharparenleft}{\kern0pt}inv\isactrlsub L\ X{\isacharparenright}{\kern0pt}\ {\isacharasterisk}{\kern0pt}{\isacharasterisk}{\kern0pt}\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ {\isacharparenleft}{\kern0pt}X{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ mat\ {\isadigit{1}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}comp\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}id{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}invertible\ {\isacharparenleft}{\kern0pt}blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ X{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isadigit{1}}{\isachardoublequoteclose}\ invertible{\isacharunderscore}{\kern0pt}def\ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ {\isacharparenleft}{\kern0pt}inv\isactrlsub L\ X{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ matrix{\isacharunderscore}{\kern0pt}inv\ {\isacharparenleft}{\kern0pt}blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ X{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isadigit{1}}\ {\isadigit{2}}\ matrix{\isacharunderscore}{\kern0pt}inv{\isacharunderscore}{\kern0pt}right\ matrix{\isacharunderscore}{\kern0pt}mul{\isacharunderscore}{\kern0pt}assoc\ matrix{\isacharunderscore}{\kern0pt}mul{\isacharunderscore}{\kern0pt}lid\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}inv{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ {\isacharparenleft}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}blinfun\ f{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ f{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ matrix{\isacharunderscore}{\kern0pt}eq\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}mult{\isacharprime}{\kern0pt}{\isacharprime}{\kern0pt}\ matrix{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}blinfun{\isacharunderscore}{\kern0pt}mult\ bfun{\isachardot}{\kern0pt}Bfun{\isacharunderscore}{\kern0pt}inverse{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ invertible{\isacharunderscore}{\kern0pt}invertible\isactrlsub L{\isacharunderscore}{\kern0pt}I{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}invertible\ {\isacharparenleft}{\kern0pt}blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ f{\isacharparenright}{\kern0pt}\ {\isasymLongrightarrow}\ invertible\isactrlsub L\ f{\isachardoublequoteclose}\isanewline
-\ \ {\isachardoublequoteopen}invertible\isactrlsub L\ {\isacharparenleft}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}blinfun\ X{\isacharparenright}{\kern0pt}\ {\isasymLongrightarrow}\ invertible\ {\isacharparenleft}{\kern0pt}X\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ real{\isacharcircum}{\kern0pt}{\isacharunderscore}{\kern0pt}{\isacharcircum}{\kern0pt}{\isacharunderscore}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ matrix{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}blinfun{\isacharunderscore}{\kern0pt}inv\isactrlsub L{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}inverse{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}\ matrix{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}blinfun{\isacharunderscore}{\kern0pt}inv\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}inv\ \isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ metis{\isacharplus}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ bounded{\isacharunderscore}{\kern0pt}linear{\isacharunderscore}{\kern0pt}blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}bounded{\isacharunderscore}{\kern0pt}linear\ {\isacharparenleft}{\kern0pt}blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}a\ {\isasymRightarrow}\isactrlsub b\ real{\isacharparenright}{\kern0pt}\ {\isasymRightarrow}\isactrlsub L\ {\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}b\ {\isasymRightarrow}\isactrlsub b\ real{\isacharparenright}{\kern0pt}\ {\isasymRightarrow}\ real{\isacharcircum}{\kern0pt}{\isacharprime}{\kern0pt}a{\isacharcircum}{\kern0pt}{\isacharprime}{\kern0pt}b{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}intro\ bounded{\isacharunderscore}{\kern0pt}linear{\isacharunderscore}{\kern0pt}intro{\isacharbrackleft}{\kern0pt}of\ {\isacharunderscore}{\kern0pt}\ {\isachardoublequoteopen}real\ CARD{\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}a{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}finite{\isacharparenright}{\kern0pt}\ {\isacharasterisk}{\kern0pt}\ real\ CARD{\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}b{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}finite{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymAnd}x\ y{\isachardot}{\kern0pt}\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ {\isacharparenleft}{\kern0pt}x\ {\isacharplus}{\kern0pt}\ y{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ x\ {\isacharplus}{\kern0pt}\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ y{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}add\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}scaleR{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymAnd}r\ x{\isachardot}{\kern0pt}\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ {\isacharparenleft}{\kern0pt}r\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ x{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ r\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}def\ matrix{\isacharunderscore}{\kern0pt}def\ blinfun{\isachardot}{\kern0pt}scaleR{\isacharunderscore}{\kern0pt}left\ vec{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}iff{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isacharasterisk}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymAnd}j{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymlambda}i{\isachardot}{\kern0pt}\ if\ i\ {\isacharequal}{\kern0pt}\ j\ then\ {\isadigit{1}}{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}real\ else\ {\isadigit{0}}{\isacharparenright}{\kern0pt}\ {\isasymin}\ bfun{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isacharasterisk}{\kern0pt}{\isacharasterisk}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymAnd}j{\isachardot}{\kern0pt}\ norm\ {\isacharparenleft}{\kern0pt}Bfun\ {\isacharparenleft}{\kern0pt}{\isasymlambda}i{\isachardot}{\kern0pt}\ if\ i\ {\isacharequal}{\kern0pt}\ j\ then\ {\isadigit{1}}{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}real\ else\ {\isadigit{0}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isadigit{1}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ Bfun{\isacharunderscore}{\kern0pt}inverse{\isacharbrackleft}{\kern0pt}OF\ {\isacharasterisk}{\kern0pt}{\isacharbrackright}{\kern0pt}\ norm{\isacharunderscore}{\kern0pt}bfun{\isacharunderscore}{\kern0pt}def{\isacharprime}{\kern0pt}\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ cSup{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}maximum\ {\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}norm\ {\isacharparenleft}{\kern0pt}blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ x{\isacharparenright}{\kern0pt}\ {\isasymle}\ norm\ x\ {\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}real\ CARD{\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}a{\isacharparenright}{\kern0pt}\ {\isacharasterisk}{\kern0pt}\ real\ CARD{\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}b{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{for}\ x\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}a\ {\isasymRightarrow}\isactrlsub b\ real{\isacharparenright}{\kern0pt}\ {\isasymRightarrow}\isactrlsub L\ {\isacharprime}{\kern0pt}b\ {\isasymRightarrow}\isactrlsub b\ real{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}norm\ {\isacharparenleft}{\kern0pt}blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ x{\isacharparenright}{\kern0pt}\ {\isasymle}\ {\isacharparenleft}{\kern0pt}{\isasymSum}i{\isasymin}UNIV{\isachardot}{\kern0pt}\ {\isasymSum}ia{\isasymin}UNIV{\isachardot}{\kern0pt}\ {\isasymbar}{\isacharparenleft}{\kern0pt}x\ {\isacharparenleft}{\kern0pt}bfun{\isachardot}{\kern0pt}Bfun\ {\isacharparenleft}{\kern0pt}{\isasymlambda}i{\isachardot}{\kern0pt}\ if\ i\ {\isacharequal}{\kern0pt}\ ia\ then\ {\isadigit{1}}\ else\ {\isadigit{0}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ i{\isasymbar}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ norm{\isacharunderscore}{\kern0pt}vec{\isacharunderscore}{\kern0pt}def\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}def\ matrix{\isacharunderscore}{\kern0pt}def\ axis{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ vec{\isacharunderscore}{\kern0pt}lambda{\isacharunderscore}{\kern0pt}inverse\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ order{\isachardot}{\kern0pt}trans{\isacharbrackleft}{\kern0pt}OF\ L{\isadigit{2}}{\isacharunderscore}{\kern0pt}set{\isacharunderscore}{\kern0pt}le{\isacharunderscore}{\kern0pt}sum{\isacharunderscore}{\kern0pt}abs{\isacharbrackright}{\kern0pt}\ order{\isachardot}{\kern0pt}trans{\isacharbrackleft}{\kern0pt}OF\ sum{\isacharunderscore}{\kern0pt}mono{\isacharbrackleft}{\kern0pt}OF\ L{\isadigit{2}}{\isacharunderscore}{\kern0pt}set{\isacharunderscore}{\kern0pt}le{\isacharunderscore}{\kern0pt}sum{\isacharunderscore}{\kern0pt}abs{\isacharbrackright}{\kern0pt}{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isasymle}\ {\isacharparenleft}{\kern0pt}{\isasymSum}i{\isasymin}{\isacharparenleft}{\kern0pt}UNIV{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}{\isacharprime}{\kern0pt}b\ set{\isacharparenright}{\kern0pt}{\isachardot}{\kern0pt}\ {\isasymSum}ia{\isasymin}{\isacharparenleft}{\kern0pt}UNIV\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isacharprime}{\kern0pt}a\ set{\isacharparenright}{\kern0pt}{\isachardot}{\kern0pt}\ norm\ x{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ norm{\isacharunderscore}{\kern0pt}blinfun\ abs{\isacharunderscore}{\kern0pt}le{\isacharunderscore}{\kern0pt}norm{\isacharunderscore}{\kern0pt}bfun\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}fastforce\ simp{\isacharcolon}{\kern0pt}\ {\isacharasterisk}{\kern0pt}{\isacharasterisk}{\kern0pt}\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ sum{\isacharunderscore}{\kern0pt}mono\ intro{\isacharcolon}{\kern0pt}\ order{\isachardot}{\kern0pt}trans{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isacharequal}{\kern0pt}\ norm\ x\ {\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}real\ CARD{\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}a{\isacharparenright}{\kern0pt}\ {\isacharasterisk}{\kern0pt}\ real\ CARD{\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}b{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \isacommand{finally}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isacommand{{\isachardot}{\kern0pt}}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ summable{\isacharunderscore}{\kern0pt}blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}summable\ {\isacharparenleft}{\kern0pt}f\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ nat\ {\isasymRightarrow}\ {\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}c{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}finite\ {\isasymRightarrow}\isactrlsub b\ {\isacharunderscore}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymRightarrow}\isactrlsub L\ {\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}c\ {\isasymRightarrow}\isactrlsub b\ {\isacharunderscore}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}summable\ {\isacharparenleft}{\kern0pt}{\isasymlambda}i{\isachardot}{\kern0pt}\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ {\isacharparenleft}{\kern0pt}f\ i{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ assms\ bounded{\isacharunderscore}{\kern0pt}linear{\isachardot}{\kern0pt}summable\ bounded{\isacharunderscore}{\kern0pt}linear{\isacharunderscore}{\kern0pt}blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{abbreviation}\isamarkupfalse%
-\ {\isachardoublequoteopen}nonneg{\isacharunderscore}{\kern0pt}blinfun\ Q\ {\isasymequiv}\ {\isadigit{0}}\ {\isasymle}\ {\isacharparenleft}{\kern0pt}blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ Q{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ nonneg{\isacharunderscore}{\kern0pt}blinfun{\isacharunderscore}{\kern0pt}mono{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}nonneg{\isacharunderscore}{\kern0pt}blinfun\ Q\ {\isasymLongrightarrow}\ u\ {\isasymle}\ v\ {\isasymLongrightarrow}\ Q\ u\ {\isasymle}\ Q\ v{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ nonneg{\isacharunderscore}{\kern0pt}mat{\isacharunderscore}{\kern0pt}mono{\isacharbrackleft}{\kern0pt}of\ {\isachardoublequoteopen}blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ Q{\isachardoublequoteclose}\ {\isachardoublequoteopen}vec{\isacharunderscore}{\kern0pt}lambda\ u{\isachardoublequoteclose}\ {\isachardoublequoteopen}vec{\isacharunderscore}{\kern0pt}lambda\ v{\isachardoublequoteclose}{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}fastforce\ simp{\isacharcolon}{\kern0pt}\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}mult{\isacharprime}{\kern0pt}{\isacharprime}{\kern0pt}\ apply{\isacharunderscore}{\kern0pt}bfun{\isacharunderscore}{\kern0pt}inverse\ Finite{\isacharunderscore}{\kern0pt}Cartesian{\isacharunderscore}{\kern0pt}Product{\isachardot}{\kern0pt}less{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}vec{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ nonneg{\isacharunderscore}{\kern0pt}blinfun{\isacharunderscore}{\kern0pt}nonneg{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}nonneg{\isacharunderscore}{\kern0pt}blinfun\ Q\ {\isasymLongrightarrow}\ {\isadigit{0}}\ {\isasymle}\ v\ {\isasymLongrightarrow}\ {\isadigit{0}}\ {\isasymle}\ Q\ v{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ nonneg{\isacharunderscore}{\kern0pt}blinfun{\isacharunderscore}{\kern0pt}mono\ blinfun{\isachardot}{\kern0pt}zero{\isacharunderscore}{\kern0pt}right\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ metis%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ nonneg{\isacharunderscore}{\kern0pt}id{\isacharunderscore}{\kern0pt}blinfun{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}nonneg{\isacharunderscore}{\kern0pt}blinfun\ id{\isacharunderscore}{\kern0pt}blinfun{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}id{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ norm{\isacharunderscore}{\kern0pt}nonneg{\isacharunderscore}{\kern0pt}blinfun{\isacharunderscore}{\kern0pt}one{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}{\isadigit{0}}\ {\isasymle}\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ X{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}norm\ X\ {\isacharequal}{\kern0pt}\ norm\ {\isacharparenleft}{\kern0pt}blinfun{\isacharunderscore}{\kern0pt}apply\ X\ {\isadigit{1}}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ norm{\isacharunderscore}{\kern0pt}blinfun{\isacharunderscore}{\kern0pt}mono{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}one\ assms\ nonneg{\isacharunderscore}{\kern0pt}blinfun{\isacharunderscore}{\kern0pt}nonneg{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ matrix{\isacharunderscore}{\kern0pt}le{\isacharunderscore}{\kern0pt}norm{\isacharunderscore}{\kern0pt}mono{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}{\isadigit{0}}\ {\isasymle}\ {\isacharparenleft}{\kern0pt}blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ C{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isakeyword{and}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ C{\isacharparenright}{\kern0pt}\ {\isasymle}\ {\isacharparenleft}{\kern0pt}blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ D{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}norm\ C\ {\isasymle}\ norm\ D{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isadigit{0}}\ {\isasymle}\ C\ {\isadigit{1}}{\isachardoublequoteclose}\ {\isachardoublequoteopen}{\isadigit{0}}\ {\isasymle}\ D\ {\isadigit{1}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assms\ zero{\isacharunderscore}{\kern0pt}le{\isacharunderscore}{\kern0pt}one\ \isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}fastforce\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ nonneg{\isacharunderscore}{\kern0pt}blinfun{\isacharunderscore}{\kern0pt}nonneg{\isacharparenright}{\kern0pt}{\isacharplus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymAnd}v{\isachardot}{\kern0pt}\ v\ {\isasymge}\ {\isadigit{0}}\ {\isasymLongrightarrow}\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ C\ {\isacharasterisk}{\kern0pt}v\ v\ {\isasymle}\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ D\ {\isacharasterisk}{\kern0pt}v\ v{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assms\ nonneg{\isacharunderscore}{\kern0pt}mat{\isacharunderscore}{\kern0pt}mono{\isacharbrackleft}{\kern0pt}of\ {\isachardoublequoteopen}blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ D\ {\isacharminus}{\kern0pt}\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ C{\isachardoublequoteclose}{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}fastforce\ simp{\isacharcolon}{\kern0pt}\ matrix{\isacharunderscore}{\kern0pt}vector{\isacharunderscore}{\kern0pt}mult{\isacharunderscore}{\kern0pt}diff{\isacharunderscore}{\kern0pt}rdistrib{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isacharasterisk}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymAnd}v{\isachardot}{\kern0pt}\ v\ {\isasymge}\ {\isadigit{0}}\ {\isasymLongrightarrow}\ C\ v\ {\isasymle}\ D\ v{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ less{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}vec{\isacharunderscore}{\kern0pt}def\ less{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}bfun{\isacharunderscore}{\kern0pt}def\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}mult{\isacharbrackleft}{\kern0pt}symmetric{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assms{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}\ assms{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}\ {\isacartoucheopen}{\isadigit{0}}\ {\isasymle}\ C\ {\isadigit{1}}{\isacartoucheclose}\ {\isacartoucheopen}{\isadigit{0}}\ {\isasymle}\ D\ {\isadigit{1}}{\isacartoucheclose}\ less{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}bfunD{\isacharbrackleft}{\kern0pt}OF\ {\isacharasterisk}{\kern0pt}{\isacharcomma}{\kern0pt}\ of\ {\isadigit{1}}{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}fastforce\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ cSUP{\isacharunderscore}{\kern0pt}mono\ simp{\isacharcolon}{\kern0pt}\ norm{\isacharunderscore}{\kern0pt}nonneg{\isacharunderscore}{\kern0pt}blinfun{\isacharunderscore}{\kern0pt}one\ norm{\isacharunderscore}{\kern0pt}bfun{\isacharunderscore}{\kern0pt}def{\isacharprime}{\kern0pt}\ less{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}bfun{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}matpow{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ {\isacharparenleft}{\kern0pt}X\ {\isacharcircum}{\kern0pt}{\isacharcircum}{\kern0pt}\ i{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ matpow\ {\isacharparenleft}{\kern0pt}blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ X{\isacharparenright}{\kern0pt}\ i{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}induction\ i{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}id\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}comp\ blinfunpow{\isacharunderscore}{\kern0pt}assoc\ simp\ del{\isacharcolon}{\kern0pt}\ blinfunpow{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ nonneg{\isacharunderscore}{\kern0pt}blinfun{\isacharunderscore}{\kern0pt}iff{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}nonneg{\isacharunderscore}{\kern0pt}blinfun\ X\ {\isasymlongleftrightarrow}\ {\isacharparenleft}{\kern0pt}{\isasymforall}v{\isasymge}{\isadigit{0}}{\isachardot}{\kern0pt}\ X\ v\ {\isasymge}\ {\isadigit{0}}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ nonneg{\isacharunderscore}{\kern0pt}mat{\isacharunderscore}{\kern0pt}iff{\isacharbrackleft}{\kern0pt}of\ {\isachardoublequoteopen}blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ X{\isachardoublequoteclose}{\isacharbrackright}{\kern0pt}\ nonneg{\isacharunderscore}{\kern0pt}blinfun{\isacharunderscore}{\kern0pt}nonneg\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}mult{\isacharprime}{\kern0pt}{\isacharprime}{\kern0pt}\ bfun{\isachardot}{\kern0pt}Bfun{\isacharunderscore}{\kern0pt}inverse\ less{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}vec{\isacharunderscore}{\kern0pt}def\ less{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}bfun{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ blinfun{\isacharunderscore}{\kern0pt}apply{\isacharunderscore}{\kern0pt}mono{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isadigit{0}}{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}real{\isacharcircum}{\kern0pt}{\isacharunderscore}{\kern0pt}{\isacharcircum}{\kern0pt}{\isacharunderscore}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymle}\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ X\ {\isasymLongrightarrow}\ {\isadigit{0}}\ {\isasymle}\ v\ {\isasymLongrightarrow}\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ X\ {\isasymle}\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ Y\ {\isasymLongrightarrow}\ X\ v\ {\isasymle}\ Y\ v{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ blinfun{\isachardot}{\kern0pt}diff{\isacharunderscore}{\kern0pt}left\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}diff\ diff{\isacharunderscore}{\kern0pt}ge{\isacharunderscore}{\kern0pt}{\isadigit{0}}{\isacharunderscore}{\kern0pt}iff{\isacharunderscore}{\kern0pt}ge\ nonneg{\isacharunderscore}{\kern0pt}blinfun{\isacharunderscore}{\kern0pt}nonneg{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-%
-\isadelimtheory
-\isanewline
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{end}\isamarkupfalse%
-%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\end{isabellebody}%
-\endinput
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diff --git a/thys/MDP-Algorithms/output/document/Code_DP.tex b/thys/MDP-Algorithms/output/document/Code_DP.tex
deleted file mode 100644
--- a/thys/MDP-Algorithms/output/document/Code_DP.tex
+++ /dev/null
@@ -1,3010 +0,0 @@
-%
-\begin{isabellebody}%
-\setisabellecontext{Code{\isacharunderscore}{\kern0pt}DP}%
-%
-\isadelimtheory
-\isanewline
-\isanewline
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{theory}\isamarkupfalse%
-\ Code{\isacharunderscore}{\kern0pt}DP\isanewline
-\ \ \isakeyword{imports}\ \isanewline
-\ \ \ \ Value{\isacharunderscore}{\kern0pt}Iteration\isanewline
-\ \ \ \ Policy{\isacharunderscore}{\kern0pt}Iteration\isanewline
-\ \ \ \ Modified{\isacharunderscore}{\kern0pt}Policy{\isacharunderscore}{\kern0pt}Iteration\isanewline
-\ \ \ \ Splitting{\isacharunderscore}{\kern0pt}Methods\isanewline
-\isanewline
-{\isachardoublequoteopen}HOL{\isacharminus}{\kern0pt}Library{\isachardot}{\kern0pt}Code{\isacharunderscore}{\kern0pt}Target{\isacharunderscore}{\kern0pt}Numeral{\isachardoublequoteclose}\isanewline
-{\isachardoublequoteopen}Gauss{\isacharunderscore}{\kern0pt}Jordan{\isachardot}{\kern0pt}Code{\isacharunderscore}{\kern0pt}Generation{\isacharunderscore}{\kern0pt}IArrays{\isachardoublequoteclose}\isanewline
-\isakeyword{begin}%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsection{Code Generation for MDP Algorithms%
-}
-\isamarkuptrue%
-%
-\isamarkupsubsection{Least Argmax%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{lemma}\isamarkupfalse%
-\ least{\isacharunderscore}{\kern0pt}list{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}sorted\ xs{\isachardoublequoteclose}\ {\isachardoublequoteopen}{\isasymexists}x\ {\isasymin}\ set\ xs{\isachardot}{\kern0pt}\ P\ x{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}LEAST\ x\ {\isasymin}\ set\ xs{\isachardot}{\kern0pt}\ P\ x{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ the\ {\isacharparenleft}{\kern0pt}find\ P\ xs{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ assms\isanewline
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}induction\ xs{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{case}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}Cons\ a\ xs{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}case\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}cases\ {\isachardoublequoteopen}P\ a{\isachardoublequoteclose}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{case}\isamarkupfalse%
-\ False\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}LEAST\ x\ {\isasymin}\ set\ {\isacharparenleft}{\kern0pt}a\ {\isacharhash}{\kern0pt}\ xs{\isacharparenright}{\kern0pt}{\isachardot}{\kern0pt}\ P\ x{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}LEAST\ x\ {\isasymin}\ set\ xs{\isachardot}{\kern0pt}\ P\ x{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ False\ Cons{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\ metis\isanewline
-\ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ False\ Cons\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ intro{\isacharcolon}{\kern0pt}\ Least{\isacharunderscore}{\kern0pt}equality{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ {\isachardoublequoteopen}least{\isacharunderscore}{\kern0pt}enum\ P\ {\isacharequal}{\kern0pt}\ the\ {\isacharparenleft}{\kern0pt}find\ P\ {\isacharparenleft}{\kern0pt}sorted{\isacharunderscore}{\kern0pt}list{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}set\ {\isacharparenleft}{\kern0pt}UNIV\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}b{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}finite{\isacharcomma}{\kern0pt}\ linorder{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ set{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ least{\isacharunderscore}{\kern0pt}enum{\isacharunderscore}{\kern0pt}eq{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymexists}x{\isachardot}{\kern0pt}\ P\ x\ {\isasymLongrightarrow}\ least{\isacharunderscore}{\kern0pt}enum\ P\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}LEAST\ x{\isachardot}{\kern0pt}\ P\ x{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ least{\isacharunderscore}{\kern0pt}list{\isacharbrackleft}{\kern0pt}symmetric{\isacharbrackright}{\kern0pt}\ least{\isacharunderscore}{\kern0pt}enum{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ {\isachardoublequoteopen}least{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}list\ f\ init\ xs\ {\isacharequal}{\kern0pt}\ \isanewline
-\ \ foldl\ {\isacharparenleft}{\kern0pt}{\isasymlambda}{\isacharparenleft}{\kern0pt}am{\isacharcomma}{\kern0pt}\ m{\isacharparenright}{\kern0pt}\ x{\isachardot}{\kern0pt}\ if\ f\ x\ {\isachargreater}{\kern0pt}\ m\ then\ {\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ f\ x{\isacharparenright}{\kern0pt}\ else\ {\isacharparenleft}{\kern0pt}am{\isacharcomma}{\kern0pt}\ m{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ init\ xs{\isachardoublequoteclose}\ \ \isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ snd{\isacharunderscore}{\kern0pt}least{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}list{\isacharcolon}{\kern0pt}\ \isanewline
-\ \ {\isachardoublequoteopen}snd\ {\isacharparenleft}{\kern0pt}least{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}list\ f\ {\isacharparenleft}{\kern0pt}n{\isacharcomma}{\kern0pt}\ f\ n{\isacharparenright}{\kern0pt}\ xs{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}MAX\ x\ {\isasymin}\ insert\ n\ {\isacharparenleft}{\kern0pt}set\ xs{\isacharparenright}{\kern0pt}{\isachardot}{\kern0pt}\ f\ x{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ least{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}list{\isacharunderscore}{\kern0pt}def\isanewline
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}induction\ xs\ arbitrary{\isacharcolon}{\kern0pt}\ n{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{case}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}Cons\ a\ xs{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{then}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}case\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}cases\ {\isachardoublequoteopen}xs\ {\isacharequal}{\kern0pt}\ {\isacharbrackleft}{\kern0pt}{\isacharbrackright}{\kern0pt}{\isachardoublequoteclose}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}fastforce\ simp{\isacharcolon}{\kern0pt}\ max{\isachardot}{\kern0pt}assoc{\isacharbrackleft}{\kern0pt}symmetric{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharplus}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ least{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}list{\isacharunderscore}{\kern0pt}snd{\isacharunderscore}{\kern0pt}fst{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}snd\ {\isacharparenleft}{\kern0pt}least{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}list\ f\ {\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ f\ x{\isacharparenright}{\kern0pt}\ xs{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ f\ {\isacharparenleft}{\kern0pt}fst\ {\isacharparenleft}{\kern0pt}least{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}list\ f\ {\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ f\ x{\isacharparenright}{\kern0pt}\ xs{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}induction\ xs\ arbitrary{\isacharcolon}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ least{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}list{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ fst{\isacharunderscore}{\kern0pt}least{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}list{\isacharcolon}{\kern0pt}\ \isanewline
-\ \ \isakeyword{fixes}\ f\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharunderscore}{\kern0pt}\ {\isasymRightarrow}\ {\isacharunderscore}{\kern0pt}\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ linorder{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}sorted\ {\isacharparenleft}{\kern0pt}n{\isacharhash}{\kern0pt}xs{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}fst\ {\isacharparenleft}{\kern0pt}least{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}list\ f\ {\isacharparenleft}{\kern0pt}n{\isacharcomma}{\kern0pt}\ f\ n{\isacharparenright}{\kern0pt}\ xs{\isacharparenright}{\kern0pt}\ \isanewline
-\ \ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}LEAST\ x{\isachardot}{\kern0pt}\ is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max\ f\ {\isacharparenleft}{\kern0pt}{\isasymlambda}x{\isachardot}{\kern0pt}\ x\ {\isasymin}\ insert\ n\ {\isacharparenleft}{\kern0pt}set\ xs{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ x{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ least{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}list{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \isacommand{using}\isamarkupfalse%
-\ assms\ \isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}induction\ xs\ arbitrary{\isacharcolon}{\kern0pt}\ n{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{case}\isamarkupfalse%
-\ Nil\isanewline
-\ \ \isacommand{then}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}case\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}def\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ Least{\isacharunderscore}{\kern0pt}equality{\isacharbrackleft}{\kern0pt}symmetric{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{case}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}Cons\ a\ xs{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}sorted\ {\isacharparenleft}{\kern0pt}a{\isacharhash}{\kern0pt}xs{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{then}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}case\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}cases\ {\isachardoublequoteopen}f\ a\ {\isachargreater}{\kern0pt}\ f\ n{\isachardoublequoteclose}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{case}\isamarkupfalse%
-\ True\isanewline
-\ \ \ \ \isacommand{then}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}fastforce\ simp{\isacharcolon}{\kern0pt}\ is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}def\ Cons{\isachardot}{\kern0pt}IH{\isacharbrackleft}{\kern0pt}OF\ {\isacartoucheopen}sorted\ {\isacharparenleft}{\kern0pt}a{\isacharhash}{\kern0pt}xs{\isacharparenright}{\kern0pt}{\isacartoucheclose}{\isacharbrackright}{\kern0pt}\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ cong{\isacharbrackleft}{\kern0pt}of\ Least{\isacharcomma}{\kern0pt}\ OF\ refl{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{case}\isamarkupfalse%
-\ False\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}LEAST\ b{\isachardot}{\kern0pt}\ is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max\ f\ {\isacharparenleft}{\kern0pt}{\isasymlambda}x{\isachardot}{\kern0pt}\ x\ {\isasymin}\ insert\ n\ {\isacharparenleft}{\kern0pt}set\ {\isacharparenleft}{\kern0pt}a\ {\isacharhash}{\kern0pt}\ xs{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ b{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}LEAST\ b{\isachardot}{\kern0pt}\ is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max\ f\ {\isacharparenleft}{\kern0pt}{\isasymlambda}x{\isachardot}{\kern0pt}\ x\ {\isasymin}\ {\isacharparenleft}{\kern0pt}set\ {\isacharparenleft}{\kern0pt}n\ {\isacharhash}{\kern0pt}\ xs{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ b{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}cases\ {\isachardoublequoteopen}is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max\ f\ {\isacharparenleft}{\kern0pt}{\isasymlambda}x{\isachardot}{\kern0pt}\ x\ {\isasymin}\ set\ {\isacharparenleft}{\kern0pt}n\ {\isacharhash}{\kern0pt}a{\isacharhash}{\kern0pt}\ xs{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ a{\isachardoublequoteclose}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{case}\isamarkupfalse%
-\ True\isanewline
-\ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}LEAST\ b{\isachardot}{\kern0pt}\ is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max\ f\ {\isacharparenleft}{\kern0pt}{\isasymlambda}x{\isachardot}{\kern0pt}\ x\ {\isasymin}\ \ {\isacharparenleft}{\kern0pt}set\ {\isacharparenleft}{\kern0pt}n{\isacharhash}{\kern0pt}a\ {\isacharhash}{\kern0pt}\ xs{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ b{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ n{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ Cons\ False\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}fastforce\ simp{\isacharcolon}{\kern0pt}\ is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}linorder\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ Least{\isacharunderscore}{\kern0pt}equality{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ False\ True\ Cons\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}fastforce\ simp{\isacharcolon}{\kern0pt}\ is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}linorder\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ Least{\isacharunderscore}{\kern0pt}equality{\isacharbrackleft}{\kern0pt}symmetric{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \isacommand{case}\isamarkupfalse%
-\ False\isanewline
-\ \ \ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}fastforce\ simp{\isacharcolon}{\kern0pt}\ not{\isacharunderscore}{\kern0pt}less\ is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}linorder\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ cong{\isacharbrackleft}{\kern0pt}of\ Least{\isacharcomma}{\kern0pt}\ OF\ refl{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ False\ Cons\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp\ add{\isacharcolon}{\kern0pt}\ Cons{\isachardot}{\kern0pt}IH{\isacharbrackleft}{\kern0pt}OF\ {\isacartoucheopen}sorted\ {\isacharparenleft}{\kern0pt}a{\isacharhash}{\kern0pt}xs{\isacharparenright}{\kern0pt}{\isacartoucheclose}{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ {\isachardoublequoteopen}least{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}enum\ f\ X\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}\isanewline
-\ \ let\ xs\ {\isacharequal}{\kern0pt}\ sorted{\isacharunderscore}{\kern0pt}list{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}set\ {\isacharparenleft}{\kern0pt}X\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharunderscore}{\kern0pt}\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}finite{\isacharcomma}{\kern0pt}\ linorder{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ set{\isacharparenright}{\kern0pt}\ in\isanewline
-\ \ fst\ {\isacharparenleft}{\kern0pt}least{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}list\ f\ {\isacharparenleft}{\kern0pt}hd\ xs{\isacharcomma}{\kern0pt}\ f\ {\isacharparenleft}{\kern0pt}hd\ xs{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}tl\ xs{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ {\isachardoublequoteopen}least{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}enum\ f\ X\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}\isanewline
-\ \ let\ xs\ {\isacharequal}{\kern0pt}\ sorted{\isacharunderscore}{\kern0pt}list{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}set\ {\isacharparenleft}{\kern0pt}X\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharunderscore}{\kern0pt}\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}finite{\isacharcomma}{\kern0pt}\ linorder{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ set{\isacharparenright}{\kern0pt}\ in\isanewline
-\ \ {\isacharparenleft}{\kern0pt}least{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}list\ f\ {\isacharparenleft}{\kern0pt}hd\ xs{\isacharcomma}{\kern0pt}\ f\ {\isacharparenleft}{\kern0pt}hd\ xs{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}tl\ xs{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ least{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}enum{\isacharunderscore}{\kern0pt}correct{\isacharcolon}{\kern0pt}\ \isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}X\ {\isasymnoteq}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}\isanewline
-\ \ {\isacharparenleft}{\kern0pt}least{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}enum\ {\isacharparenleft}{\kern0pt}f\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isacharunderscore}{\kern0pt}\ {\isasymRightarrow}\ {\isacharparenleft}{\kern0pt}{\isacharunderscore}{\kern0pt}\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ linorder{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ X{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}LEAST\ x{\isachardot}{\kern0pt}\ is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max\ f\ {\isacharparenleft}{\kern0pt}{\isasymlambda}x{\isachardot}{\kern0pt}\ x\ {\isasymin}\ X{\isacharparenright}{\kern0pt}\ x{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isacharasterisk}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}xs\ {\isasymnoteq}\ {\isacharbrackleft}{\kern0pt}{\isacharbrackright}{\kern0pt}\ {\isasymLongrightarrow}\ {\isacharparenleft}{\kern0pt}x\ {\isacharequal}{\kern0pt}\ hd\ xs\ {\isasymor}\ x\ {\isasymin}\ set\ {\isacharparenleft}{\kern0pt}tl\ xs{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymlongleftrightarrow}\ x\ {\isasymin}\ set\ xs{\isachardoublequoteclose}\ \isakeyword{for}\ x\ xs\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ list{\isachardot}{\kern0pt}set{\isacharunderscore}{\kern0pt}sel\ list{\isachardot}{\kern0pt}exhaust{\isacharunderscore}{\kern0pt}sel\ set{\isacharunderscore}{\kern0pt}ConsD\ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ least{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}enum{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ \ assms\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ Let{\isacharunderscore}{\kern0pt}def\ fst{\isacharunderscore}{\kern0pt}least{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}list\ {\isacharasterisk}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ least{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}enum{\isacharunderscore}{\kern0pt}correct{\isadigit{1}}{\isacharcolon}{\kern0pt}\ \isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}X\ {\isasymnoteq}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\ \isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}fst\ {\isacharparenleft}{\kern0pt}least{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}enum\ {\isacharparenleft}{\kern0pt}f\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isacharunderscore}{\kern0pt}\ {\isasymRightarrow}\ {\isacharparenleft}{\kern0pt}{\isacharunderscore}{\kern0pt}\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ linorder{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ X{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}LEAST\ x{\isachardot}{\kern0pt}\ is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max\ f\ {\isacharparenleft}{\kern0pt}{\isasymlambda}x{\isachardot}{\kern0pt}\ x\ {\isasymin}\ X{\isacharparenright}{\kern0pt}\ x{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isacharasterisk}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}xs\ {\isasymnoteq}\ {\isacharbrackleft}{\kern0pt}{\isacharbrackright}{\kern0pt}\ {\isasymLongrightarrow}\ {\isacharparenleft}{\kern0pt}x\ {\isacharequal}{\kern0pt}\ hd\ xs\ {\isasymor}\ x\ {\isasymin}\ set\ {\isacharparenleft}{\kern0pt}tl\ xs{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymlongleftrightarrow}\ x\ {\isasymin}\ set\ xs{\isachardoublequoteclose}\ \isakeyword{for}\ x\ xs\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ list{\isachardot}{\kern0pt}set{\isacharunderscore}{\kern0pt}sel\ list{\isachardot}{\kern0pt}exhaust{\isacharunderscore}{\kern0pt}sel\ set{\isacharunderscore}{\kern0pt}ConsD\ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assms\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ least{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}enum{\isacharunderscore}{\kern0pt}def\ Let{\isacharunderscore}{\kern0pt}def\ fst{\isacharunderscore}{\kern0pt}least{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}list\ {\isacharasterisk}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ least{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}enum{\isacharunderscore}{\kern0pt}correct{\isadigit{2}}{\isacharcolon}{\kern0pt}\ \isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}X\ {\isasymnoteq}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}snd\ {\isacharparenleft}{\kern0pt}least{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}enum\ {\isacharparenleft}{\kern0pt}f\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isacharunderscore}{\kern0pt}\ {\isasymRightarrow}\ {\isacharparenleft}{\kern0pt}{\isacharunderscore}{\kern0pt}\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ linorder{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ X{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}MAX\ x\ {\isasymin}\ X{\isachardot}{\kern0pt}\ f\ x{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isacharasterisk}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}xs\ {\isasymnoteq}\ {\isacharbrackleft}{\kern0pt}{\isacharbrackright}{\kern0pt}\ {\isasymLongrightarrow}\ insert\ {\isacharparenleft}{\kern0pt}hd\ xs{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}set\ {\isacharparenleft}{\kern0pt}tl\ xs{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ set\ xs{\isachardoublequoteclose}\ \isakeyword{for}\ xs\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ list{\isachardot}{\kern0pt}exhaust{\isacharunderscore}{\kern0pt}sel\ list{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isadigit{5}}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assms\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ least{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}enum{\isacharunderscore}{\kern0pt}def\ Let{\isacharunderscore}{\kern0pt}def\ snd{\isacharunderscore}{\kern0pt}least{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}list\ {\isacharasterisk}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Functions as Vectors%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{typedef}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}a{\isacharcomma}{\kern0pt}\ {\isacharprime}{\kern0pt}b{\isacharparenright}{\kern0pt}\ Fun\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}UNIV\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}a\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}b{\isacharparenright}{\kern0pt}\ set{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ blast%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{setup{\isacharunderscore}{\kern0pt}lifting}\isamarkupfalse%
-\ type{\isacharunderscore}{\kern0pt}definition{\isacharunderscore}{\kern0pt}Fun\isanewline
-\isanewline
-\isacommand{lift{\isacharunderscore}{\kern0pt}definition}\isamarkupfalse%
-\ to{\isacharunderscore}{\kern0pt}Fun\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}a\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}b{\isacharparenright}{\kern0pt}\ {\isasymRightarrow}\ {\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}a{\isacharcomma}{\kern0pt}\ {\isacharprime}{\kern0pt}b{\isacharparenright}{\kern0pt}\ Fun{\isachardoublequoteclose}\ \isakeyword{is}\ id%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{{\isachardot}{\kern0pt}}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ {\isachardoublequoteopen}fun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}vec\ {\isacharparenleft}{\kern0pt}v\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}a{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}finite{\isacharcomma}{\kern0pt}\ {\isacharprime}{\kern0pt}b{\isacharparenright}{\kern0pt}\ Fun{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ vec{\isacharunderscore}{\kern0pt}lambda\ {\isacharparenleft}{\kern0pt}Rep{\isacharunderscore}{\kern0pt}Fun\ v{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{lift{\isacharunderscore}{\kern0pt}definition}\isamarkupfalse%
-\ vec{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}fun\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}b{\isacharcircum}{\kern0pt}{\isacharprime}{\kern0pt}a\ {\isasymRightarrow}\ {\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}a{\isacharcomma}{\kern0pt}\ {\isacharprime}{\kern0pt}b{\isacharparenright}{\kern0pt}\ Fun{\isachardoublequoteclose}\ \isakeyword{is}\ vec{\isacharunderscore}{\kern0pt}nth%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{{\isachardot}{\kern0pt}}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ Fun{\isacharunderscore}{\kern0pt}inverse{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}Rep{\isacharunderscore}{\kern0pt}Fun\ {\isacharparenleft}{\kern0pt}Abs{\isacharunderscore}{\kern0pt}Fun\ f{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ f{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ Abs{\isacharunderscore}{\kern0pt}Fun{\isacharunderscore}{\kern0pt}inverse\ \isacommand{by}\isamarkupfalse%
-\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lift{\isacharunderscore}{\kern0pt}definition}\isamarkupfalse%
-\ zero{\isacharunderscore}{\kern0pt}Fun\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}a{\isacharcomma}{\kern0pt}\ {\isacharprime}{\kern0pt}b{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}zero{\isacharparenright}{\kern0pt}\ Fun{\isachardoublequoteclose}\ \isakeyword{is}\ {\isadigit{0}}%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{{\isachardot}{\kern0pt}}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{code{\isacharunderscore}{\kern0pt}datatype}\isamarkupfalse%
-\ vec{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}fun\isanewline
-\isanewline
-\isacommand{lemmas}\isamarkupfalse%
-\ vec{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}fun{\isachardot}{\kern0pt}rep{\isacharunderscore}{\kern0pt}eq{\isacharbrackleft}{\kern0pt}code{\isacharbrackright}{\kern0pt}\isanewline
-\isanewline
-\isacommand{instantiation}\isamarkupfalse%
-\ Fun\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isacharparenleft}{\kern0pt}enum{\isacharcomma}{\kern0pt}\ equal{\isacharparenright}{\kern0pt}\ equal\isanewline
-\isakeyword{begin}\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ {\isachardoublequoteopen}equal{\isacharunderscore}{\kern0pt}Fun\ {\isacharparenleft}{\kern0pt}f\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}a{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}enum{\isacharcomma}{\kern0pt}\ {\isacharprime}{\kern0pt}b{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}equal{\isacharparenright}{\kern0pt}\ Fun{\isacharparenright}{\kern0pt}\ g\ {\isacharequal}{\kern0pt}\ \ {\isacharparenleft}{\kern0pt}Rep{\isacharunderscore}{\kern0pt}Fun\ f\ {\isacharequal}{\kern0pt}\ Rep{\isacharunderscore}{\kern0pt}Fun\ g{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isacommand{instance}\isamarkupfalse%
-\ \isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ standard\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ equal{\isacharunderscore}{\kern0pt}Fun{\isacharunderscore}{\kern0pt}def\ Rep{\isacharunderscore}{\kern0pt}Fun{\isacharunderscore}{\kern0pt}inject{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isacommand{end}\isamarkupfalse%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Bounded Functions as Vectors%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{lemma}\isamarkupfalse%
-\ Bfun{\isacharunderscore}{\kern0pt}inverse{\isacharunderscore}{\kern0pt}fin{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}apply{\isacharunderscore}{\kern0pt}bfun\ {\isacharparenleft}{\kern0pt}Bfun\ {\isacharparenleft}{\kern0pt}f\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isacharprime}{\kern0pt}c\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ finite\ {\isasymRightarrow}\ {\isacharunderscore}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ f{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ finite\ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}fastforce\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ Bfun{\isacharunderscore}{\kern0pt}inverse\ simp{\isacharcolon}{\kern0pt}\ bfun{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ {\isachardoublequoteopen}bfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}vec\ {\isacharparenleft}{\kern0pt}v\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}a{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}finite{\isacharparenright}{\kern0pt}\ {\isasymRightarrow}\isactrlsub b\ {\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}b{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}metric{\isacharunderscore}{\kern0pt}space{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ vec{\isacharunderscore}{\kern0pt}lambda\ v{\isachardoublequoteclose}\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ {\isachardoublequoteopen}vec{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}bfun\ v\ {\isacharequal}{\kern0pt}\ Bfun\ {\isacharparenleft}{\kern0pt}vec{\isacharunderscore}{\kern0pt}nth\ v{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{code{\isacharunderscore}{\kern0pt}datatype}\isamarkupfalse%
-\ vec{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}bfun\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ apply{\isacharunderscore}{\kern0pt}bfun{\isacharunderscore}{\kern0pt}vec{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}bfun{\isacharbrackleft}{\kern0pt}code{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}apply{\isacharunderscore}{\kern0pt}bfun\ {\isacharparenleft}{\kern0pt}vec{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}bfun\ f{\isacharparenright}{\kern0pt}\ x\ {\isacharequal}{\kern0pt}\ f\ {\isachardollar}{\kern0pt}\ x{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ vec{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}bfun{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ {\isacharbrackleft}{\kern0pt}code{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isadigit{0}}\ {\isacharequal}{\kern0pt}\ vec{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}bfun\ {\isadigit{0}}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ vec{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}bfun{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{IArrays with Lengths in the Type%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{typedef}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}s\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ mod{\isacharunderscore}{\kern0pt}type{\isacharcomma}{\kern0pt}\ {\isacharprime}{\kern0pt}a{\isacharparenright}{\kern0pt}\ iarray{\isacharunderscore}{\kern0pt}type\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}arr\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isacharprime}{\kern0pt}a\ iarray{\isachardot}{\kern0pt}\ IArray{\isachardot}{\kern0pt}length\ arr\ {\isacharequal}{\kern0pt}\ CARD{\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}s{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ someI{\isacharunderscore}{\kern0pt}ex{\isacharbrackleft}{\kern0pt}OF\ Ex{\isacharunderscore}{\kern0pt}list{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}length{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ exI{\isacharbrackleft}{\kern0pt}of\ {\isacharunderscore}{\kern0pt}\ {\isachardoublequoteopen}IArray\ {\isacharparenleft}{\kern0pt}SOME\ xs{\isachardot}{\kern0pt}\ length\ xs\ {\isacharequal}{\kern0pt}\ CARD{\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}s{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{setup{\isacharunderscore}{\kern0pt}lifting}\isamarkupfalse%
-\ type{\isacharunderscore}{\kern0pt}definition{\isacharunderscore}{\kern0pt}iarray{\isacharunderscore}{\kern0pt}type\isanewline
-\isanewline
-\isacommand{lift{\isacharunderscore}{\kern0pt}definition}\isamarkupfalse%
-\ fun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}iarray{\isacharunderscore}{\kern0pt}t\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}s{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}{\isacharbraceleft}{\kern0pt}mod{\isacharunderscore}{\kern0pt}type{\isacharbraceright}{\kern0pt}\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a{\isacharparenright}{\kern0pt}\ {\isasymRightarrow}\ {\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}s{\isacharcomma}{\kern0pt}\ {\isacharprime}{\kern0pt}a{\isacharparenright}{\kern0pt}\ iarray{\isacharunderscore}{\kern0pt}type{\isachardoublequoteclose}\ \isakeyword{is}\ {\isachardoublequoteopen}{\isasymlambda}f{\isachardot}{\kern0pt}\ IArray{\isachardot}{\kern0pt}of{\isacharunderscore}{\kern0pt}fun\ {\isacharparenleft}{\kern0pt}{\isasymlambda}s{\isachardot}{\kern0pt}\ f\ {\isacharparenleft}{\kern0pt}from{\isacharunderscore}{\kern0pt}nat\ s{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}CARD{\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}s{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lift{\isacharunderscore}{\kern0pt}definition}\isamarkupfalse%
-\ iarray{\isacharunderscore}{\kern0pt}t{\isacharunderscore}{\kern0pt}sub\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}s{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}mod{\isacharunderscore}{\kern0pt}type{\isacharcomma}{\kern0pt}\ {\isacharprime}{\kern0pt}a{\isacharparenright}{\kern0pt}\ iarray{\isacharunderscore}{\kern0pt}type\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}s\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\ \isakeyword{is}\ {\isachardoublequoteopen}{\isasymlambda}v\ x{\isachardot}{\kern0pt}\ IArray{\isachardot}{\kern0pt}sub\ v\ {\isacharparenleft}{\kern0pt}to{\isacharunderscore}{\kern0pt}nat\ x{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{{\isachardot}{\kern0pt}}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{lift{\isacharunderscore}{\kern0pt}definition}\isamarkupfalse%
-\ iarray{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}vec\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}s{\isacharcomma}{\kern0pt}\ {\isacharprime}{\kern0pt}a{\isacharparenright}{\kern0pt}\ iarray{\isacharunderscore}{\kern0pt}type\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a{\isacharcircum}{\kern0pt}{\isacharprime}{\kern0pt}s{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}{\isacharbraceleft}{\kern0pt}mod{\isacharunderscore}{\kern0pt}type{\isacharcomma}{\kern0pt}\ finite{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\ \isanewline
-\ \ \isakeyword{is}\ {\isachardoublequoteopen}{\isasymlambda}v{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymchi}\ s{\isachardot}{\kern0pt}\ IArray{\isachardot}{\kern0pt}sub\ v\ {\isacharparenleft}{\kern0pt}to{\isacharunderscore}{\kern0pt}nat\ s{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{{\isachardot}{\kern0pt}}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{lift{\isacharunderscore}{\kern0pt}definition}\isamarkupfalse%
-\ vec{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}iarray\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isacharcircum}{\kern0pt}{\isacharprime}{\kern0pt}s{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}{\isacharbraceleft}{\kern0pt}mod{\isacharunderscore}{\kern0pt}type{\isacharcomma}{\kern0pt}\ finite{\isacharbraceright}{\kern0pt}\ {\isasymRightarrow}\ {\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}s{\isacharcomma}{\kern0pt}\ {\isacharprime}{\kern0pt}a{\isacharparenright}{\kern0pt}\ iarray{\isacharunderscore}{\kern0pt}type{\isachardoublequoteclose}\ \isanewline
-\ \ \isakeyword{is}\ {\isachardoublequoteopen}{\isasymlambda}v{\isachardot}{\kern0pt}\ IArray{\isachardot}{\kern0pt}of{\isacharunderscore}{\kern0pt}fun\ {\isacharparenleft}{\kern0pt}{\isasymlambda}s{\isachardot}{\kern0pt}\ v\ {\isachardollar}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}from{\isacharunderscore}{\kern0pt}nat\ s{\isacharparenright}{\kern0pt}\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isacharprime}{\kern0pt}s{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}CARD{\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}s{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ length{\isacharunderscore}{\kern0pt}iarray{\isacharunderscore}{\kern0pt}type\ {\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}length\ {\isacharparenleft}{\kern0pt}IArray{\isachardot}{\kern0pt}list{\isacharunderscore}{\kern0pt}of\ {\isacharparenleft}{\kern0pt}Rep{\isacharunderscore}{\kern0pt}iarray{\isacharunderscore}{\kern0pt}type\ {\isacharparenleft}{\kern0pt}v{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}s{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}{\isacharbraceleft}{\kern0pt}mod{\isacharunderscore}{\kern0pt}type{\isacharbraceright}{\kern0pt}{\isacharcomma}{\kern0pt}\ {\isacharprime}{\kern0pt}a{\isacharparenright}{\kern0pt}\ iarray{\isacharunderscore}{\kern0pt}type{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ CARD{\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}s{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ Rep{\isacharunderscore}{\kern0pt}iarray{\isacharunderscore}{\kern0pt}type\ \isacommand{by}\isamarkupfalse%
-\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ iarray{\isacharunderscore}{\kern0pt}t{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}iff{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}v\ {\isacharequal}{\kern0pt}\ w{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymforall}x{\isachardot}{\kern0pt}\ iarray{\isacharunderscore}{\kern0pt}t{\isacharunderscore}{\kern0pt}sub\ v\ x\ {\isacharequal}{\kern0pt}\ iarray{\isacharunderscore}{\kern0pt}t{\isacharunderscore}{\kern0pt}sub\ w\ x{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ iarray{\isacharunderscore}{\kern0pt}t{\isacharunderscore}{\kern0pt}sub{\isachardot}{\kern0pt}rep{\isacharunderscore}{\kern0pt}eq\ IArray{\isachardot}{\kern0pt}sub{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ Rep{\isacharunderscore}{\kern0pt}iarray{\isacharunderscore}{\kern0pt}type{\isacharunderscore}{\kern0pt}inject\ iarray{\isacharunderscore}{\kern0pt}exhaust{\isadigit{2}}\ length{\isacharunderscore}{\kern0pt}iarray{\isacharunderscore}{\kern0pt}type\ list{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}iff{\isacharunderscore}{\kern0pt}nth{\isacharunderscore}{\kern0pt}eq\ to{\isacharunderscore}{\kern0pt}nat{\isacharunderscore}{\kern0pt}from{\isacharunderscore}{\kern0pt}nat{\isacharunderscore}{\kern0pt}id{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ iarray{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}vec{\isacharunderscore}{\kern0pt}inv{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}iarray{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}vec\ {\isacharparenleft}{\kern0pt}vec{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}iarray\ v{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ v{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ to{\isacharunderscore}{\kern0pt}nat{\isacharunderscore}{\kern0pt}less{\isacharunderscore}{\kern0pt}card\ iarray{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}vec{\isachardot}{\kern0pt}rep{\isacharunderscore}{\kern0pt}eq\ vec{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}iarray{\isachardot}{\kern0pt}rep{\isacharunderscore}{\kern0pt}eq\ vec{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}iff{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ vec{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}iarray{\isacharunderscore}{\kern0pt}inv{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}vec{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}iarray\ {\isacharparenleft}{\kern0pt}iarray{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}vec\ v{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ v{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ to{\isacharunderscore}{\kern0pt}nat{\isacharunderscore}{\kern0pt}less{\isacharunderscore}{\kern0pt}card\ iarray{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}vec{\isachardot}{\kern0pt}rep{\isacharunderscore}{\kern0pt}eq\ vec{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}iarray{\isachardot}{\kern0pt}rep{\isacharunderscore}{\kern0pt}eq\ iarray{\isacharunderscore}{\kern0pt}t{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}iff\ iarray{\isacharunderscore}{\kern0pt}t{\isacharunderscore}{\kern0pt}sub{\isachardot}{\kern0pt}rep{\isacharunderscore}{\kern0pt}eq{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{code{\isacharunderscore}{\kern0pt}datatype}\isamarkupfalse%
-\ iarray{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}vec\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ vec{\isacharunderscore}{\kern0pt}nth{\isacharunderscore}{\kern0pt}iarray{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}vec{\isacharbrackleft}{\kern0pt}code{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}vec{\isacharunderscore}{\kern0pt}nth\ {\isacharparenleft}{\kern0pt}iarray{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}vec\ v{\isacharparenright}{\kern0pt}\ x\ {\isacharequal}{\kern0pt}\ iarray{\isacharunderscore}{\kern0pt}t{\isacharunderscore}{\kern0pt}sub\ v\ x{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ iarray{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}vec{\isachardot}{\kern0pt}rep{\isacharunderscore}{\kern0pt}eq\ iarray{\isacharunderscore}{\kern0pt}t{\isacharunderscore}{\kern0pt}sub{\isachardot}{\kern0pt}rep{\isacharunderscore}{\kern0pt}eq{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ vec{\isacharunderscore}{\kern0pt}lambda{\isacharunderscore}{\kern0pt}iarray{\isacharunderscore}{\kern0pt}t{\isacharbrackleft}{\kern0pt}code{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}vec{\isacharunderscore}{\kern0pt}lambda\ v\ {\isacharequal}{\kern0pt}\ iarray{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}vec\ {\isacharparenleft}{\kern0pt}fun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}iarray{\isacharunderscore}{\kern0pt}t\ v{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ iarray{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}vec{\isachardot}{\kern0pt}rep{\isacharunderscore}{\kern0pt}eq\ fun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}iarray{\isacharunderscore}{\kern0pt}t{\isachardot}{\kern0pt}rep{\isacharunderscore}{\kern0pt}eq\ to{\isacharunderscore}{\kern0pt}nat{\isacharunderscore}{\kern0pt}less{\isacharunderscore}{\kern0pt}card{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ zero{\isacharunderscore}{\kern0pt}iarray{\isacharbrackleft}{\kern0pt}code{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isadigit{0}}\ {\isacharequal}{\kern0pt}\ iarray{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}vec\ {\isacharparenleft}{\kern0pt}fun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}iarray{\isacharunderscore}{\kern0pt}t\ {\isadigit{0}}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ iarray{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}vec{\isachardot}{\kern0pt}rep{\isacharunderscore}{\kern0pt}eq\ fun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}iarray{\isacharunderscore}{\kern0pt}t{\isachardot}{\kern0pt}rep{\isacharunderscore}{\kern0pt}eq\ to{\isacharunderscore}{\kern0pt}nat{\isacharunderscore}{\kern0pt}less{\isacharunderscore}{\kern0pt}card\ vec{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}iff{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Value Iteration%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{locale}\isamarkupfalse%
-\ vi{\isacharunderscore}{\kern0pt}code\ {\isacharequal}{\kern0pt}\ \isanewline
-\ \ MDP{\isacharunderscore}{\kern0pt}ord\ A\ K\ r\ l\ \isakeyword{for}\ A\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}s{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}mod{\isacharunderscore}{\kern0pt}type\ {\isasymRightarrow}\ {\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}a{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}{\isacharbraceleft}{\kern0pt}finite{\isacharcomma}{\kern0pt}\ wellorder{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ set{\isachardoublequoteclose}\ \isanewline
-\ \ \isakeyword{and}\ K\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}s{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}{\isacharbraceleft}{\kern0pt}finite{\isacharcomma}{\kern0pt}\ mod{\isacharunderscore}{\kern0pt}type{\isacharbraceright}{\kern0pt}\ {\isasymtimes}\ {\isacharprime}{\kern0pt}a{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}{\isacharbraceleft}{\kern0pt}finite{\isacharcomma}{\kern0pt}\ wellorder{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}s\ pmf{\isachardoublequoteclose}\ \isakeyword{and}\ r\ l\isanewline
-\isakeyword{begin}\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ {\isachardoublequoteopen}vi{\isacharunderscore}{\kern0pt}test\ {\isacharparenleft}{\kern0pt}v{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}{\isacharprime}{\kern0pt}s{\isasymRightarrow}\isactrlsub b\ real{\isacharparenright}{\kern0pt}\ v{\isacharprime}{\kern0pt}\ eps\ {\isacharequal}{\kern0pt}\ {\isadigit{2}}\ {\isacharasterisk}{\kern0pt}\ l\ {\isacharasterisk}{\kern0pt}\ dist\ v\ v{\isacharprime}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{partial{\isacharunderscore}{\kern0pt}function}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}tailrec{\isacharparenright}{\kern0pt}\ value{\isacharunderscore}{\kern0pt}iteration{\isacharunderscore}{\kern0pt}partial\ \isakeyword{where}\ {\isacharbrackleft}{\kern0pt}code{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}value{\isacharunderscore}{\kern0pt}iteration{\isacharunderscore}{\kern0pt}partial\ eps\ v\ {\isacharequal}{\kern0pt}\ \isanewline
-\ \ {\isacharparenleft}{\kern0pt}let\ v{\isacharprime}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isasymL}\isactrlsub b\ v\ in\isanewline
-\ \ {\isacharparenleft}{\kern0pt}if\ {\isadigit{2}}\ {\isacharasterisk}{\kern0pt}\ l\ {\isacharasterisk}{\kern0pt}\ dist\ v\ v{\isacharprime}{\kern0pt}\ {\isacharless}{\kern0pt}\ eps\ {\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}\ {\isacharminus}{\kern0pt}\ l{\isacharparenright}{\kern0pt}\ then\ v{\isacharprime}{\kern0pt}\ else\ {\isacharparenleft}{\kern0pt}value{\isacharunderscore}{\kern0pt}iteration{\isacharunderscore}{\kern0pt}partial\ eps\ v{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ vi{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}partial{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}eps\ {\isachargreater}{\kern0pt}\ {\isadigit{0}}\ {\isasymLongrightarrow}\ value{\isacharunderscore}{\kern0pt}iteration{\isacharunderscore}{\kern0pt}partial\ eps\ v\ {\isacharequal}{\kern0pt}\ value{\isacharunderscore}{\kern0pt}iteration\ eps\ v{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}induction\ eps\ v\ rule{\isacharcolon}{\kern0pt}\ value{\isacharunderscore}{\kern0pt}iteration{\isachardot}{\kern0pt}induct{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{case}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}\ eps\ v{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{then}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}case\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ Let{\isacharunderscore}{\kern0pt}def\ value{\isacharunderscore}{\kern0pt}iteration{\isachardot}{\kern0pt}simps\ value{\isacharunderscore}{\kern0pt}iteration{\isacharunderscore}{\kern0pt}partial{\isachardot}{\kern0pt}simps{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ {\isachardoublequoteopen}L{\isacharunderscore}{\kern0pt}det\ d\ {\isacharequal}{\kern0pt}\ L\ {\isacharparenleft}{\kern0pt}mk{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\ d{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ code{\isacharunderscore}{\kern0pt}L{\isacharunderscore}{\kern0pt}det\ {\isacharbrackleft}{\kern0pt}code{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}L{\isacharunderscore}{\kern0pt}det\ d\ {\isacharparenleft}{\kern0pt}vec{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}bfun\ v{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ vec{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}bfun\ {\isacharparenleft}{\kern0pt}{\isasymchi}\ s{\isachardot}{\kern0pt}\ L\isactrlsub a\ {\isacharparenleft}{\kern0pt}d\ s{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}vec{\isacharunderscore}{\kern0pt}nth\ v{\isacharparenright}{\kern0pt}\ s{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ L{\isacharunderscore}{\kern0pt}det{\isacharunderscore}{\kern0pt}def\ vec{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}bfun{\isacharunderscore}{\kern0pt}def\ L{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}L\isactrlsub a{\isacharunderscore}{\kern0pt}det{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ code{\isacharunderscore}{\kern0pt}{\isasymL}\isactrlsub b\ {\isacharbrackleft}{\kern0pt}code{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymL}\isactrlsub b\ {\isacharparenleft}{\kern0pt}vec{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}bfun\ v{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ vec{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}bfun\ {\isacharparenleft}{\kern0pt}{\isasymchi}\ s{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}MAX\ a\ {\isasymin}\ A\ s{\isachardot}{\kern0pt}\ r\ {\isacharparenleft}{\kern0pt}s{\isacharcomma}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isacharplus}{\kern0pt}\ l\ {\isacharasterisk}{\kern0pt}\ measure{\isacharunderscore}{\kern0pt}pmf{\isachardot}{\kern0pt}expectation\ {\isacharparenleft}{\kern0pt}K\ {\isacharparenleft}{\kern0pt}s{\isacharcomma}{\kern0pt}\ a{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}vec{\isacharunderscore}{\kern0pt}nth\ v{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ vec{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}bfun{\isacharunderscore}{\kern0pt}def\ {\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}fin{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}det\ A{\isacharunderscore}{\kern0pt}ne\ cSup{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}Max{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ code{\isacharunderscore}{\kern0pt}value{\isacharunderscore}{\kern0pt}iteration{\isacharbrackleft}{\kern0pt}code{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}value{\isacharunderscore}{\kern0pt}iteration\ eps\ {\isacharparenleft}{\kern0pt}vec{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}bfun\ v{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ \isanewline
-\ \ {\isacharparenleft}{\kern0pt}if\ eps\ {\isasymle}\ {\isadigit{0}}\ then\ {\isasymL}\isactrlsub b\ {\isacharparenleft}{\kern0pt}vec{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}bfun\ v{\isacharparenright}{\kern0pt}\ else\ value{\isacharunderscore}{\kern0pt}iteration{\isacharunderscore}{\kern0pt}partial\ eps\ {\isacharparenleft}{\kern0pt}vec{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}bfun\ v{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ value{\isacharunderscore}{\kern0pt}iteration{\isachardot}{\kern0pt}simps\ vi{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}partial{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lift{\isacharunderscore}{\kern0pt}definition}\isamarkupfalse%
-\ find{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}impl\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}s\ {\isasymRightarrow}\isactrlsub b\ real{\isacharparenright}{\kern0pt}\ {\isasymRightarrow}\ {\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}s{\isacharcomma}{\kern0pt}\ {\isacharprime}{\kern0pt}a{\isacharparenright}{\kern0pt}\ Fun{\isachardoublequoteclose}\ \isakeyword{is}\ {\isachardoublequoteopen}{\isasymlambda}v{\isachardot}{\kern0pt}\ find{\isacharunderscore}{\kern0pt}policy{\isacharprime}{\kern0pt}\ v{\isachardoublequoteclose}%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{{\isachardot}{\kern0pt}}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ code{\isacharunderscore}{\kern0pt}find{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}impl{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}find{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}impl\ v\ {\isacharequal}{\kern0pt}\ vec{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}fun\ {\isacharparenleft}{\kern0pt}{\isasymchi}\ s{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}LEAST\ x{\isachardot}{\kern0pt}\ x\ {\isasymin}\ opt{\isacharunderscore}{\kern0pt}acts\ v\ s{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ vec{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}fun{\isacharunderscore}{\kern0pt}def\ find{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}impl{\isacharunderscore}{\kern0pt}def\ find{\isacharunderscore}{\kern0pt}policy{\isacharprime}{\kern0pt}{\isacharunderscore}{\kern0pt}def\ Abs{\isacharunderscore}{\kern0pt}Fun{\isacharunderscore}{\kern0pt}inject{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ code{\isacharunderscore}{\kern0pt}find{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}impl{\isacharunderscore}{\kern0pt}opt{\isacharbrackleft}{\kern0pt}code{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}find{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}impl\ v\ {\isacharequal}{\kern0pt}\ vec{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}fun\ {\isacharparenleft}{\kern0pt}{\isasymchi}\ s{\isachardot}{\kern0pt}\ least{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}enum\ {\isacharparenleft}{\kern0pt}{\isasymlambda}a{\isachardot}{\kern0pt}\ L\isactrlsub a\ a\ v\ s{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}A\ s{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ is{\isacharunderscore}{\kern0pt}opt{\isacharunderscore}{\kern0pt}act{\isacharunderscore}{\kern0pt}def\ code{\isacharunderscore}{\kern0pt}find{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}impl\ least{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}enum{\isacharunderscore}{\kern0pt}correct{\isacharbrackleft}{\kern0pt}OF\ A{\isacharunderscore}{\kern0pt}ne{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ code{\isacharunderscore}{\kern0pt}vi{\isacharunderscore}{\kern0pt}policy{\isacharprime}{\kern0pt}{\isacharbrackleft}{\kern0pt}code{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}vi{\isacharunderscore}{\kern0pt}policy{\isacharprime}{\kern0pt}\ eps\ v\ {\isacharequal}{\kern0pt}\ Rep{\isacharunderscore}{\kern0pt}Fun\ {\isacharparenleft}{\kern0pt}find{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}impl\ {\isacharparenleft}{\kern0pt}value{\isacharunderscore}{\kern0pt}iteration\ eps\ v{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ vi{\isacharunderscore}{\kern0pt}policy{\isacharprime}{\kern0pt}{\isacharunderscore}{\kern0pt}def\ find{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}impl{\isacharunderscore}{\kern0pt}def\ \isacommand{by}\isamarkupfalse%
-\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Policy Iteration%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{partial{\isacharunderscore}{\kern0pt}function}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}tailrec{\isacharparenright}{\kern0pt}\ policy{\isacharunderscore}{\kern0pt}iteration{\isacharunderscore}{\kern0pt}partial\ \isakeyword{where}\ {\isacharbrackleft}{\kern0pt}code{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}policy{\isacharunderscore}{\kern0pt}iteration{\isacharunderscore}{\kern0pt}partial\ d\ {\isacharequal}{\kern0pt}\ \isanewline
-\ \ {\isacharparenleft}{\kern0pt}let\ d{\isacharprime}{\kern0pt}\ {\isacharequal}{\kern0pt}\ policy{\isacharunderscore}{\kern0pt}step\ d\ in\ if\ d\ {\isacharequal}{\kern0pt}\ d{\isacharprime}{\kern0pt}\ then\ d\ else\ policy{\isacharunderscore}{\kern0pt}iteration{\isacharunderscore}{\kern0pt}partial\ d{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ pi{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}partial{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}d\ {\isasymin}\ D\isactrlsub D\ {\isasymLongrightarrow}\ policy{\isacharunderscore}{\kern0pt}iteration{\isacharunderscore}{\kern0pt}partial\ d\ {\isacharequal}{\kern0pt}\ policy{\isacharunderscore}{\kern0pt}iteration\ d{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}induction\ d\ rule{\isacharcolon}{\kern0pt}\ policy{\isacharunderscore}{\kern0pt}iteration{\isachardot}{\kern0pt}induct{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{case}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}\ d{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{then}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}case\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ Let{\isacharunderscore}{\kern0pt}def\ is{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det{\isacharunderscore}{\kern0pt}pi\ policy{\isacharunderscore}{\kern0pt}step{\isacharunderscore}{\kern0pt}def\ policy{\isacharunderscore}{\kern0pt}iteration{\isacharunderscore}{\kern0pt}partial{\isachardot}{\kern0pt}simps{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ {\isachardoublequoteopen}P{\isacharunderscore}{\kern0pt}mat\ d\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymchi}\ i\ j{\isachardot}{\kern0pt}\ pmf\ {\isacharparenleft}{\kern0pt}K\ {\isacharparenleft}{\kern0pt}i{\isacharcomma}{\kern0pt}\ Rep{\isacharunderscore}{\kern0pt}Fun\ d\ i{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ j{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ {\isachardoublequoteopen}r{\isacharunderscore}{\kern0pt}vec{\isacharprime}{\kern0pt}\ d\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymchi}\ i{\isachardot}{\kern0pt}\ r{\isacharparenleft}{\kern0pt}i{\isacharcomma}{\kern0pt}\ Rep{\isacharunderscore}{\kern0pt}Fun\ d\ i{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{lift{\isacharunderscore}{\kern0pt}definition}\isamarkupfalse%
-\ policy{\isacharunderscore}{\kern0pt}eval{\isacharprime}{\kern0pt}\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}s{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}{\isacharbraceleft}{\kern0pt}mod{\isacharunderscore}{\kern0pt}type{\isacharcomma}{\kern0pt}\ finite{\isacharbraceright}{\kern0pt}{\isacharcomma}{\kern0pt}\ {\isacharprime}{\kern0pt}a{\isacharparenright}{\kern0pt}\ Fun\ {\isasymRightarrow}\ {\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}s\ {\isasymRightarrow}\isactrlsub b\ real{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{is}\ {\isachardoublequoteopen}policy{\isacharunderscore}{\kern0pt}eval{\isachardoublequoteclose}%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{{\isachardot}{\kern0pt}}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ mat{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}blinfun{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}mat\ {\isadigit{1}}\ {\isacharminus}{\kern0pt}\ l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isacharparenleft}{\kern0pt}P{\isacharunderscore}{\kern0pt}mat\ {\isacharparenleft}{\kern0pt}vec{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}fun\ d{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ {\isacharparenleft}{\kern0pt}id{\isacharunderscore}{\kern0pt}blinfun\ {\isacharminus}{\kern0pt}\ l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isasymP}\isactrlsub {\isadigit{1}}\ {\isacharparenleft}{\kern0pt}mk{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\ {\isacharparenleft}{\kern0pt}vec{\isacharunderscore}{\kern0pt}nth\ d{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}diff\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}id\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}scaleR\isanewline
-\ \ \isacommand{unfolding}\isamarkupfalse%
-\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}def\ P{\isacharunderscore}{\kern0pt}mat{\isacharunderscore}{\kern0pt}def\ {\isasymP}\isactrlsub {\isadigit{1}}{\isachardot}{\kern0pt}rep{\isacharunderscore}{\kern0pt}eq\ K{\isacharunderscore}{\kern0pt}st{\isacharunderscore}{\kern0pt}def\ push{\isacharunderscore}{\kern0pt}exp{\isacharunderscore}{\kern0pt}def\ matrix{\isacharunderscore}{\kern0pt}def\ axis{\isacharunderscore}{\kern0pt}def\ vec{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}fun{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ if{\isacharunderscore}{\kern0pt}distrib\ mk{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det{\isacharunderscore}{\kern0pt}def\ integral{\isacharunderscore}{\kern0pt}measure{\isacharunderscore}{\kern0pt}pmf{\isacharbrackleft}{\kern0pt}of\ UNIV{\isacharbrackright}{\kern0pt}\ \ pmf{\isacharunderscore}{\kern0pt}expectation{\isacharunderscore}{\kern0pt}bind{\isacharbrackleft}{\kern0pt}of\ UNIV{\isacharbrackright}{\kern0pt}\ pmf{\isacharunderscore}{\kern0pt}bind\ cong{\isacharcolon}{\kern0pt}\ if{\isacharunderscore}{\kern0pt}cong{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}vec{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}policy{\isacharunderscore}{\kern0pt}eval{\isacharprime}{\kern0pt}\ {\isacharparenleft}{\kern0pt}vec{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}fun\ d{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ vec{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}bfun\ {\isacharparenleft}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}inv\ {\isacharparenleft}{\kern0pt}mat\ {\isadigit{1}}\ {\isacharminus}{\kern0pt}\ l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isacharparenleft}{\kern0pt}P{\isacharunderscore}{\kern0pt}mat\ {\isacharparenleft}{\kern0pt}vec{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}fun\ d{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharasterisk}{\kern0pt}v\ {\isacharparenleft}{\kern0pt}r{\isacharunderscore}{\kern0pt}vec{\isacharprime}{\kern0pt}\ {\isacharparenleft}{\kern0pt}vec{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}fun\ d{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}d\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}Rep{\isacharunderscore}{\kern0pt}Fun\ {\isacharparenleft}{\kern0pt}vec{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}fun\ d{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}vec{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}bfun\ {\isacharparenleft}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}inv\ {\isacharparenleft}{\kern0pt}mat\ {\isadigit{1}}\ {\isacharminus}{\kern0pt}\ l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isacharparenleft}{\kern0pt}P{\isacharunderscore}{\kern0pt}mat\ {\isacharparenleft}{\kern0pt}vec{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}fun\ d{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharasterisk}{\kern0pt}v\ {\isacharparenleft}{\kern0pt}r{\isacharunderscore}{\kern0pt}vec{\isacharprime}{\kern0pt}\ {\isacharparenleft}{\kern0pt}vec{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}fun\ d{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ matrix{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}blinfun\ {\isacharparenleft}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}inv\ {\isacharparenleft}{\kern0pt}mat\ {\isadigit{1}}\ {\isacharminus}{\kern0pt}\ l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isacharparenleft}{\kern0pt}P{\isacharunderscore}{\kern0pt}mat\ {\isacharparenleft}{\kern0pt}vec{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}fun\ d{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}vec{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}bfun\ {\isacharparenleft}{\kern0pt}r{\isacharunderscore}{\kern0pt}vec{\isacharprime}{\kern0pt}\ {\isacharparenleft}{\kern0pt}vec{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}fun\ d{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ matrix{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}blinfun{\isacharunderscore}{\kern0pt}mult\ vec{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}bfun{\isacharunderscore}{\kern0pt}def\ r{\isacharunderscore}{\kern0pt}vec{\isacharprime}{\kern0pt}{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isacharequal}{\kern0pt}\ matrix{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}blinfun\ {\isacharparenleft}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}inv\ {\isacharparenleft}{\kern0pt}blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ {\isacharparenleft}{\kern0pt}id{\isacharunderscore}{\kern0pt}blinfun\ {\isacharminus}{\kern0pt}\ l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isasymP}\isactrlsub {\isadigit{1}}\ {\isacharparenleft}{\kern0pt}mk{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\ {\isacharquery}{\kern0pt}d{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}r{\isacharunderscore}{\kern0pt}dec\isactrlsub b\ {\isacharparenleft}{\kern0pt}mk{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\ {\isacharquery}{\kern0pt}d{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ mat{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}blinfun\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ \ r{\isacharunderscore}{\kern0pt}vec{\isacharprime}{\kern0pt}{\isacharunderscore}{\kern0pt}def\ vec{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}bfun{\isacharunderscore}{\kern0pt}def\ vec{\isacharunderscore}{\kern0pt}lambda{\isacharunderscore}{\kern0pt}inverse\ r{\isacharunderscore}{\kern0pt}dec\isactrlsub b{\isacharunderscore}{\kern0pt}def\ vec{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}fun{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isacharequal}{\kern0pt}\ inv\isactrlsub L\ {\isacharparenleft}{\kern0pt}id{\isacharunderscore}{\kern0pt}blinfun\ {\isacharminus}{\kern0pt}\ l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isasymP}\isactrlsub {\isadigit{1}}\ {\isacharparenleft}{\kern0pt}mk{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\ {\isacharquery}{\kern0pt}d{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}r{\isacharunderscore}{\kern0pt}dec\isactrlsub b\ {\isacharparenleft}{\kern0pt}mk{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\ {\isacharquery}{\kern0pt}d{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}inverse{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isacharbrackleft}{\kern0pt}symmetric{\isacharbrackright}{\kern0pt}\ invertible\isactrlsub L{\isacharunderscore}{\kern0pt}inf{\isacharunderscore}{\kern0pt}sum\ matrix{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}blinfun{\isacharunderscore}{\kern0pt}inv{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{finally}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}vec{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}bfun\ {\isacharparenleft}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}inv\ {\isacharparenleft}{\kern0pt}mat\ {\isadigit{1}}\ {\isacharminus}{\kern0pt}\ l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isacharparenleft}{\kern0pt}P{\isacharunderscore}{\kern0pt}mat\ {\isacharparenleft}{\kern0pt}vec{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}fun\ d{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharasterisk}{\kern0pt}v\ {\isacharparenleft}{\kern0pt}r{\isacharunderscore}{\kern0pt}vec{\isacharprime}{\kern0pt}\ {\isacharparenleft}{\kern0pt}vec{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}fun\ d{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ inv\isactrlsub L\ {\isacharparenleft}{\kern0pt}id{\isacharunderscore}{\kern0pt}blinfun\ {\isacharminus}{\kern0pt}\ l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isasymP}\isactrlsub {\isadigit{1}}\ {\isacharparenleft}{\kern0pt}mk{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\ {\isacharquery}{\kern0pt}d{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}r{\isacharunderscore}{\kern0pt}dec\isactrlsub b\ {\isacharparenleft}{\kern0pt}mk{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\ {\isacharquery}{\kern0pt}d{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isacommand{{\isachardot}{\kern0pt}}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ {\isasymnu}{\isacharunderscore}{\kern0pt}stationary\ policy{\isacharunderscore}{\kern0pt}eval{\isacharprime}{\kern0pt}{\isachardot}{\kern0pt}rep{\isacharunderscore}{\kern0pt}eq\ policy{\isacharunderscore}{\kern0pt}eval{\isacharunderscore}{\kern0pt}def\ inv\isactrlsub L{\isacharunderscore}{\kern0pt}inf{\isacharunderscore}{\kern0pt}sum\ blincomp{\isacharunderscore}{\kern0pt}scaleR{\isacharunderscore}{\kern0pt}right{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}vec{\isacharunderscore}{\kern0pt}opt{\isacharbrackleft}{\kern0pt}code{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}policy{\isacharunderscore}{\kern0pt}eval{\isacharprime}{\kern0pt}\ {\isacharparenleft}{\kern0pt}vec{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}fun\ d{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ vec{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}bfun\ {\isacharparenleft}{\kern0pt}Matrix{\isacharunderscore}{\kern0pt}To{\isacharunderscore}{\kern0pt}IArray{\isachardot}{\kern0pt}iarray{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}vec\ {\isacharparenleft}{\kern0pt}Matrix{\isacharunderscore}{\kern0pt}To{\isacharunderscore}{\kern0pt}IArray{\isachardot}{\kern0pt}vec{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}iarray\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}fst\ {\isacharparenleft}{\kern0pt}Gauss{\isacharunderscore}{\kern0pt}Jordan{\isacharunderscore}{\kern0pt}PA\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}mat\ {\isadigit{1}}\ {\isacharminus}{\kern0pt}\ l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isacharparenleft}{\kern0pt}P{\isacharunderscore}{\kern0pt}mat\ {\isacharparenleft}{\kern0pt}vec{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}fun\ d{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharasterisk}{\kern0pt}v\ {\isacharparenleft}{\kern0pt}r{\isacharunderscore}{\kern0pt}vec{\isacharprime}{\kern0pt}\ {\isacharparenleft}{\kern0pt}vec{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}fun\ d{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}vec\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ mat{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}blinfun\ matrix{\isacharunderscore}{\kern0pt}inv{\isacharunderscore}{\kern0pt}Gauss{\isacharunderscore}{\kern0pt}Jordan{\isacharunderscore}{\kern0pt}PA\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}inverse{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}\ invertible\isactrlsub L{\isacharunderscore}{\kern0pt}inf{\isacharunderscore}{\kern0pt}sum\ iarray{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}vec{\isacharunderscore}{\kern0pt}vec{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}iarray{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lift{\isacharunderscore}{\kern0pt}definition}\isamarkupfalse%
-\ policy{\isacharunderscore}{\kern0pt}improvement{\isacharprime}{\kern0pt}\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}s{\isacharcomma}{\kern0pt}\ {\isacharprime}{\kern0pt}a{\isacharparenright}{\kern0pt}\ Fun\ {\isasymRightarrow}\ {\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}s\ {\isasymRightarrow}\isactrlsub b\ real{\isacharparenright}{\kern0pt}\ {\isasymRightarrow}\ {\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}s{\isacharcomma}{\kern0pt}\ {\isacharprime}{\kern0pt}a{\isacharparenright}{\kern0pt}\ Fun{\isachardoublequoteclose}\ \isanewline
-\ \ \isakeyword{is}\ policy{\isacharunderscore}{\kern0pt}improvement%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{{\isachardot}{\kern0pt}}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ {\isacharbrackleft}{\kern0pt}code{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}policy{\isacharunderscore}{\kern0pt}improvement{\isacharprime}{\kern0pt}\ {\isacharparenleft}{\kern0pt}vec{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}fun\ d{\isacharparenright}{\kern0pt}\ v\ {\isacharequal}{\kern0pt}\ vec{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}fun\ {\isacharparenleft}{\kern0pt}{\isasymchi}\ s{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}if\ is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max\ {\isacharparenleft}{\kern0pt}{\isasymlambda}a{\isachardot}{\kern0pt}\ L\isactrlsub a\ a\ v\ s{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymlambda}a{\isachardot}{\kern0pt}\ a\ {\isasymin}\ A\ s{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}d\ {\isachardollar}{\kern0pt}\ s{\isacharparenright}{\kern0pt}\ then\ d\ {\isachardollar}{\kern0pt}\ s\ else\ LEAST\ x{\isachardot}{\kern0pt}\ is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max\ {\isacharparenleft}{\kern0pt}{\isasymlambda}a{\isachardot}{\kern0pt}\ L\isactrlsub a\ a\ v\ s{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymlambda}a{\isachardot}{\kern0pt}\ a\ {\isasymin}\ A\ s{\isacharparenright}{\kern0pt}\ x{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ is{\isacharunderscore}{\kern0pt}opt{\isacharunderscore}{\kern0pt}act{\isacharunderscore}{\kern0pt}def\ policy{\isacharunderscore}{\kern0pt}improvement{\isacharprime}{\kern0pt}{\isacharunderscore}{\kern0pt}def\ vec{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}fun{\isacharunderscore}{\kern0pt}def\ vec{\isacharunderscore}{\kern0pt}lambda{\isacharunderscore}{\kern0pt}inverse\ policy{\isacharunderscore}{\kern0pt}improvement{\isacharunderscore}{\kern0pt}def\ Abs{\isacharunderscore}{\kern0pt}Fun{\isacharunderscore}{\kern0pt}inject{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lift{\isacharunderscore}{\kern0pt}definition}\isamarkupfalse%
-\ policy{\isacharunderscore}{\kern0pt}step{\isacharprime}{\kern0pt}\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}s{\isacharcomma}{\kern0pt}\ {\isacharprime}{\kern0pt}a{\isacharparenright}{\kern0pt}\ Fun\ {\isasymRightarrow}\ {\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}s{\isacharcomma}{\kern0pt}\ {\isacharprime}{\kern0pt}a{\isacharparenright}{\kern0pt}\ Fun{\isachardoublequoteclose}\ \isanewline
-\ \ \isakeyword{is}\ policy{\isacharunderscore}{\kern0pt}step%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{{\isachardot}{\kern0pt}}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ {\isacharbrackleft}{\kern0pt}code{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}policy{\isacharunderscore}{\kern0pt}step{\isacharprime}{\kern0pt}\ d\ {\isacharequal}{\kern0pt}\ policy{\isacharunderscore}{\kern0pt}improvement{\isacharprime}{\kern0pt}\ d\ {\isacharparenleft}{\kern0pt}policy{\isacharunderscore}{\kern0pt}eval{\isacharprime}{\kern0pt}\ d{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ policy{\isacharunderscore}{\kern0pt}step{\isacharprime}{\kern0pt}{\isacharunderscore}{\kern0pt}def\ policy{\isacharunderscore}{\kern0pt}step{\isacharunderscore}{\kern0pt}def\ policy{\isacharunderscore}{\kern0pt}improvement{\isacharprime}{\kern0pt}{\isacharunderscore}{\kern0pt}def\ policy{\isacharunderscore}{\kern0pt}eval{\isacharprime}{\kern0pt}{\isacharunderscore}{\kern0pt}def\ apply{\isacharunderscore}{\kern0pt}bfun{\isacharunderscore}{\kern0pt}inverse{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lift{\isacharunderscore}{\kern0pt}definition}\isamarkupfalse%
-\ policy{\isacharunderscore}{\kern0pt}iteration{\isacharunderscore}{\kern0pt}partial{\isacharprime}{\kern0pt}\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}s{\isacharcomma}{\kern0pt}\ {\isacharprime}{\kern0pt}a{\isacharparenright}{\kern0pt}\ Fun\ {\isasymRightarrow}\ {\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}s{\isacharcomma}{\kern0pt}\ {\isacharprime}{\kern0pt}a{\isacharparenright}{\kern0pt}\ Fun{\isachardoublequoteclose}\ \isanewline
-\ \ \isakeyword{is}\ policy{\isacharunderscore}{\kern0pt}iteration{\isacharunderscore}{\kern0pt}partial%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{{\isachardot}{\kern0pt}}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ {\isacharbrackleft}{\kern0pt}code{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}policy{\isacharunderscore}{\kern0pt}iteration{\isacharunderscore}{\kern0pt}partial{\isacharprime}{\kern0pt}\ d\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}let\ d{\isacharprime}{\kern0pt}\ {\isacharequal}{\kern0pt}\ policy{\isacharunderscore}{\kern0pt}step{\isacharprime}{\kern0pt}\ d\ in\ if\ d\ {\isacharequal}{\kern0pt}\ d{\isacharprime}{\kern0pt}\ then\ d\ else\ policy{\isacharunderscore}{\kern0pt}iteration{\isacharunderscore}{\kern0pt}partial{\isacharprime}{\kern0pt}\ d{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ policy{\isacharunderscore}{\kern0pt}iteration{\isacharunderscore}{\kern0pt}partial{\isacharprime}{\kern0pt}{\isachardot}{\kern0pt}rep{\isacharunderscore}{\kern0pt}eq\ policy{\isacharunderscore}{\kern0pt}step{\isacharprime}{\kern0pt}{\isachardot}{\kern0pt}rep{\isacharunderscore}{\kern0pt}eq\ Let{\isacharunderscore}{\kern0pt}def\ policy{\isacharunderscore}{\kern0pt}iteration{\isacharunderscore}{\kern0pt}partial{\isachardot}{\kern0pt}simps\ Rep{\isacharunderscore}{\kern0pt}Fun{\isacharunderscore}{\kern0pt}inject{\isacharbrackleft}{\kern0pt}symmetric{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lift{\isacharunderscore}{\kern0pt}definition}\isamarkupfalse%
-\ policy{\isacharunderscore}{\kern0pt}iteration{\isacharprime}{\kern0pt}\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}s{\isacharcomma}{\kern0pt}\ {\isacharprime}{\kern0pt}a{\isacharparenright}{\kern0pt}\ Fun\ {\isasymRightarrow}\ {\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}s{\isacharcomma}{\kern0pt}\ {\isacharprime}{\kern0pt}a{\isacharparenright}{\kern0pt}\ Fun{\isachardoublequoteclose}\ \isakeyword{is}\ policy{\isacharunderscore}{\kern0pt}iteration%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{{\isachardot}{\kern0pt}}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ code{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}iteration{\isacharprime}{\kern0pt}{\isacharbrackleft}{\kern0pt}code{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}policy{\isacharunderscore}{\kern0pt}iteration{\isacharprime}{\kern0pt}\ d\ {\isacharequal}{\kern0pt}\isanewline
-\ \ {\isacharparenleft}{\kern0pt}if\ Rep{\isacharunderscore}{\kern0pt}Fun\ d\ {\isasymnotin}\ D\isactrlsub D\ then\ d\ else\ \ {\isacharparenleft}{\kern0pt}policy{\isacharunderscore}{\kern0pt}iteration{\isacharunderscore}{\kern0pt}partial{\isacharprime}{\kern0pt}\ d{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ transfer\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ pi{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}partial{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ code{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}iteration{\isacharbrackleft}{\kern0pt}code{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}policy{\isacharunderscore}{\kern0pt}iteration\ d\ {\isacharequal}{\kern0pt}\ Rep{\isacharunderscore}{\kern0pt}Fun\ {\isacharparenleft}{\kern0pt}policy{\isacharunderscore}{\kern0pt}iteration{\isacharprime}{\kern0pt}\ {\isacharparenleft}{\kern0pt}vec{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}fun\ {\isacharparenleft}{\kern0pt}vec{\isacharunderscore}{\kern0pt}lambda\ d{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp\ add{\isacharcolon}{\kern0pt}\ vec{\isacharunderscore}{\kern0pt}lambda{\isacharunderscore}{\kern0pt}inverse\ policy{\isacharunderscore}{\kern0pt}iteration{\isacharprime}{\kern0pt}{\isachardot}{\kern0pt}rep{\isacharunderscore}{\kern0pt}eq\ vec{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}fun{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Gauss-Seidel Iteration%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{partial{\isacharunderscore}{\kern0pt}function}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}tailrec{\isacharparenright}{\kern0pt}\ gs{\isacharunderscore}{\kern0pt}iteration{\isacharunderscore}{\kern0pt}partial\ \isakeyword{where}\isanewline
-\ \ {\isacharbrackleft}{\kern0pt}code{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}gs{\isacharunderscore}{\kern0pt}iteration{\isacharunderscore}{\kern0pt}partial\ eps\ v\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}\isanewline
-\ \ let\ v{\isacharprime}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}fun\isactrlsub b\ v{\isacharparenright}{\kern0pt}\ in\ \isanewline
-\ \ {\isacharparenleft}{\kern0pt}if\ {\isadigit{2}}\ {\isacharasterisk}{\kern0pt}\ l\ {\isacharasterisk}{\kern0pt}\ dist\ v\ v{\isacharprime}{\kern0pt}\ {\isacharless}{\kern0pt}\ eps\ {\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}\ {\isacharminus}{\kern0pt}\ l{\isacharparenright}{\kern0pt}\ then\ v{\isacharprime}{\kern0pt}\ else\ gs{\isacharunderscore}{\kern0pt}iteration{\isacharunderscore}{\kern0pt}partial\ eps\ v{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ gs{\isacharunderscore}{\kern0pt}iteration{\isacharunderscore}{\kern0pt}partial{\isacharunderscore}{\kern0pt}eq{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}eps\ {\isachargreater}{\kern0pt}\ {\isadigit{0}}\ {\isasymLongrightarrow}\ gs{\isacharunderscore}{\kern0pt}iteration{\isacharunderscore}{\kern0pt}partial\ eps\ v\ {\isacharequal}{\kern0pt}\ gs{\isacharunderscore}{\kern0pt}iteration\ eps\ v{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}induction\ eps\ v\ rule{\isacharcolon}{\kern0pt}\ gs{\isacharunderscore}{\kern0pt}iteration{\isachardot}{\kern0pt}induct{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ gs{\isacharunderscore}{\kern0pt}iteration{\isacharunderscore}{\kern0pt}partial{\isachardot}{\kern0pt}simps\ Let{\isacharunderscore}{\kern0pt}def\ gs{\isacharunderscore}{\kern0pt}iteration{\isachardot}{\kern0pt}simps{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ gs{\isacharunderscore}{\kern0pt}iteration{\isacharunderscore}{\kern0pt}code{\isacharunderscore}{\kern0pt}opt{\isacharbrackleft}{\kern0pt}code{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}gs{\isacharunderscore}{\kern0pt}iteration\ eps\ v\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}if\ eps\ {\isasymle}\ {\isadigit{0}}\ then\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}fun\isactrlsub b\ v\ else\ gs{\isacharunderscore}{\kern0pt}iteration{\isacharunderscore}{\kern0pt}partial\ eps\ v{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ gs{\isacharunderscore}{\kern0pt}iteration{\isacharunderscore}{\kern0pt}partial{\isacharunderscore}{\kern0pt}eq\ gs{\isacharunderscore}{\kern0pt}iteration{\isachardot}{\kern0pt}simps{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ {\isachardoublequoteopen}vec{\isacharunderscore}{\kern0pt}upd\ v\ i\ x\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymchi}\ j{\isachardot}{\kern0pt}\ if\ i\ {\isacharequal}{\kern0pt}\ j\ then\ x\ else\ v\ {\isachardollar}{\kern0pt}\ j{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}fold{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}GS{\isacharunderscore}{\kern0pt}rec\ v\ {\isacharequal}{\kern0pt}\ foldl\ {\isacharparenleft}{\kern0pt}{\isasymlambda}v\ s{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}vec{\isacharunderscore}{\kern0pt}upd\ v\ s\ {\isacharparenleft}{\kern0pt}GS{\isacharunderscore}{\kern0pt}iter{\isacharunderscore}{\kern0pt}max\ v\ s{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ v\ {\isacharparenleft}{\kern0pt}sorted{\isacharunderscore}{\kern0pt}list{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}set\ UNIV{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}vec{\isacharunderscore}{\kern0pt}lambda\ {\isacharparenleft}{\kern0pt}foldl\ {\isacharparenleft}{\kern0pt}{\isasymlambda}v\ s{\isachardot}{\kern0pt}\ v{\isacharparenleft}{\kern0pt}s\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}iter\ v\ s{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}{\isachardollar}{\kern0pt}{\isacharparenright}{\kern0pt}\ v{\isacharparenright}{\kern0pt}\ xs{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ foldl\ {\isacharparenleft}{\kern0pt}{\isasymlambda}v\ s{\isachardot}{\kern0pt}\ vec{\isacharunderscore}{\kern0pt}upd\ v\ s\ {\isacharparenleft}{\kern0pt}GS{\isacharunderscore}{\kern0pt}iter{\isacharunderscore}{\kern0pt}max\ v\ s{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ v\ xs{\isachardoublequoteclose}\ \isakeyword{for}\ xs\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}induction\ xs\ arbitrary{\isacharcolon}{\kern0pt}\ v{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{case}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}Cons\ a\ xs{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}case\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ vec{\isacharunderscore}{\kern0pt}upd{\isacharunderscore}{\kern0pt}def{\isacharbrackleft}{\kern0pt}of\ v{\isacharbrackright}{\kern0pt}\ Cons{\isacharbrackleft}{\kern0pt}symmetric{\isacharbrackright}{\kern0pt}\ eq{\isacharunderscore}{\kern0pt}commute\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}iter{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}iter{\isacharunderscore}{\kern0pt}max\ cong{\isacharcolon}{\kern0pt}\ if{\isacharunderscore}{\kern0pt}cong{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}def\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}fun{\isacharunderscore}{\kern0pt}code{\isacharprime}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}fun{\isacharunderscore}{\kern0pt}code{\isacharprime}{\kern0pt}{\isacharprime}{\kern0pt}{\isacharprime}{\kern0pt}{\isacharprime}{\kern0pt}{\isacharbrackleft}{\kern0pt}code{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}fun\isactrlsub b\ {\isacharparenleft}{\kern0pt}vec{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}bfun\ v{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ vec{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}bfun\ {\isacharparenleft}{\kern0pt}foldl\ {\isacharparenleft}{\kern0pt}{\isasymlambda}v\ s{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}vec{\isacharunderscore}{\kern0pt}upd\ v\ s\ {\isacharparenleft}{\kern0pt}GS{\isacharunderscore}{\kern0pt}iter{\isacharunderscore}{\kern0pt}max\ v\ s{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ v\ {\isacharparenleft}{\kern0pt}sorted{\isacharunderscore}{\kern0pt}list{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}set\ UNIV{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp\ add{\isacharcolon}{\kern0pt}\ \ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}fold{\isacharbrackleft}{\kern0pt}symmetric{\isacharbrackright}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}elem\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}fun\isactrlsub b{\isachardot}{\kern0pt}rep{\isacharunderscore}{\kern0pt}eq\ vec{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}bfun{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ GS{\isacharunderscore}{\kern0pt}iter{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}code\ {\isacharbrackleft}{\kern0pt}code{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}GS{\isacharunderscore}{\kern0pt}iter{\isacharunderscore}{\kern0pt}max\ v\ s\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}MAX\ a\ {\isasymin}\ A\ s{\isachardot}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}iter\ a\ v\ s{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ A{\isacharunderscore}{\kern0pt}ne\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}iter{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}def\ cSup{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}Max{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lift{\isacharunderscore}{\kern0pt}definition}\isamarkupfalse%
-\ opt{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}gs{\isacharprime}{\kern0pt}{\isacharprime}{\kern0pt}\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}s\ {\isasymRightarrow}\isactrlsub b\ real{\isacharparenright}{\kern0pt}\ {\isasymRightarrow}\ {\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}s{\isacharcomma}{\kern0pt}\ {\isacharprime}{\kern0pt}a{\isacharparenright}{\kern0pt}\ Fun{\isachardoublequoteclose}\ \isakeyword{is}\ opt{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}gs%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{{\isachardot}{\kern0pt}}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{declare}\isamarkupfalse%
-\ opt{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}gs{\isacharprime}{\kern0pt}{\isacharprime}{\kern0pt}{\isachardot}{\kern0pt}rep{\isacharunderscore}{\kern0pt}eq{\isacharbrackleft}{\kern0pt}symmetric{\isacharcomma}{\kern0pt}\ code{\isacharbrackright}{\kern0pt}\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}am{\isacharunderscore}{\kern0pt}code{\isacharprime}{\kern0pt}{\isacharunderscore}{\kern0pt}prod{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}am{\isacharunderscore}{\kern0pt}code{\isacharprime}{\kern0pt}\ v\ d\ {\isacharequal}{\kern0pt}\ \isanewline
-\ \ {\isacharparenleft}{\kern0pt}{\isasymlambda}s{\isacharprime}{\kern0pt}{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}\isanewline
-\ \ \ \ let\ {\isacharparenleft}{\kern0pt}v{\isacharprime}{\kern0pt}{\isacharcomma}{\kern0pt}\ d{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ foldl\ {\isacharparenleft}{\kern0pt}{\isasymlambda}{\isacharparenleft}{\kern0pt}v{\isacharcomma}{\kern0pt}d{\isacharparenright}{\kern0pt}\ s{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}v{\isacharparenleft}{\kern0pt}s\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}GS{\isacharunderscore}{\kern0pt}iter{\isacharunderscore}{\kern0pt}max\ {\isacharparenleft}{\kern0pt}vec{\isacharunderscore}{\kern0pt}lambda\ v{\isacharparenright}{\kern0pt}\ s{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharcomma}{\kern0pt}\ \ d{\isacharparenleft}{\kern0pt}s\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}iter{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max\ {\isacharparenleft}{\kern0pt}vec{\isacharunderscore}{\kern0pt}lambda\ v{\isacharparenright}{\kern0pt}\ s{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}vec{\isacharunderscore}{\kern0pt}nth\ v{\isacharcomma}{\kern0pt}\ d{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}sorted{\isacharunderscore}{\kern0pt}list{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}set\ UNIV{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ in\ {\isacharparenleft}{\kern0pt}v{\isacharprime}{\kern0pt}\ s{\isacharprime}{\kern0pt}{\isacharcomma}{\kern0pt}\ d{\isacharprime}{\kern0pt}\ s{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isasymlambda}x{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}f\ x{\isacharcomma}{\kern0pt}\ g\ x{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenleft}{\kern0pt}y\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}z{\isacharcomma}{\kern0pt}\ w{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymlambda}x{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}f{\isacharparenleft}{\kern0pt}y\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ z{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ x{\isacharcomma}{\kern0pt}\ {\isacharparenleft}{\kern0pt}g{\isacharparenleft}{\kern0pt}y\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ w{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ x{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{for}\ f\ g\ y\ z\ w\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{2}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}{\isachardollar}{\kern0pt}{\isacharparenright}{\kern0pt}\ f{\isacharparenright}{\kern0pt}{\isacharparenleft}{\kern0pt}a\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ y{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isachardollar}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}vec{\isacharunderscore}{\kern0pt}lambda\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}vec{\isacharunderscore}{\kern0pt}nth\ f{\isacharparenright}{\kern0pt}{\isacharparenleft}{\kern0pt}a\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ y{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{for}\ f\ a\ y\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}foldl\ {\isacharparenleft}{\kern0pt}{\isasymlambda}vd\ s{\isachardot}{\kern0pt}\ vd{\isacharparenleft}{\kern0pt}s\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}GS{\isacharunderscore}{\kern0pt}iter{\isacharunderscore}{\kern0pt}max\ {\isacharparenleft}{\kern0pt}{\isasymchi}\ s{\isachardot}{\kern0pt}\ fst\ {\isacharparenleft}{\kern0pt}vd\ s{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ s{\isacharcomma}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}iter{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max\ {\isacharparenleft}{\kern0pt}{\isasymchi}\ s{\isachardot}{\kern0pt}\ fst\ {\isacharparenleft}{\kern0pt}vd\ s{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ s{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymlambda}s{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}v\ {\isachardollar}{\kern0pt}\ s{\isacharcomma}{\kern0pt}\ d\ s{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ xs\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymlambda}s{\isacharprime}{\kern0pt}{\isachardot}{\kern0pt}\ let\ {\isacharparenleft}{\kern0pt}v{\isacharprime}{\kern0pt}{\isacharcomma}{\kern0pt}\ d{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ foldl\ {\isacharparenleft}{\kern0pt}{\isasymlambda}{\isacharparenleft}{\kern0pt}v{\isacharcomma}{\kern0pt}\ d{\isacharparenright}{\kern0pt}\ s{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}v{\isacharparenleft}{\kern0pt}s\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}iter{\isacharunderscore}{\kern0pt}max\ {\isacharparenleft}{\kern0pt}vec{\isacharunderscore}{\kern0pt}lambda\ v{\isacharparenright}{\kern0pt}\ s{\isacharparenright}{\kern0pt}{\isacharcomma}{\kern0pt}\ d{\isacharparenleft}{\kern0pt}s\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}iter{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max\ {\isacharparenleft}{\kern0pt}vec{\isacharunderscore}{\kern0pt}lambda\ v{\isacharparenright}{\kern0pt}\ s{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}{\isachardollar}{\kern0pt}{\isacharparenright}{\kern0pt}\ v{\isacharcomma}{\kern0pt}\ d{\isacharparenright}{\kern0pt}\ xs\ in\ {\isacharparenleft}{\kern0pt}v{\isacharprime}{\kern0pt}\ s{\isacharprime}{\kern0pt}{\isacharcomma}{\kern0pt}\ d{\isacharprime}{\kern0pt}\ s{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{for}\ xs\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}induction\ xs\ arbitrary{\isacharcolon}{\kern0pt}\ v\ d{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{case}\isamarkupfalse%
-\ Nil\isanewline
-\ \ \ \ \isacommand{then}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}case\ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{case}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}Cons\ a\ xs{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}case\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ {\isadigit{1}}\ Cons{\isachardot}{\kern0pt}IH{\isacharbrackleft}{\kern0pt}of\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}vec{\isacharunderscore}{\kern0pt}lambda\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}{\isachardollar}{\kern0pt}{\isacharparenright}{\kern0pt}\ v{\isacharparenright}{\kern0pt}{\isacharparenleft}{\kern0pt}a\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}iter{\isacharunderscore}{\kern0pt}max\ v\ a{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}{\isacharcomma}{\kern0pt}\ unfolded\ {\isadigit{2}}{\isacharbrackleft}{\kern0pt}symmetric{\isacharbrackright}{\kern0pt}{\isacharbrackright}{\kern0pt}\ del{\isacharcolon}{\kern0pt}\ fun{\isacharunderscore}{\kern0pt}upd{\isacharunderscore}{\kern0pt}apply{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}am{\isacharunderscore}{\kern0pt}code{\isacharprime}{\kern0pt}{\isacharunderscore}{\kern0pt}def\ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ code{\isacharunderscore}{\kern0pt}GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}am{\isacharunderscore}{\kern0pt}arr{\isacharunderscore}{\kern0pt}opt{\isacharbrackleft}{\kern0pt}code{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}opt{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}gs{\isacharprime}{\kern0pt}{\isacharprime}{\kern0pt}\ {\isacharparenleft}{\kern0pt}vec{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}bfun\ v{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ vec{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}fun\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}snd\ {\isacharparenleft}{\kern0pt}foldl\ {\isacharparenleft}{\kern0pt}{\isasymlambda}{\isacharparenleft}{\kern0pt}v{\isacharcomma}{\kern0pt}\ d{\isacharparenright}{\kern0pt}\ s{\isachardot}{\kern0pt}\isanewline
-\ \ let\ {\isacharparenleft}{\kern0pt}am{\isacharcomma}{\kern0pt}\ m{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ least{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}enum\ {\isacharparenleft}{\kern0pt}{\isasymlambda}a{\isachardot}{\kern0pt}\ r\ {\isacharparenleft}{\kern0pt}s{\isacharcomma}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isacharplus}{\kern0pt}\ l\ {\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSum}s{\isacharprime}{\kern0pt}\ {\isasymin}\ UNIV{\isachardot}{\kern0pt}\ pmf\ {\isacharparenleft}{\kern0pt}K\ {\isacharparenleft}{\kern0pt}s{\isacharcomma}{\kern0pt}a{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ s{\isacharprime}{\kern0pt}\ {\isacharasterisk}{\kern0pt}\ v\ {\isachardollar}{\kern0pt}\ s{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}A\ s{\isacharparenright}{\kern0pt}\ in\isanewline
-\ \ {\isacharparenleft}{\kern0pt}vec{\isacharunderscore}{\kern0pt}upd\ v\ s\ m{\isacharcomma}{\kern0pt}\ vec{\isacharunderscore}{\kern0pt}upd\ d\ s\ am{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ \isanewline
-\ \ {\isacharparenleft}{\kern0pt}v{\isacharcomma}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymchi}\ s{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}least{\isacharunderscore}{\kern0pt}enum\ {\isacharparenleft}{\kern0pt}{\isasymlambda}a{\isachardot}{\kern0pt}\ a\ {\isasymin}\ A\ s{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}sorted{\isacharunderscore}{\kern0pt}list{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}set\ UNIV{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}opt{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}gs{\isacharprime}{\kern0pt}{\isacharprime}{\kern0pt}\ v{\isacharprime}{\kern0pt}\ {\isacharequal}{\kern0pt}\ Abs{\isacharunderscore}{\kern0pt}Fun\ {\isacharparenleft}{\kern0pt}opt{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}gs\ v{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{for}\ v{\isacharprime}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ opt{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}gs{\isacharprime}{\kern0pt}{\isacharprime}{\kern0pt}{\isachardot}{\kern0pt}abs{\isacharunderscore}{\kern0pt}eq{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{2}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}opt{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}gs\ {\isacharparenleft}{\kern0pt}vec{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}bfun\ v{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ opt{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}gs{\isacharprime}{\kern0pt}\ d\ v{\isachardoublequoteclose}\ \isakeyword{for}\ v\ d\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ Bfun{\isacharunderscore}{\kern0pt}inverse{\isacharunderscore}{\kern0pt}fin\ opt{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}gs{\isacharunderscore}{\kern0pt}eq{\isacharprime}{\kern0pt}\ vec{\isacharunderscore}{\kern0pt}lambda{\isacharunderscore}{\kern0pt}eta\ vec{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}bfun{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{3}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}opt{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}gs{\isacharprime}{\kern0pt}\ d\ v\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymlambda}s{\isachardot}{\kern0pt}\ snd\ {\isacharparenleft}{\kern0pt}GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}am{\isacharunderscore}{\kern0pt}code\ v\ d\ s{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{for}\ d\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}am{\isacharunderscore}{\kern0pt}code{\isacharunderscore}{\kern0pt}eq{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{4}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}am{\isacharunderscore}{\kern0pt}code\ v\ d\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymlambda}s{\isacharprime}{\kern0pt}{\isachardot}{\kern0pt}\ let\ {\isacharparenleft}{\kern0pt}v{\isacharprime}{\kern0pt}{\isacharcomma}{\kern0pt}\ d{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ foldl\ {\isacharparenleft}{\kern0pt}{\isasymlambda}{\isacharparenleft}{\kern0pt}v{\isacharcomma}{\kern0pt}\ d{\isacharparenright}{\kern0pt}\ s{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}v{\isacharparenleft}{\kern0pt}s\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}iter{\isacharunderscore}{\kern0pt}max\ {\isacharparenleft}{\kern0pt}vec{\isacharunderscore}{\kern0pt}lambda\ v{\isacharparenright}{\kern0pt}\ s{\isacharparenright}{\kern0pt}{\isacharcomma}{\kern0pt}\ d{\isacharparenleft}{\kern0pt}s\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}iter{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max\ {\isacharparenleft}{\kern0pt}vec{\isacharunderscore}{\kern0pt}lambda\ v{\isacharparenright}{\kern0pt}\ s{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}{\isachardollar}{\kern0pt}{\isacharparenright}{\kern0pt}\ v{\isacharcomma}{\kern0pt}\ d{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}sorted{\isacharunderscore}{\kern0pt}list{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}set\ UNIV{\isacharparenright}{\kern0pt}\ in\ {\isacharparenleft}{\kern0pt}v{\isacharprime}{\kern0pt}\ s{\isacharprime}{\kern0pt}{\isacharcomma}{\kern0pt}\ d{\isacharprime}{\kern0pt}\ s{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{for}\ d\ s\ v\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}am{\isacharunderscore}{\kern0pt}code{\isacharprime}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}am{\isacharunderscore}{\kern0pt}code{\isacharprime}{\kern0pt}{\isacharunderscore}{\kern0pt}prod\ \isacommand{by}\isamarkupfalse%
-\ presburger\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{5}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}foldl\ {\isacharparenleft}{\kern0pt}{\isasymlambda}{\isacharparenleft}{\kern0pt}v{\isacharcomma}{\kern0pt}\ d{\isacharparenright}{\kern0pt}\ s{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}v{\isacharparenleft}{\kern0pt}s\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}iter{\isacharunderscore}{\kern0pt}max\ {\isacharparenleft}{\kern0pt}vec{\isacharunderscore}{\kern0pt}lambda\ v{\isacharparenright}{\kern0pt}\ s{\isacharparenright}{\kern0pt}{\isacharcomma}{\kern0pt}\ d{\isacharparenleft}{\kern0pt}s\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}iter{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max\ {\isacharparenleft}{\kern0pt}vec{\isacharunderscore}{\kern0pt}lambda\ v{\isacharparenright}{\kern0pt}\ s{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}{\isachardollar}{\kern0pt}{\isacharparenright}{\kern0pt}\ v{\isacharcomma}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isachardollar}{\kern0pt}{\isacharparenright}{\kern0pt}\ d{\isacharparenright}{\kern0pt}\ xs\ {\isacharequal}{\kern0pt}\ \isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}let\ {\isacharparenleft}{\kern0pt}v{\isacharprime}{\kern0pt}{\isacharcomma}{\kern0pt}\ d{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ foldl\ {\isacharparenleft}{\kern0pt}{\isasymlambda}{\isacharparenleft}{\kern0pt}v{\isacharcomma}{\kern0pt}\ d{\isacharparenright}{\kern0pt}\ s{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}vec{\isacharunderscore}{\kern0pt}upd\ v\ s\ {\isacharparenleft}{\kern0pt}GS{\isacharunderscore}{\kern0pt}iter{\isacharunderscore}{\kern0pt}max\ v\ s{\isacharparenright}{\kern0pt}{\isacharcomma}{\kern0pt}\ vec{\isacharunderscore}{\kern0pt}upd\ d\ s\ {\isacharparenleft}{\kern0pt}GS{\isacharunderscore}{\kern0pt}iter{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max\ v\ s{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}v{\isacharcomma}{\kern0pt}\ d{\isacharparenright}{\kern0pt}\ xs\ in\ {\isacharparenleft}{\kern0pt}vec{\isacharunderscore}{\kern0pt}nth\ v{\isacharprime}{\kern0pt}{\isacharcomma}{\kern0pt}\ vec{\isacharunderscore}{\kern0pt}nth\ d{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{for}\ d\ v\ xs\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}induction\ xs\ arbitrary{\isacharcolon}{\kern0pt}\ d\ v{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{case}\isamarkupfalse%
-\ Nil\isanewline
-\ \ \ \ \isacommand{then}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}case\ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{case}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}Cons\ a\ xs{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}case\isanewline
-\ \ \ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ \ vec{\isacharunderscore}{\kern0pt}lambda{\isacharunderscore}{\kern0pt}inverse\ Let{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ Cons{\isacharbrackleft}{\kern0pt}symmetric{\isacharcomma}{\kern0pt}\ unfolded\ Let{\isacharunderscore}{\kern0pt}def{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ vec{\isacharunderscore}{\kern0pt}lambda{\isacharunderscore}{\kern0pt}inverse\ vec{\isacharunderscore}{\kern0pt}upd{\isacharunderscore}{\kern0pt}def\ Let{\isacharunderscore}{\kern0pt}def\ eq{\isacharunderscore}{\kern0pt}commute\ cong{\isacharcolon}{\kern0pt}\ if{\isacharunderscore}{\kern0pt}cong{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{6}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}opt{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}gs{\isacharprime}{\kern0pt}{\isacharprime}{\kern0pt}\ {\isacharparenleft}{\kern0pt}vec{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}bfun\ v{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ vec{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}fun\ {\isacharparenleft}{\kern0pt}snd\ {\isacharparenleft}{\kern0pt}foldl\ {\isacharparenleft}{\kern0pt}{\isasymlambda}{\isacharparenleft}{\kern0pt}v{\isacharcomma}{\kern0pt}\ d{\isacharparenright}{\kern0pt}\ s{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}vec{\isacharunderscore}{\kern0pt}upd\ v\ s\ {\isacharparenleft}{\kern0pt}GS{\isacharunderscore}{\kern0pt}iter{\isacharunderscore}{\kern0pt}max\ v\ s{\isacharparenright}{\kern0pt}{\isacharcomma}{\kern0pt}\ vec{\isacharunderscore}{\kern0pt}upd\ d\ s\ {\isacharparenleft}{\kern0pt}GS{\isacharunderscore}{\kern0pt}iter{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max\ v\ s{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}v{\isacharcomma}{\kern0pt}\ d{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}sorted{\isacharunderscore}{\kern0pt}list{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}set\ UNIV{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{for}\ d\isanewline
-\ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ {\isadigit{1}}\isanewline
-\ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ {\isadigit{2}}{\isacharbrackleft}{\kern0pt}of\ {\isacharunderscore}{\kern0pt}\ {\isachardoublequoteopen}Rep{\isacharunderscore}{\kern0pt}Fun\ {\isacharparenleft}{\kern0pt}vec{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}fun\ d{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ {\isadigit{3}}\isanewline
-\ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ {\isadigit{4}}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isadigit{5}}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ Let{\isacharunderscore}{\kern0pt}def\ case{\isacharunderscore}{\kern0pt}prod{\isacharunderscore}{\kern0pt}beta\ vec{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}fun{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ Let{\isacharunderscore}{\kern0pt}def\ case{\isacharunderscore}{\kern0pt}prod{\isacharunderscore}{\kern0pt}beta\isanewline
-\ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ least{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}enum{\isacharunderscore}{\kern0pt}correct{\isadigit{1}}{\isacharbrackleft}{\kern0pt}OF\ A{\isacharunderscore}{\kern0pt}ne{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ least{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}enum{\isacharunderscore}{\kern0pt}correct{\isadigit{2}}{\isacharbrackleft}{\kern0pt}OF\ A{\isacharunderscore}{\kern0pt}ne{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ least{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}enum{\isacharunderscore}{\kern0pt}correct{\isadigit{2}}{\isacharbrackleft}{\kern0pt}OF\ A{\isacharunderscore}{\kern0pt}ne{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isadigit{6}}\ fin{\isacharunderscore}{\kern0pt}actions\ A{\isacharunderscore}{\kern0pt}ne\ \isanewline
-\ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ GS{\isacharunderscore}{\kern0pt}iter{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}def\ GS{\isacharunderscore}{\kern0pt}iter{\isacharunderscore}{\kern0pt}def\ GS{\isacharunderscore}{\kern0pt}iter{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}def\ \isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ cSup{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}Max\ split{\isacharunderscore}{\kern0pt}beta{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Modified Policy Iteration%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{sublocale}\isamarkupfalse%
-\ MDP{\isacharunderscore}{\kern0pt}MPI\ A\ K\ r\ l\ {\isachardoublequoteopen}{\isasymlambda}X{\isachardot}{\kern0pt}\ Least\ {\isacharparenleft}{\kern0pt}{\isasymlambda}x{\isachardot}{\kern0pt}\ x\ {\isasymin}\ X{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ MDP{\isacharunderscore}{\kern0pt}act{\isacharunderscore}{\kern0pt}axioms\ MDP{\isacharunderscore}{\kern0pt}reward{\isacharunderscore}{\kern0pt}axioms\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ unfold{\isacharunderscore}{\kern0pt}locales\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ {\isachardoublequoteopen}d{\isadigit{0}}\ s\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}LEAST\ a{\isachardot}{\kern0pt}\ a\ {\isasymin}\ A\ s{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isanewline
-\isacommand{lift{\isacharunderscore}{\kern0pt}definition}\isamarkupfalse%
-\ d{\isadigit{0}}{\isacharprime}{\kern0pt}\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}s{\isacharcomma}{\kern0pt}\ {\isacharprime}{\kern0pt}a{\isacharparenright}{\kern0pt}\ Fun{\isachardoublequoteclose}\ \isakeyword{is}\ d{\isadigit{0}}%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{{\isachardot}{\kern0pt}}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ d{\isadigit{0}}{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}is{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\ d{\isadigit{0}}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ A{\isacharunderscore}{\kern0pt}ne\ \isacommand{unfolding}\isamarkupfalse%
-\ d{\isadigit{0}}{\isacharunderscore}{\kern0pt}def\ is{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ simp%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ v{\isadigit{0}}{\isacharunderscore}{\kern0pt}code{\isacharbrackleft}{\kern0pt}code{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}v{\isadigit{0}}{\isacharunderscore}{\kern0pt}mpi\isactrlsub b\ {\isacharequal}{\kern0pt}\ vec{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}bfun\ {\isacharparenleft}{\kern0pt}{\isasymchi}\ s{\isachardot}{\kern0pt}\ r{\isacharunderscore}{\kern0pt}min\ {\isacharslash}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}\ {\isacharminus}{\kern0pt}\ l{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ v{\isadigit{0}}{\isacharunderscore}{\kern0pt}mpi\isactrlsub b{\isacharunderscore}{\kern0pt}def\ v{\isadigit{0}}{\isacharunderscore}{\kern0pt}mpi{\isacharunderscore}{\kern0pt}def\ vec{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}bfun{\isacharunderscore}{\kern0pt}def\ {\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ d{\isadigit{0}}{\isacharprime}{\kern0pt}{\isacharunderscore}{\kern0pt}code{\isacharbrackleft}{\kern0pt}code{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}d{\isadigit{0}}{\isacharprime}{\kern0pt}\ {\isacharequal}{\kern0pt}\ vec{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}fun\ {\isacharparenleft}{\kern0pt}{\isasymchi}\ s{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}LEAST\ a{\isachardot}{\kern0pt}\ a\ {\isasymin}\ A\ s{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ d{\isadigit{0}}{\isacharprime}{\kern0pt}{\isachardot}{\kern0pt}rep{\isacharunderscore}{\kern0pt}eq\ d{\isadigit{0}}{\isacharunderscore}{\kern0pt}def\ Rep{\isacharunderscore}{\kern0pt}Fun{\isacharunderscore}{\kern0pt}inject{\isacharbrackleft}{\kern0pt}symmetric{\isacharbrackright}{\kern0pt}\ vec{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}fun{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ step{\isacharunderscore}{\kern0pt}value{\isacharunderscore}{\kern0pt}code{\isacharbrackleft}{\kern0pt}code{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}L{\isacharunderscore}{\kern0pt}pow\ v\ d\ m\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}L{\isacharunderscore}{\kern0pt}det\ d\ {\isacharcircum}{\kern0pt}{\isacharcircum}{\kern0pt}\ Suc\ m{\isacharparenright}{\kern0pt}\ v{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ L{\isacharunderscore}{\kern0pt}pow{\isacharunderscore}{\kern0pt}def\ L{\isacharunderscore}{\kern0pt}det{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{partial{\isacharunderscore}{\kern0pt}function}\isamarkupfalse%
-\ \ {\isacharparenleft}{\kern0pt}tailrec{\isacharparenright}{\kern0pt}\ mpi{\isacharunderscore}{\kern0pt}partial\ \isakeyword{where}\ {\isacharbrackleft}{\kern0pt}code{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}mpi{\isacharunderscore}{\kern0pt}partial\ eps\ d\ v\ m\ {\isacharequal}{\kern0pt}\ \isanewline
-\ \ {\isacharparenleft}{\kern0pt}let\ d{\isacharprime}{\kern0pt}\ {\isacharequal}{\kern0pt}\ policy{\isacharunderscore}{\kern0pt}improvement\ d\ v\ in\ {\isacharparenleft}{\kern0pt}\isanewline
-\ \ \ \ if\ {\isadigit{2}}\ {\isacharasterisk}{\kern0pt}\ l\ {\isacharasterisk}{\kern0pt}\ dist\ v\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ v{\isacharparenright}{\kern0pt}\ {\isacharless}{\kern0pt}\ \ eps\ {\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}\ {\isacharminus}{\kern0pt}\ l{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ then\ {\isacharparenleft}{\kern0pt}d{\isacharprime}{\kern0pt}{\isacharcomma}{\kern0pt}\ v{\isacharparenright}{\kern0pt}\ \isanewline
-\ \ \ \ else\ mpi{\isacharunderscore}{\kern0pt}partial\ eps\ d{\isacharprime}{\kern0pt}\ {\isacharparenleft}{\kern0pt}L{\isacharunderscore}{\kern0pt}pow\ v\ d{\isacharprime}{\kern0pt}\ {\isacharparenleft}{\kern0pt}m\ {\isadigit{0}}\ v{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymlambda}n{\isachardot}{\kern0pt}\ m\ {\isacharparenleft}{\kern0pt}Suc\ n{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ mpi{\isacharunderscore}{\kern0pt}partial{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}algo{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}eps\ {\isachargreater}{\kern0pt}\ {\isadigit{0}}{\isachardoublequoteclose}\ {\isachardoublequoteopen}d\ {\isasymin}\ D\isactrlsub D{\isachardoublequoteclose}\ {\isachardoublequoteopen}v\ {\isasymle}\ {\isasymL}\isactrlsub b\ v{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}mpi{\isacharunderscore}{\kern0pt}partial\ eps\ d\ v\ m\ {\isacharequal}{\kern0pt}\ mpi{\isacharunderscore}{\kern0pt}algo\ eps\ d\ v\ m{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}mpi{\isacharunderscore}{\kern0pt}algo{\isacharunderscore}{\kern0pt}dom\ eps\ {\isacharparenleft}{\kern0pt}d{\isacharcomma}{\kern0pt}v{\isacharcomma}{\kern0pt}m{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assms\ termination{\isacharunderscore}{\kern0pt}mpi{\isacharunderscore}{\kern0pt}algo\ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}induction\ rule{\isacharcolon}{\kern0pt}\ mpi{\isacharunderscore}{\kern0pt}algo{\isachardot}{\kern0pt}pinduct{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ Let{\isacharunderscore}{\kern0pt}def\ mpi{\isacharunderscore}{\kern0pt}algo{\isachardot}{\kern0pt}psimps\ mpi{\isacharunderscore}{\kern0pt}partial{\isachardot}{\kern0pt}simps{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lift{\isacharunderscore}{\kern0pt}definition}\isamarkupfalse%
-\ mpi{\isacharunderscore}{\kern0pt}partial{\isacharprime}{\kern0pt}\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}real\ {\isasymRightarrow}\ {\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}s{\isacharcomma}{\kern0pt}\ {\isacharprime}{\kern0pt}a{\isacharparenright}{\kern0pt}\ Fun\ {\isasymRightarrow}\ {\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}s\ {\isasymRightarrow}\isactrlsub b\ real{\isacharparenright}{\kern0pt}\ {\isasymRightarrow}\ {\isacharparenleft}{\kern0pt}nat\ {\isasymRightarrow}\ {\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}s\ {\isasymRightarrow}\isactrlsub b\ real{\isacharparenright}{\kern0pt}\ {\isasymRightarrow}\ nat{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ {\isasymRightarrow}\ {\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}s{\isacharcomma}{\kern0pt}\ {\isacharprime}{\kern0pt}a{\isacharparenright}{\kern0pt}\ Fun\ {\isasymtimes}\ {\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}s\ {\isasymRightarrow}\isactrlsub b\ real{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{is}\ mpi{\isacharunderscore}{\kern0pt}partial%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{{\isachardot}{\kern0pt}}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ mpi{\isacharunderscore}{\kern0pt}partial{\isacharprime}{\kern0pt}{\isacharunderscore}{\kern0pt}code{\isacharbrackleft}{\kern0pt}code{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}mpi{\isacharunderscore}{\kern0pt}partial{\isacharprime}{\kern0pt}\ eps\ d\ v\ m\ {\isacharequal}{\kern0pt}\isanewline
-\ \ {\isacharparenleft}{\kern0pt}let\ d{\isacharprime}{\kern0pt}\ {\isacharequal}{\kern0pt}\ policy{\isacharunderscore}{\kern0pt}improvement{\isacharprime}{\kern0pt}\ d\ v\ in\ {\isacharparenleft}{\kern0pt}\isanewline
-\ \ \ \ if\ {\isadigit{2}}\ {\isacharasterisk}{\kern0pt}\ l\ {\isacharasterisk}{\kern0pt}\ dist\ v\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ v{\isacharparenright}{\kern0pt}\ {\isacharless}{\kern0pt}\ \ eps\ {\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}\ {\isacharminus}{\kern0pt}\ l{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ then\ {\isacharparenleft}{\kern0pt}d{\isacharprime}{\kern0pt}{\isacharcomma}{\kern0pt}\ v{\isacharparenright}{\kern0pt}\ \isanewline
-\ \ \ \ else\ mpi{\isacharunderscore}{\kern0pt}partial{\isacharprime}{\kern0pt}\ eps\ d{\isacharprime}{\kern0pt}\ {\isacharparenleft}{\kern0pt}L{\isacharunderscore}{\kern0pt}pow\ v\ {\isacharparenleft}{\kern0pt}Rep{\isacharunderscore}{\kern0pt}Fun\ d{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}m\ {\isadigit{0}}\ v{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymlambda}n{\isachardot}{\kern0pt}\ m\ {\isacharparenleft}{\kern0pt}Suc\ n{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ mpi{\isacharunderscore}{\kern0pt}partial{\isacharprime}{\kern0pt}{\isacharunderscore}{\kern0pt}def\ mpi{\isacharunderscore}{\kern0pt}partial{\isachardot}{\kern0pt}simps\ Let{\isacharunderscore}{\kern0pt}def\ policy{\isacharunderscore}{\kern0pt}improvement{\isacharprime}{\kern0pt}{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ r{\isacharunderscore}{\kern0pt}min{\isacharunderscore}{\kern0pt}code{\isacharbrackleft}{\kern0pt}code{\isacharunderscore}{\kern0pt}unfold{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}r{\isacharunderscore}{\kern0pt}min\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}MIN\ s{\isachardot}{\kern0pt}\ MIN\ a{\isachardot}{\kern0pt}\ r{\isacharparenleft}{\kern0pt}s{\isacharcomma}{\kern0pt}a{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ cInf{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}Min{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ mpi{\isacharunderscore}{\kern0pt}user{\isacharunderscore}{\kern0pt}code{\isacharbrackleft}{\kern0pt}code{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}mpi{\isacharunderscore}{\kern0pt}user\ eps\ m\ {\isacharequal}{\kern0pt}\ \isanewline
-\ \ {\isacharparenleft}{\kern0pt}if\ eps\ {\isasymle}\ {\isadigit{0}}\ then\ undefined\ else\ \isanewline
-\ \ \ \ let\ {\isacharparenleft}{\kern0pt}d{\isacharcomma}{\kern0pt}\ v{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ mpi{\isacharunderscore}{\kern0pt}partial{\isacharprime}{\kern0pt}\ eps\ d{\isadigit{0}}{\isacharprime}{\kern0pt}\ v{\isadigit{0}}{\isacharunderscore}{\kern0pt}mpi\isactrlsub b\ m\ in\ {\isacharparenleft}{\kern0pt}Rep{\isacharunderscore}{\kern0pt}Fun\ d{\isacharcomma}{\kern0pt}\ v{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ mpi{\isacharunderscore}{\kern0pt}user{\isacharunderscore}{\kern0pt}def\ case{\isacharunderscore}{\kern0pt}prod{\isacharunderscore}{\kern0pt}beta\ mpi{\isacharunderscore}{\kern0pt}partial{\isacharprime}{\kern0pt}{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \isacommand{using}\isamarkupfalse%
-\ mpi{\isacharunderscore}{\kern0pt}partial{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}algo\ A{\isacharunderscore}{\kern0pt}ne\ v{\isadigit{0}}{\isacharunderscore}{\kern0pt}mpi\isactrlsub b{\isacharunderscore}{\kern0pt}le{\isacharunderscore}{\kern0pt}{\isasymL}\isactrlsub b\ d{\isadigit{0}}{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ d{\isadigit{0}}{\isacharprime}{\kern0pt}{\isachardot}{\kern0pt}rep{\isacharunderscore}{\kern0pt}eq\ d{\isadigit{0}}{\isacharunderscore}{\kern0pt}def{\isacharbrackleft}{\kern0pt}symmetric{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isacommand{end}\isamarkupfalse%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Auxiliary Equations%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{lemma}\isamarkupfalse%
-\ {\isacharbrackleft}{\kern0pt}code{\isacharunderscore}{\kern0pt}unfold{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}dist\ {\isacharparenleft}{\kern0pt}f{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}{\isacharprime}{\kern0pt}a{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}finite\ {\isasymRightarrow}\isactrlsub b\ {\isacharprime}{\kern0pt}b{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}metric{\isacharunderscore}{\kern0pt}space{\isacharparenright}{\kern0pt}\ g\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}MAX\ a{\isachardot}{\kern0pt}\ dist\ {\isacharparenleft}{\kern0pt}apply{\isacharunderscore}{\kern0pt}bfun\ f\ a{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}g\ a{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ dist{\isacharunderscore}{\kern0pt}bfun{\isacharunderscore}{\kern0pt}def\ cSup{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}Max{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ member{\isacharunderscore}{\kern0pt}code{\isacharbrackleft}{\kern0pt}code\ del{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ List{\isachardot}{\kern0pt}coset\ xs\ {\isasymlongleftrightarrow}\ {\isasymnot}\ List{\isachardot}{\kern0pt}member\ xs\ x{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ in{\isacharunderscore}{\kern0pt}set{\isacharunderscore}{\kern0pt}member{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ {\isacharbrackleft}{\kern0pt}code{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}iarray{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}vec\ v\ {\isacharplus}{\kern0pt}\ iarray{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}vec\ u\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}Matrix{\isacharunderscore}{\kern0pt}To{\isacharunderscore}{\kern0pt}IArray{\isachardot}{\kern0pt}iarray{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}vec\ {\isacharparenleft}{\kern0pt}Rep{\isacharunderscore}{\kern0pt}iarray{\isacharunderscore}{\kern0pt}type\ v\ {\isacharplus}{\kern0pt}\ Rep{\isacharunderscore}{\kern0pt}iarray{\isacharunderscore}{\kern0pt}type\ u{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharcomma}{\kern0pt}\ lifting{\isacharparenright}{\kern0pt}\ Matrix{\isacharunderscore}{\kern0pt}To{\isacharunderscore}{\kern0pt}IArray{\isachardot}{\kern0pt}vec{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}iarray{\isacharunderscore}{\kern0pt}def\ iarray{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}vec{\isacharunderscore}{\kern0pt}vec{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}iarray\ vec{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}iarray{\isachardot}{\kern0pt}rep{\isacharunderscore}{\kern0pt}eq\ vec{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}iarray{\isacharunderscore}{\kern0pt}inv\ vec{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}iarray{\isacharunderscore}{\kern0pt}plus{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ {\isacharbrackleft}{\kern0pt}code{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}iarray{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}vec\ v\ {\isacharminus}{\kern0pt}\ iarray{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}vec\ u\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}Matrix{\isacharunderscore}{\kern0pt}To{\isacharunderscore}{\kern0pt}IArray{\isachardot}{\kern0pt}iarray{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}vec\ {\isacharparenleft}{\kern0pt}Rep{\isacharunderscore}{\kern0pt}iarray{\isacharunderscore}{\kern0pt}type\ v\ {\isacharminus}{\kern0pt}\ Rep{\isacharunderscore}{\kern0pt}iarray{\isacharunderscore}{\kern0pt}type\ u{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ minus{\isacharunderscore}{\kern0pt}iarray{\isacharunderscore}{\kern0pt}def\ Matrix{\isacharunderscore}{\kern0pt}To{\isacharunderscore}{\kern0pt}IArray{\isachardot}{\kern0pt}iarray{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}vec{\isacharunderscore}{\kern0pt}def\ iarray{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}vec{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ vec{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}iff\ to{\isacharunderscore}{\kern0pt}nat{\isacharunderscore}{\kern0pt}less{\isacharunderscore}{\kern0pt}card{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ matrix{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}iarray{\isacharunderscore}{\kern0pt}minus{\isacharbrackleft}{\kern0pt}code{\isacharunderscore}{\kern0pt}unfold{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}matrix{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}iarray\ {\isacharparenleft}{\kern0pt}A\ {\isacharminus}{\kern0pt}\ B{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ matrix{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}iarray\ A\ {\isacharminus}{\kern0pt}\ matrix{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}iarray\ B{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ matrix{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}iarray{\isacharunderscore}{\kern0pt}def\ o{\isacharunderscore}{\kern0pt}def\ minus{\isacharunderscore}{\kern0pt}iarray{\isacharunderscore}{\kern0pt}def\ Matrix{\isacharunderscore}{\kern0pt}To{\isacharunderscore}{\kern0pt}IArray{\isachardot}{\kern0pt}vec{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}iarray{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ simp%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{declare}\isamarkupfalse%
-\ matrix{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}iarray{\isacharunderscore}{\kern0pt}fst{\isacharunderscore}{\kern0pt}Gauss{\isacharunderscore}{\kern0pt}Jordan{\isacharunderscore}{\kern0pt}PA{\isacharbrackleft}{\kern0pt}code{\isacharunderscore}{\kern0pt}unfold{\isacharbrackright}{\kern0pt}\ \isanewline
-%
-\isadelimtheory
-\isanewline
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{end}\isamarkupfalse%
-%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\end{isabellebody}%
-\endinput
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-%:%1973=516%:%
-%:%1976=517%:%
-%:%1980=517%:%
-%:%1981=517%:%
-%:%1982=518%:%
-%:%1983=518%:%
-%:%1988=518%:%
-%:%1991=519%:%
-%:%1992=520%:%
-%:%1993=520%:%
-%:%1996=521%:%
-%:%2000=521%:%
-%:%2001=521%:%
-%:%2002=522%:%
-%:%2003=522%:%
-%:%2008=522%:%
-%:%2011=523%:%
-%:%2012=524%:%
-%:%2013=524%:%
-%:%2016=525%:%
-%:%2021=526%:%
diff --git a/thys/MDP-Algorithms/output/document/Code_Gridworld.tex b/thys/MDP-Algorithms/output/document/Code_Gridworld.tex
deleted file mode 100644
--- a/thys/MDP-Algorithms/output/document/Code_Gridworld.tex
+++ /dev/null
@@ -1,493 +0,0 @@
-%
-\begin{isabellebody}%
-\setisabellecontext{Code{\isacharunderscore}{\kern0pt}Gridworld}%
-%
-\isadelimtheory
-\isanewline
-\isanewline
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{theory}\isamarkupfalse%
-\ Code{\isacharunderscore}{\kern0pt}Gridworld\isanewline
-\ \ \isakeyword{imports}\ \isanewline
-\ \ \ \ Code{\isacharunderscore}{\kern0pt}Mod\isanewline
-\isakeyword{begin}%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsection{Gridworld Example%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{lemma}\isamarkupfalse%
-\ {\isacharbrackleft}{\kern0pt}code\ abstype{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}embed{\isacharunderscore}{\kern0pt}pmf\ {\isacharparenleft}{\kern0pt}pmf\ P{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ P{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharcomma}{\kern0pt}\ lifting{\isacharparenright}{\kern0pt}\ td{\isacharunderscore}{\kern0pt}pmf{\isacharunderscore}{\kern0pt}embed{\isacharunderscore}{\kern0pt}pmf\ type{\isacharunderscore}{\kern0pt}definition{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemmas}\isamarkupfalse%
-\ {\isacharbrackleft}{\kern0pt}code{\isacharunderscore}{\kern0pt}abbrev\ del{\isacharbrackright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ pmf{\isacharunderscore}{\kern0pt}integral{\isacharunderscore}{\kern0pt}code{\isacharunderscore}{\kern0pt}unfold\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ {\isacharbrackleft}{\kern0pt}code{\isacharunderscore}{\kern0pt}unfold{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\ \isanewline
-\ \ {\isachardoublequoteopen}measure{\isacharunderscore}{\kern0pt}pmf{\isachardot}{\kern0pt}expectation\ P\ {\isacharparenleft}{\kern0pt}f\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isacharprime}{\kern0pt}a\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ enum\ {\isasymRightarrow}\ real{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSum}x\ {\isasymin}\ UNIV{\isachardot}{\kern0pt}\ pmf\ P\ x\ {\isacharasterisk}{\kern0pt}\ f\ x{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharcomma}{\kern0pt}\ lifting{\isacharparenright}{\kern0pt}\ UNIV{\isacharunderscore}{\kern0pt}I\ finite{\isacharunderscore}{\kern0pt}class{\isachardot}{\kern0pt}finite{\isacharunderscore}{\kern0pt}UNIV\ integral{\isacharunderscore}{\kern0pt}measure{\isacharunderscore}{\kern0pt}pmf\ \isanewline
-\ \ \ \ \ \ real{\isacharunderscore}{\kern0pt}scaleR{\isacharunderscore}{\kern0pt}def\ sum{\isachardot}{\kern0pt}cong{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ {\isacharbrackleft}{\kern0pt}code{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}pmf\ {\isacharparenleft}{\kern0pt}return{\isacharunderscore}{\kern0pt}pmf\ x{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymlambda}y{\isachardot}{\kern0pt}\ indicat{\isacharunderscore}{\kern0pt}real\ {\isacharbraceleft}{\kern0pt}y{\isacharbraceright}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isachardoublequoteclose}\ \isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ {\isacharbrackleft}{\kern0pt}code{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\ \isanewline
-\ \ {\isachardoublequoteopen}pmf\ {\isacharparenleft}{\kern0pt}bind{\isacharunderscore}{\kern0pt}pmf\ N\ f{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymlambda}i\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isacharprime}{\kern0pt}a{\isachardot}{\kern0pt}\ measure{\isacharunderscore}{\kern0pt}pmf{\isachardot}{\kern0pt}expectation\ N\ {\isacharparenleft}{\kern0pt}{\isasymlambda}{\isacharparenleft}{\kern0pt}x\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isacharprime}{\kern0pt}b\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}enum{\isacharparenright}{\kern0pt}{\isachardot}{\kern0pt}\ pmf\ {\isacharparenleft}{\kern0pt}f\ x{\isacharparenright}{\kern0pt}\ i{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ Probability{\isacharunderscore}{\kern0pt}Mass{\isacharunderscore}{\kern0pt}Function{\isachardot}{\kern0pt}pmf{\isacharunderscore}{\kern0pt}bind\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ fast%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{type{\isacharunderscore}{\kern0pt}synonym}\isamarkupfalse%
-\ state{\isacharunderscore}{\kern0pt}robot\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}{\isadigit{1}}{\isadigit{3}}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ {\isachardoublequoteopen}from{\isacharunderscore}{\kern0pt}state\ x\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}Rep\ x\ div\ {\isadigit{4}}{\isacharcomma}{\kern0pt}\ Rep\ x\ mod\ {\isadigit{4}}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ {\isachardoublequoteopen}to{\isacharunderscore}{\kern0pt}state\ x\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}Abs\ {\isacharparenleft}{\kern0pt}fst\ x\ {\isacharasterisk}{\kern0pt}\ {\isadigit{4}}\ {\isacharplus}{\kern0pt}\ snd\ x{\isacharparenright}{\kern0pt}\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ state{\isacharunderscore}{\kern0pt}robot{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isanewline
-\isacommand{type{\isacharunderscore}{\kern0pt}synonym}\isamarkupfalse%
-\ action{\isacharunderscore}{\kern0pt}robot\ {\isacharequal}{\kern0pt}\ {\isadigit{4}}\isanewline
-\isanewline
-\isacommand{fun}\isamarkupfalse%
-\ A{\isacharunderscore}{\kern0pt}robot\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}state{\isacharunderscore}{\kern0pt}robot\ {\isasymRightarrow}\ action{\isacharunderscore}{\kern0pt}robot\ set{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}A{\isacharunderscore}{\kern0pt}robot\ pos\ {\isacharequal}{\kern0pt}\ UNIV{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{abbreviation}\isamarkupfalse%
-\ {\isachardoublequoteopen}noise\ {\isasymequiv}\ {\isacharparenleft}{\kern0pt}{\isadigit{0}}{\isachardot}{\kern0pt}{\isadigit{2}}\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ real{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{lift{\isacharunderscore}{\kern0pt}definition}\isamarkupfalse%
-\ add{\isacharunderscore}{\kern0pt}noise\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}action{\isacharunderscore}{\kern0pt}robot\ {\isasymRightarrow}\ action{\isacharunderscore}{\kern0pt}robot\ pmf{\isachardoublequoteclose}\ \isakeyword{is}\ {\isachardoublequoteopen}{\isasymlambda}det\ rnd{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}\isanewline
-\ \ if\ det\ {\isacharequal}{\kern0pt}\ rnd\ then\ {\isadigit{1}}\ {\isacharminus}{\kern0pt}\ noise\ else\ if\ det\ {\isacharequal}{\kern0pt}\ rnd\ {\isacharminus}{\kern0pt}\ {\isadigit{1}}\ {\isasymor}\ det\ {\isacharequal}{\kern0pt}\ rnd\ {\isacharplus}{\kern0pt}\ {\isadigit{1}}\ then\ noise\ {\isacharslash}{\kern0pt}\ {\isadigit{2}}\ else\ {\isadigit{0}}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{subgoal}\isamarkupfalse%
-\ \isakeyword{for}\ n\isanewline
-\ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ nn{\isacharunderscore}{\kern0pt}integral{\isacharunderscore}{\kern0pt}count{\isacharunderscore}{\kern0pt}space{\isacharunderscore}{\kern0pt}finite{\isacharbrackleft}{\kern0pt}OF\ finite{\isacharbrackright}{\kern0pt}\ UNIV{\isacharunderscore}{\kern0pt}{\isadigit{4}}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ exhaust{\isacharunderscore}{\kern0pt}{\isadigit{4}}{\isacharbrackleft}{\kern0pt}of\ n{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ fastforce\isanewline
-\ \ \isacommand{done}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{fun}\isamarkupfalse%
-\ r{\isacharunderscore}{\kern0pt}robot\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}state{\isacharunderscore}{\kern0pt}robot\ {\isasymtimes}\ action{\isacharunderscore}{\kern0pt}robot{\isacharparenright}{\kern0pt}\ {\isasymRightarrow}\ real{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}r{\isacharunderscore}{\kern0pt}robot\ {\isacharparenleft}{\kern0pt}s{\isacharcomma}{\kern0pt}a{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}\isanewline
-\ \ if\ from{\isacharunderscore}{\kern0pt}state\ s\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharcomma}{\kern0pt}{\isadigit{3}}{\isacharparenright}{\kern0pt}\ then\ {\isadigit{1}}\ else\isanewline
-\ \ if\ from{\isacharunderscore}{\kern0pt}state\ s\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharcomma}{\kern0pt}{\isadigit{3}}{\isacharparenright}{\kern0pt}\ then\ {\isacharminus}{\kern0pt}{\isadigit{1}}\ else\ \isanewline
-\ \ if\ from{\isacharunderscore}{\kern0pt}state\ s\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isadigit{3}}{\isacharcomma}{\kern0pt}{\isadigit{0}}{\isacharparenright}{\kern0pt}\ then\ {\isadigit{0}}\ else\isanewline
-\ \ {\isadigit{0}}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{fun}\isamarkupfalse%
-\ K{\isacharunderscore}{\kern0pt}robot\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}state{\isacharunderscore}{\kern0pt}robot\ {\isasymtimes}\ action{\isacharunderscore}{\kern0pt}robot{\isacharparenright}{\kern0pt}\ {\isasymRightarrow}\ state{\isacharunderscore}{\kern0pt}robot\ pmf{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}K{\isacharunderscore}{\kern0pt}robot\ {\isacharparenleft}{\kern0pt}loc{\isacharcomma}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ \isanewline
-\ \ do\ {\isacharbraceleft}{\kern0pt}\isanewline
-\ \ a\ {\isasymleftarrow}\ add{\isacharunderscore}{\kern0pt}noise\ a{\isacharsemicolon}{\kern0pt}\isanewline
-\ \ let\ {\isacharparenleft}{\kern0pt}y{\isacharcomma}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ from{\isacharunderscore}{\kern0pt}state\ loc{\isacharsemicolon}{\kern0pt}\isanewline
-\ \ let\ {\isacharparenleft}{\kern0pt}y{\isacharprime}{\kern0pt}{\isacharcomma}{\kern0pt}\ x{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ {\isacharparenleft}{\kern0pt}if\ a\ {\isacharequal}{\kern0pt}\ {\isadigit{0}}\ then\ {\isacharparenleft}{\kern0pt}y\ {\isacharplus}{\kern0pt}\ {\isadigit{1}}\ {\isacharcomma}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ else\ if\ a\ {\isacharequal}{\kern0pt}\ {\isadigit{1}}\ then\ {\isacharparenleft}{\kern0pt}y{\isacharcomma}{\kern0pt}\ x{\isacharplus}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ else\ if\ a\ {\isacharequal}{\kern0pt}\ {\isadigit{2}}\ then\ {\isacharparenleft}{\kern0pt}y{\isacharminus}{\kern0pt}{\isadigit{1}}{\isacharcomma}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ else\ if\ a\ {\isacharequal}{\kern0pt}\ {\isadigit{3}}\ then\ {\isacharparenleft}{\kern0pt}y{\isacharcomma}{\kern0pt}\ x{\isacharminus}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ else\ undefined{\isacharparenright}{\kern0pt}{\isacharsemicolon}{\kern0pt}\isanewline
-\ \ \ return{\isacharunderscore}{\kern0pt}pmf\ {\isacharparenleft}{\kern0pt}\isanewline
-\ \ \ \ \ \ if\ {\isacharparenleft}{\kern0pt}y{\isacharcomma}{\kern0pt}x{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharcomma}{\kern0pt}{\isadigit{3}}{\isacharparenright}{\kern0pt}\ {\isasymor}\ {\isacharparenleft}{\kern0pt}y{\isacharcomma}{\kern0pt}x{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharcomma}{\kern0pt}{\isadigit{3}}{\isacharparenright}{\kern0pt}\ {\isasymor}\ {\isacharparenleft}{\kern0pt}y{\isacharcomma}{\kern0pt}x{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isadigit{3}}{\isacharcomma}{\kern0pt}{\isadigit{0}}{\isacharparenright}{\kern0pt}\ \isanewline
-\ \ \ \ \ \ \ \ then\ to{\isacharunderscore}{\kern0pt}state\ {\isacharparenleft}{\kern0pt}{\isadigit{3}}{\isacharcomma}{\kern0pt}{\isadigit{0}}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ else\ if\ y{\isacharprime}{\kern0pt}\ {\isacharless}{\kern0pt}\ {\isadigit{0}}\ {\isasymor}\ y{\isacharprime}{\kern0pt}\ {\isachargreater}{\kern0pt}\ {\isadigit{2}}\ {\isasymor}\ x{\isacharprime}{\kern0pt}\ {\isacharless}{\kern0pt}\ {\isadigit{0}}\ {\isasymor}\ x{\isacharprime}{\kern0pt}\ {\isachargreater}{\kern0pt}\ {\isadigit{3}}\ {\isasymor}\ {\isacharparenleft}{\kern0pt}y{\isacharprime}{\kern0pt}{\isacharcomma}{\kern0pt}x{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharcomma}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ then\ to{\isacharunderscore}{\kern0pt}state\ {\isacharparenleft}{\kern0pt}y{\isacharcomma}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ else\ to{\isacharunderscore}{\kern0pt}state\ {\isacharparenleft}{\kern0pt}y{\isacharprime}{\kern0pt}{\isacharcomma}{\kern0pt}\ x{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ {\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ {\isachardoublequoteopen}l{\isacharunderscore}{\kern0pt}robot\ {\isacharequal}{\kern0pt}\ {\isadigit{0}}{\isachardot}{\kern0pt}{\isadigit{9}}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ {\isachardoublequoteopen}vi{\isacharunderscore}{\kern0pt}code\ A{\isacharunderscore}{\kern0pt}robot\ r{\isacharunderscore}{\kern0pt}robot\ l{\isacharunderscore}{\kern0pt}robot{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ standard\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ l{\isacharunderscore}{\kern0pt}robot{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{abbreviation}\isamarkupfalse%
-\ {\isachardoublequoteopen}to{\isacharunderscore}{\kern0pt}gridworld\ f\ {\isasymequiv}\ f\ K{\isacharunderscore}{\kern0pt}robot\ r{\isacharunderscore}{\kern0pt}robot\ l{\isacharunderscore}{\kern0pt}robot{\isachardoublequoteclose}\isanewline
-\isacommand{abbreviation}\isamarkupfalse%
-\ {\isachardoublequoteopen}to{\isacharunderscore}{\kern0pt}gridworld{\isacharprime}{\kern0pt}\ f\ {\isasymequiv}\ f\ K{\isacharunderscore}{\kern0pt}robot\ A{\isacharunderscore}{\kern0pt}robot\ r{\isacharunderscore}{\kern0pt}robot\ l{\isacharunderscore}{\kern0pt}robot{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{abbreviation}\isamarkupfalse%
-\ {\isachardoublequoteopen}gridworld{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}eval{\isacharprime}{\kern0pt}\ {\isasymequiv}\ to{\isacharunderscore}{\kern0pt}gridworld\ mod{\isacharunderscore}{\kern0pt}MDP{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}eval{\isacharprime}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isacommand{abbreviation}\isamarkupfalse%
-\ {\isachardoublequoteopen}gridworld{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}step{\isacharprime}{\kern0pt}\ {\isasymequiv}\ to{\isacharunderscore}{\kern0pt}gridworld{\isacharprime}{\kern0pt}\ mod{\isacharunderscore}{\kern0pt}MDP{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}iteration{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}step{\isacharprime}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isacommand{abbreviation}\isamarkupfalse%
-\ {\isachardoublequoteopen}gridworld{\isacharunderscore}{\kern0pt}mpi{\isacharunderscore}{\kern0pt}user\ {\isasymequiv}\ to{\isacharunderscore}{\kern0pt}gridworld{\isacharprime}{\kern0pt}\ mod{\isacharunderscore}{\kern0pt}MDP{\isacharunderscore}{\kern0pt}mpi{\isacharunderscore}{\kern0pt}user{\isachardoublequoteclose}\isanewline
-\isacommand{abbreviation}\isamarkupfalse%
-\ {\isachardoublequoteopen}gridworld{\isacharunderscore}{\kern0pt}opt{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}gs\ {\isasymequiv}\ to{\isacharunderscore}{\kern0pt}gridworld{\isacharprime}{\kern0pt}\ mod{\isacharunderscore}{\kern0pt}MDP{\isacharunderscore}{\kern0pt}opt{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}gs{\isachardoublequoteclose}\isanewline
-\isacommand{abbreviation}\isamarkupfalse%
-\ {\isachardoublequoteopen}gridworld{\isacharunderscore}{\kern0pt}{\isasymL}\isactrlsub b\ {\isasymequiv}\ to{\isacharunderscore}{\kern0pt}gridworld{\isacharprime}{\kern0pt}\ mod{\isacharunderscore}{\kern0pt}MDP{\isacharunderscore}{\kern0pt}{\isasymL}\isactrlsub b{\isachardoublequoteclose}\isanewline
-\isacommand{abbreviation}\isamarkupfalse%
-\ {\isachardoublequoteopen}gridworld{\isacharunderscore}{\kern0pt}find{\isacharunderscore}{\kern0pt}policy{\isacharprime}{\kern0pt}\ {\isasymequiv}\ to{\isacharunderscore}{\kern0pt}gridworld{\isacharprime}{\kern0pt}\ mod{\isacharunderscore}{\kern0pt}MDP{\isacharunderscore}{\kern0pt}find{\isacharunderscore}{\kern0pt}policy{\isacharprime}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isacommand{abbreviation}\isamarkupfalse%
-\ {\isachardoublequoteopen}gridworld{\isacharunderscore}{\kern0pt}GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}fun\isactrlsub b\ {\isasymequiv}\ to{\isacharunderscore}{\kern0pt}gridworld{\isacharprime}{\kern0pt}\ mod{\isacharunderscore}{\kern0pt}MDP{\isacharunderscore}{\kern0pt}GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}fun\isactrlsub b{\isachardoublequoteclose}\isanewline
-\isacommand{abbreviation}\isamarkupfalse%
-\ {\isachardoublequoteopen}gridworld{\isacharunderscore}{\kern0pt}vi{\isacharunderscore}{\kern0pt}policy{\isacharprime}{\kern0pt}\ {\isasymequiv}\ to{\isacharunderscore}{\kern0pt}gridworld{\isacharprime}{\kern0pt}\ mod{\isacharunderscore}{\kern0pt}MDP{\isacharunderscore}{\kern0pt}vi{\isacharunderscore}{\kern0pt}policy{\isacharprime}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isacommand{abbreviation}\isamarkupfalse%
-\ {\isachardoublequoteopen}gridworld{\isacharunderscore}{\kern0pt}vi{\isacharunderscore}{\kern0pt}gs{\isacharunderscore}{\kern0pt}policy\ {\isasymequiv}\ to{\isacharunderscore}{\kern0pt}gridworld{\isacharprime}{\kern0pt}\ mod{\isacharunderscore}{\kern0pt}MDP{\isacharunderscore}{\kern0pt}vi{\isacharunderscore}{\kern0pt}gs{\isacharunderscore}{\kern0pt}policy{\isachardoublequoteclose}\isanewline
-\isacommand{abbreviation}\isamarkupfalse%
-\ {\isachardoublequoteopen}gridworld{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}iteration\ {\isasymequiv}\ to{\isacharunderscore}{\kern0pt}gridworld{\isacharprime}{\kern0pt}\ mod{\isacharunderscore}{\kern0pt}MDP{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}iteration{\isachardoublequoteclose}\isanewline
-\isanewline
-\isanewline
-\isacommand{fun}\isamarkupfalse%
-\ pi{\isacharunderscore}{\kern0pt}robot{\isacharunderscore}{\kern0pt}n\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}pi{\isacharunderscore}{\kern0pt}robot{\isacharunderscore}{\kern0pt}n\ {\isadigit{0}}\ d\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}d{\isacharcomma}{\kern0pt}\ gridworld{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}eval{\isacharprime}{\kern0pt}\ d{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ {\isacharbar}{\kern0pt}\ \isanewline
-\ \ {\isachardoublequoteopen}pi{\isacharunderscore}{\kern0pt}robot{\isacharunderscore}{\kern0pt}n\ {\isacharparenleft}{\kern0pt}Suc\ n{\isacharparenright}{\kern0pt}\ d\ {\isacharequal}{\kern0pt}\ pi{\isacharunderscore}{\kern0pt}robot{\isacharunderscore}{\kern0pt}n\ n\ {\isacharparenleft}{\kern0pt}gridworld{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}step{\isacharprime}{\kern0pt}\ d{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ {\isachardoublequoteopen}mpi{\isacharunderscore}{\kern0pt}robot\ eps\ {\isacharequal}{\kern0pt}\ gridworld{\isacharunderscore}{\kern0pt}mpi{\isacharunderscore}{\kern0pt}user\ eps\ {\isacharparenleft}{\kern0pt}{\isasymlambda}{\isacharunderscore}{\kern0pt}{\isachardot}{\kern0pt}\ {\isadigit{3}}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{fun}\isamarkupfalse%
-\ gs{\isacharunderscore}{\kern0pt}robot{\isacharunderscore}{\kern0pt}n\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}gs{\isacharunderscore}{\kern0pt}robot{\isacharunderscore}{\kern0pt}n\ {\isacharparenleft}{\kern0pt}{\isadigit{0}}\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ nat{\isacharparenright}{\kern0pt}\ v\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}gridworld{\isacharunderscore}{\kern0pt}opt{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}gs\ v{\isacharcomma}{\kern0pt}\ v{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ {\isacharbar}{\kern0pt}\isanewline
-\ \ {\isachardoublequoteopen}gs{\isacharunderscore}{\kern0pt}robot{\isacharunderscore}{\kern0pt}n\ {\isacharparenleft}{\kern0pt}Suc\ n\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ nat{\isacharparenright}{\kern0pt}\ v\ {\isacharequal}{\kern0pt}\ gs{\isacharunderscore}{\kern0pt}robot{\isacharunderscore}{\kern0pt}n\ n\ {\isacharparenleft}{\kern0pt}gridworld{\isacharunderscore}{\kern0pt}GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}fun\isactrlsub b\ v{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{fun}\isamarkupfalse%
-\ vi{\isacharunderscore}{\kern0pt}robot{\isacharunderscore}{\kern0pt}n\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}vi{\isacharunderscore}{\kern0pt}robot{\isacharunderscore}{\kern0pt}n\ {\isacharparenleft}{\kern0pt}{\isadigit{0}}\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ nat{\isacharparenright}{\kern0pt}\ v\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}gridworld{\isacharunderscore}{\kern0pt}find{\isacharunderscore}{\kern0pt}policy{\isacharprime}{\kern0pt}\ v{\isacharcomma}{\kern0pt}\ v{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ {\isacharbar}{\kern0pt}\isanewline
-\ \ {\isachardoublequoteopen}vi{\isacharunderscore}{\kern0pt}robot{\isacharunderscore}{\kern0pt}n\ {\isacharparenleft}{\kern0pt}Suc\ n\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ nat{\isacharparenright}{\kern0pt}\ v\ {\isacharequal}{\kern0pt}\ vi{\isacharunderscore}{\kern0pt}robot{\isacharunderscore}{\kern0pt}n\ n\ {\isacharparenleft}{\kern0pt}gridworld{\isacharunderscore}{\kern0pt}{\isasymL}\isactrlsub b\ v{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ {\isachardoublequoteopen}mpi{\isacharunderscore}{\kern0pt}result\ eps\ {\isacharequal}{\kern0pt}\ \isanewline
-\ \ {\isacharparenleft}{\kern0pt}let\ {\isacharparenleft}{\kern0pt}d{\isacharcomma}{\kern0pt}\ v{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ mpi{\isacharunderscore}{\kern0pt}robot\ eps\ in\ {\isacharparenleft}{\kern0pt}d{\isacharcomma}{\kern0pt}v{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ {\isachardoublequoteopen}gs{\isacharunderscore}{\kern0pt}result\ n\ {\isacharequal}{\kern0pt}\ \isanewline
-\ \ {\isacharparenleft}{\kern0pt}let\ {\isacharparenleft}{\kern0pt}d{\isacharcomma}{\kern0pt}v{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ gs{\isacharunderscore}{\kern0pt}robot{\isacharunderscore}{\kern0pt}n\ n\ {\isadigit{0}}\ in\ {\isacharparenleft}{\kern0pt}d{\isacharcomma}{\kern0pt}v{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ {\isachardoublequoteopen}vi{\isacharunderscore}{\kern0pt}result{\isacharunderscore}{\kern0pt}n\ n\ {\isacharequal}{\kern0pt}\ \isanewline
-\ \ {\isacharparenleft}{\kern0pt}let\ {\isacharparenleft}{\kern0pt}d{\isacharcomma}{\kern0pt}\ v{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ vi{\isacharunderscore}{\kern0pt}robot{\isacharunderscore}{\kern0pt}n\ n\ {\isadigit{0}}\ in\ {\isacharparenleft}{\kern0pt}d{\isacharcomma}{\kern0pt}v{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ {\isachardoublequoteopen}pi{\isacharunderscore}{\kern0pt}result{\isacharunderscore}{\kern0pt}n\ n\ {\isacharequal}{\kern0pt}\ \isanewline
-\ \ {\isacharparenleft}{\kern0pt}let\ {\isacharparenleft}{\kern0pt}d{\isacharcomma}{\kern0pt}\ v{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ pi{\isacharunderscore}{\kern0pt}robot{\isacharunderscore}{\kern0pt}n\ n\ {\isacharparenleft}{\kern0pt}vec{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}fun\ {\isadigit{0}}{\isacharparenright}{\kern0pt}\ in\ {\isacharparenleft}{\kern0pt}d{\isacharcomma}{\kern0pt}v{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ {\isachardoublequoteopen}fun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}list\ f\ {\isacharequal}{\kern0pt}\ map\ f\ {\isacharparenleft}{\kern0pt}sorted{\isacharunderscore}{\kern0pt}list{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}set\ UNIV{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ {\isachardoublequoteopen}benchmark{\isacharunderscore}{\kern0pt}gs\ {\isacharequal}{\kern0pt}\ fun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}list\ {\isacharparenleft}{\kern0pt}gridworld{\isacharunderscore}{\kern0pt}vi{\isacharunderscore}{\kern0pt}policy{\isacharprime}{\kern0pt}\ {\isadigit{0}}{\isachardot}{\kern0pt}{\isadigit{1}}\ {\isadigit{0}}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ {\isachardoublequoteopen}benchmark{\isacharunderscore}{\kern0pt}vi\ {\isacharequal}{\kern0pt}\ fun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}list\ {\isacharparenleft}{\kern0pt}gridworld{\isacharunderscore}{\kern0pt}vi{\isacharunderscore}{\kern0pt}gs{\isacharunderscore}{\kern0pt}policy\ {\isadigit{0}}{\isachardot}{\kern0pt}{\isadigit{1}}\ {\isadigit{0}}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ {\isachardoublequoteopen}benchmark{\isacharunderscore}{\kern0pt}mpi\ {\isacharequal}{\kern0pt}\ fun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}list\ {\isacharparenleft}{\kern0pt}fst\ {\isacharparenleft}{\kern0pt}gridworld{\isacharunderscore}{\kern0pt}mpi{\isacharunderscore}{\kern0pt}user\ {\isadigit{0}}{\isachardot}{\kern0pt}{\isadigit{1}}\ {\isacharparenleft}{\kern0pt}{\isasymlambda}{\isacharunderscore}{\kern0pt}\ {\isacharunderscore}{\kern0pt}{\isachardot}{\kern0pt}\ {\isadigit{3}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ {\isachardoublequoteopen}benchmark{\isacharunderscore}{\kern0pt}pi\ {\isacharequal}{\kern0pt}\ fun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}list\ {\isacharparenleft}{\kern0pt}gridworld{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}iteration\ {\isadigit{0}}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isanewline
-\isanewline
-\isacommand{export{\isacharunderscore}{\kern0pt}code}\isamarkupfalse%
-\ benchmark{\isacharunderscore}{\kern0pt}gs\ benchmark{\isacharunderscore}{\kern0pt}vi\ benchmark{\isacharunderscore}{\kern0pt}mpi\ benchmark{\isacharunderscore}{\kern0pt}pi\ \isakeyword{in}\ Haskell\ \isakeyword{module{\isacharunderscore}{\kern0pt}name}\ DP\isanewline
-\isacommand{export{\isacharunderscore}{\kern0pt}code}\isamarkupfalse%
-\ benchmark{\isacharunderscore}{\kern0pt}gs\ benchmark{\isacharunderscore}{\kern0pt}vi\ benchmark{\isacharunderscore}{\kern0pt}mpi\ benchmark{\isacharunderscore}{\kern0pt}pi\ \isakeyword{in}\ SML\ \isakeyword{module{\isacharunderscore}{\kern0pt}name}\ DP\isanewline
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{end}\isamarkupfalse%
-%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\end{isabellebody}%
-\endinput
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diff --git a/thys/MDP-Algorithms/output/document/Code_Inventory.tex b/thys/MDP-Algorithms/output/document/Code_Inventory.tex
deleted file mode 100644
--- a/thys/MDP-Algorithms/output/document/Code_Inventory.tex
+++ /dev/null
@@ -1,877 +0,0 @@
-%
-\begin{isabellebody}%
-\setisabellecontext{Code{\isacharunderscore}{\kern0pt}Inventory}%
-%
-\isadelimtheory
-\isanewline
-\isanewline
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{theory}\isamarkupfalse%
-\ Code{\isacharunderscore}{\kern0pt}Inventory\isanewline
-\ \ \isakeyword{imports}\ \isanewline
-\ \ \ \ Code{\isacharunderscore}{\kern0pt}Mod\isanewline
-\ \ \isanewline
-\ \ \ \ Code{\isacharunderscore}{\kern0pt}Real{\isacharunderscore}{\kern0pt}Approx{\isacharunderscore}{\kern0pt}By{\isacharunderscore}{\kern0pt}Float{\isacharunderscore}{\kern0pt}Fix\isanewline
-\isakeyword{begin}%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsection{Inventory Management Example%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{lemma}\isamarkupfalse%
-\ {\isacharbrackleft}{\kern0pt}code\ abstype{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}embed{\isacharunderscore}{\kern0pt}pmf\ {\isacharparenleft}{\kern0pt}pmf\ P{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ P{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharcomma}{\kern0pt}\ lifting{\isacharparenright}{\kern0pt}\ td{\isacharunderscore}{\kern0pt}pmf{\isacharunderscore}{\kern0pt}embed{\isacharunderscore}{\kern0pt}pmf\ type{\isacharunderscore}{\kern0pt}definition{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemmas}\isamarkupfalse%
-\ {\isacharbrackleft}{\kern0pt}code{\isacharunderscore}{\kern0pt}abbrev\ del{\isacharbrackright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ pmf{\isacharunderscore}{\kern0pt}integral{\isacharunderscore}{\kern0pt}code{\isacharunderscore}{\kern0pt}unfold\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ {\isacharbrackleft}{\kern0pt}code{\isacharunderscore}{\kern0pt}unfold{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\ \isanewline
-\ \ {\isachardoublequoteopen}measure{\isacharunderscore}{\kern0pt}pmf{\isachardot}{\kern0pt}expectation\ P\ {\isacharparenleft}{\kern0pt}f\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isacharprime}{\kern0pt}a\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ enum\ {\isasymRightarrow}\ real{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSum}x\ {\isasymin}\ UNIV{\isachardot}{\kern0pt}\ pmf\ P\ x\ {\isacharasterisk}{\kern0pt}\ f\ x{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharcomma}{\kern0pt}\ lifting{\isacharparenright}{\kern0pt}\ UNIV{\isacharunderscore}{\kern0pt}I\ finite{\isacharunderscore}{\kern0pt}class{\isachardot}{\kern0pt}finite{\isacharunderscore}{\kern0pt}UNIV\ integral{\isacharunderscore}{\kern0pt}measure{\isacharunderscore}{\kern0pt}pmf\ \isanewline
-\ \ \ \ \ \ real{\isacharunderscore}{\kern0pt}scaleR{\isacharunderscore}{\kern0pt}def\ sum{\isachardot}{\kern0pt}cong{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ {\isacharbrackleft}{\kern0pt}code{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}pmf\ {\isacharparenleft}{\kern0pt}return{\isacharunderscore}{\kern0pt}pmf\ x{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymlambda}y{\isachardot}{\kern0pt}\ indicat{\isacharunderscore}{\kern0pt}real\ {\isacharbraceleft}{\kern0pt}y{\isacharbraceright}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isachardoublequoteclose}\ \isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ {\isacharbrackleft}{\kern0pt}code{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\ \isanewline
-\ \ {\isachardoublequoteopen}pmf\ {\isacharparenleft}{\kern0pt}bind{\isacharunderscore}{\kern0pt}pmf\ N\ f{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymlambda}i\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isacharprime}{\kern0pt}a{\isachardot}{\kern0pt}\ measure{\isacharunderscore}{\kern0pt}pmf{\isachardot}{\kern0pt}expectation\ N\ {\isacharparenleft}{\kern0pt}{\isasymlambda}{\isacharparenleft}{\kern0pt}x\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isacharprime}{\kern0pt}b\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}enum{\isacharparenright}{\kern0pt}{\isachardot}{\kern0pt}\ pmf\ {\isacharparenleft}{\kern0pt}f\ x{\isacharparenright}{\kern0pt}\ i{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ Probability{\isacharunderscore}{\kern0pt}Mass{\isacharunderscore}{\kern0pt}Function{\isachardot}{\kern0pt}pmf{\isacharunderscore}{\kern0pt}bind\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ fast%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ pmf{\isacharunderscore}{\kern0pt}finite{\isacharunderscore}{\kern0pt}le{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite\ {\isacharparenleft}{\kern0pt}X\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}a{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}finite{\isacharparenright}{\kern0pt}\ set{\isacharparenright}{\kern0pt}\ {\isasymLongrightarrow}\ sum\ {\isacharparenleft}{\kern0pt}pmf\ p{\isacharparenright}{\kern0pt}\ X\ {\isasymle}\ {\isadigit{1}}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}subst\ sum{\isacharunderscore}{\kern0pt}pmf{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}{\isadigit{1}}{\isacharbrackleft}{\kern0pt}symmetric{\isacharcomma}{\kern0pt}\ of\ UNIV\ p{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}auto\ intro{\isacharcolon}{\kern0pt}\ sum{\isacharunderscore}{\kern0pt}mono{\isadigit{2}}{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ mod{\isacharunderscore}{\kern0pt}less{\isacharunderscore}{\kern0pt}diff{\isacharcolon}{\kern0pt}\ \isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}{\isadigit{0}}\ {\isacharless}{\kern0pt}\ {\isacharparenleft}{\kern0pt}x{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}{\isacharprime}{\kern0pt}s{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}{\isacharbraceleft}{\kern0pt}mod{\isacharunderscore}{\kern0pt}type{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ {\isachardoublequoteopen}x\ {\isasymle}\ y{\isachardoublequoteclose}\ \isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}y\ {\isacharminus}{\kern0pt}\ x\ {\isacharless}{\kern0pt}\ y{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isadigit{0}}\ {\isasymle}\ Rep\ y\ {\isacharminus}{\kern0pt}\ Rep\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assms\ mono{\isacharunderscore}{\kern0pt}Rep\ \isacommand{unfolding}\isamarkupfalse%
-\ mono{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}Rep\ y\ {\isacharminus}{\kern0pt}\ Rep\ x\ {\isacharequal}{\kern0pt}\ Rep\ {\isacharparenleft}{\kern0pt}y\ {\isacharminus}{\kern0pt}\ x{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ mod{\isacharunderscore}{\kern0pt}type{\isacharunderscore}{\kern0pt}class{\isachardot}{\kern0pt}diff{\isacharunderscore}{\kern0pt}def\ Rep{\isacharunderscore}{\kern0pt}Abs{\isacharunderscore}{\kern0pt}mod\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ Rep{\isacharunderscore}{\kern0pt}ge{\isacharunderscore}{\kern0pt}{\isadigit{0}}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ mod{\isacharunderscore}{\kern0pt}pos{\isacharunderscore}{\kern0pt}pos{\isacharunderscore}{\kern0pt}trivial{\isacharbrackleft}{\kern0pt}OF\ {\isacartoucheopen}{\isadigit{0}}\ {\isasymle}\ Rep\ y\ {\isacharminus}{\kern0pt}\ Rep\ x{\isacartoucheclose}\ order{\isachardot}{\kern0pt}strict{\isacharunderscore}{\kern0pt}trans{\isadigit{1}}{\isacharbrackleft}{\kern0pt}OF\ {\isacharunderscore}{\kern0pt}\ Rep{\isacharunderscore}{\kern0pt}less{\isacharunderscore}{\kern0pt}n{\isacharcomma}{\kern0pt}\ of\ {\isacharunderscore}{\kern0pt}\ y{\isacharbrackright}{\kern0pt}{\isacharcomma}{\kern0pt}\ symmetric{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}Rep\ y\ {\isacharminus}{\kern0pt}\ Rep\ x\ {\isacharless}{\kern0pt}\ Rep\ y{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assms{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}\ strict{\isacharunderscore}{\kern0pt}mono{\isacharunderscore}{\kern0pt}Rep\ Rep{\isacharunderscore}{\kern0pt}ge{\isacharunderscore}{\kern0pt}{\isadigit{0}}\ le{\isacharunderscore}{\kern0pt}less\ not{\isacharunderscore}{\kern0pt}less\ \isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}fastforce\ simp{\isacharcolon}{\kern0pt}\ strict{\isacharunderscore}{\kern0pt}mono{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}Rep\ {\isacharparenleft}{\kern0pt}y\ {\isacharminus}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isacharless}{\kern0pt}\ Rep\ y{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ {\isadigit{1}}\ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ not{\isacharunderscore}{\kern0pt}less{\isacharunderscore}{\kern0pt}iff{\isacharunderscore}{\kern0pt}gr{\isacharunderscore}{\kern0pt}or{\isacharunderscore}{\kern0pt}eq\ strict{\isacharunderscore}{\kern0pt}mono{\isacharunderscore}{\kern0pt}Rep\ strict{\isacharunderscore}{\kern0pt}mono{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{locale}\isamarkupfalse%
-\ inventory\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \isakeyword{fixes}\ fixed{\isacharunderscore}{\kern0pt}cost\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ real\isanewline
-\ \ \ \ \isakeyword{and}\ var{\isacharunderscore}{\kern0pt}cost\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}s{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}{\isacharbraceleft}{\kern0pt}mod{\isacharunderscore}{\kern0pt}type{\isacharcomma}{\kern0pt}\ finite{\isacharbraceright}{\kern0pt}\ {\isasymRightarrow}\ real{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isakeyword{and}\ inv{\isacharunderscore}{\kern0pt}cost\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}s\ {\isasymRightarrow}\ real{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isakeyword{and}\ demand{\isacharunderscore}{\kern0pt}prob\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}s\ pmf{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isakeyword{and}\ revenue\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}s\ {\isasymRightarrow}\ real{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isakeyword{and}\ discount\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}real{\isachardoublequoteclose}\isanewline
-\isakeyword{begin}\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ {\isachardoublequoteopen}order{\isacharunderscore}{\kern0pt}cost\ u\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}if\ u\ {\isacharequal}{\kern0pt}\ {\isadigit{0}}\ then\ {\isadigit{0}}\ else\ fixed{\isacharunderscore}{\kern0pt}cost\ {\isacharplus}{\kern0pt}\ var{\isacharunderscore}{\kern0pt}cost\ u{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isanewline
-\isacommand{definition}\isamarkupfalse%
-\ {\isachardoublequoteopen}prob{\isacharunderscore}{\kern0pt}ge{\isacharunderscore}{\kern0pt}inv\ u\ {\isacharequal}{\kern0pt}\ {\isadigit{1}}\ {\isacharminus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSum}j{\isacharless}{\kern0pt}u{\isachardot}{\kern0pt}\ pmf\ demand{\isacharunderscore}{\kern0pt}prob\ j{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ {\isachardoublequoteopen}exp{\isacharunderscore}{\kern0pt}rev\ u\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSum}j{\isacharless}{\kern0pt}u{\isachardot}{\kern0pt}\ revenue\ j\ {\isacharasterisk}{\kern0pt}\ pmf\ demand{\isacharunderscore}{\kern0pt}prob\ j{\isacharparenright}{\kern0pt}\ {\isacharplus}{\kern0pt}\ revenue\ u\ {\isacharasterisk}{\kern0pt}\ prob{\isacharunderscore}{\kern0pt}ge{\isacharunderscore}{\kern0pt}inv\ u{\isachardoublequoteclose}\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ {\isachardoublequoteopen}reward\ sa\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}case\ sa\ of\ {\isacharparenleft}{\kern0pt}s{\isacharcomma}{\kern0pt}a{\isacharparenright}{\kern0pt}\ {\isasymRightarrow}\ exp{\isacharunderscore}{\kern0pt}rev\ {\isacharparenleft}{\kern0pt}s\ {\isacharplus}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isacharminus}{\kern0pt}\ order{\isacharunderscore}{\kern0pt}cost\ a\ {\isacharminus}{\kern0pt}\ inv{\isacharunderscore}{\kern0pt}cost\ {\isacharparenleft}{\kern0pt}s\ {\isacharplus}{\kern0pt}\ a{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isacommand{lift{\isacharunderscore}{\kern0pt}definition}\isamarkupfalse%
-\ transition\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}s\ {\isasymtimes}\ {\isacharprime}{\kern0pt}s{\isacharparenright}{\kern0pt}\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}s\ pmf{\isachardoublequoteclose}\ \isakeyword{is}\ {\isachardoublequoteopen}{\isasymlambda}{\isacharparenleft}{\kern0pt}s{\isacharcomma}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ s{\isacharprime}{\kern0pt}{\isachardot}{\kern0pt}\ \isanewline
-\ \ {\isacharparenleft}{\kern0pt}if\ CARD{\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}s{\isacharparenright}{\kern0pt}\ {\isasymle}\ Rep\ s\ {\isacharplus}{\kern0pt}\ Rep\ a\ \isanewline
-\ \ then\ {\isacharparenleft}{\kern0pt}if\ s{\isacharprime}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isadigit{0}}\ then\ {\isadigit{1}}\ else\ {\isadigit{0}}{\isacharparenright}{\kern0pt}\isanewline
-\ \ else\ {\isacharparenleft}{\kern0pt}if\ s\ {\isacharplus}{\kern0pt}\ a\ {\isacharless}{\kern0pt}\ s{\isacharprime}{\kern0pt}\ then\ {\isadigit{0}}\ else\isanewline
-\ \ \ if\ s{\isacharprime}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isadigit{0}}\ then\ prob{\isacharunderscore}{\kern0pt}ge{\isacharunderscore}{\kern0pt}inv\ {\isacharparenleft}{\kern0pt}s{\isacharplus}{\kern0pt}a{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ else\ pmf\ demand{\isacharunderscore}{\kern0pt}prob\ {\isacharparenleft}{\kern0pt}s\ {\isacharplus}{\kern0pt}\ a\ {\isacharminus}{\kern0pt}\ s{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ {\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}safe{\isacharcomma}{\kern0pt}\ goal{\isacharunderscore}{\kern0pt}cases{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{case}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}\ a\ b\ x{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{then}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}case\ \isacommand{unfolding}\isamarkupfalse%
-\ prob{\isacharunderscore}{\kern0pt}ge{\isacharunderscore}{\kern0pt}inv{\isacharunderscore}{\kern0pt}def\ \isacommand{using}\isamarkupfalse%
-\ pmf{\isacharunderscore}{\kern0pt}finite{\isacharunderscore}{\kern0pt}le\ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{case}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}{\isadigit{2}}\ a\ b{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{then}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}case\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}cases\ {\isachardoublequoteopen}int\ CARD{\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}s{\isacharparenright}{\kern0pt}\ {\isasymle}\ Rep\ a\ {\isacharplus}{\kern0pt}\ Rep\ b{\isachardoublequoteclose}{\isacharparenright}{\kern0pt}\ \ \isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{case}\isamarkupfalse%
-\ False\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isasymintegral}\isactrlsup {\isacharplus}{\kern0pt}\ x{\isachardot}{\kern0pt}\ ennreal\ {\isacharparenleft}{\kern0pt}if\ int\ CARD{\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}s{\isacharparenright}{\kern0pt}\ {\isasymle}\ Rep\ a\ {\isacharplus}{\kern0pt}\ Rep\ b\ then\ if\ x\ {\isacharequal}{\kern0pt}\ {\isadigit{0}}\ then\ {\isadigit{1}}\ else\ {\isadigit{0}}\ else\ if\ a\ {\isacharplus}{\kern0pt}\ b\ {\isacharless}{\kern0pt}\ x\ then\ {\isadigit{0}}\ else\ if\ x\ {\isacharequal}{\kern0pt}\ {\isadigit{0}}\ then\ prob{\isacharunderscore}{\kern0pt}ge{\isacharunderscore}{\kern0pt}inv\ {\isacharparenleft}{\kern0pt}a\ {\isacharplus}{\kern0pt}\ b{\isacharparenright}{\kern0pt}\ else\ pmf\ demand{\isacharunderscore}{\kern0pt}prob\ {\isacharparenleft}{\kern0pt}a\ {\isacharplus}{\kern0pt}\ b\ {\isacharminus}{\kern0pt}\ x{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasympartial}count{\isacharunderscore}{\kern0pt}space\ UNIV{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymSum}x\ {\isasymin}\ UNIV{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}if\ a\ {\isacharplus}{\kern0pt}\ b\ {\isacharless}{\kern0pt}\ x\ then\ {\isadigit{0}}\ else\ if\ x\ {\isacharequal}{\kern0pt}\ {\isadigit{0}}\ then\ prob{\isacharunderscore}{\kern0pt}ge{\isacharunderscore}{\kern0pt}inv\ {\isacharparenleft}{\kern0pt}a\ {\isacharplus}{\kern0pt}\ b{\isacharparenright}{\kern0pt}\ else\ pmf\ demand{\isacharunderscore}{\kern0pt}prob\ {\isacharparenleft}{\kern0pt}a\ {\isacharplus}{\kern0pt}\ b\ {\isacharminus}{\kern0pt}\ x{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ pmf{\isacharunderscore}{\kern0pt}nonneg\ prob{\isacharunderscore}{\kern0pt}ge{\isacharunderscore}{\kern0pt}inv{\isacharunderscore}{\kern0pt}def\ pmf{\isacharunderscore}{\kern0pt}finite{\isacharunderscore}{\kern0pt}le\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ nn{\isacharunderscore}{\kern0pt}integral{\isacharunderscore}{\kern0pt}count{\isacharunderscore}{\kern0pt}space{\isacharunderscore}{\kern0pt}finite\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ sum{\isacharunderscore}{\kern0pt}ennreal{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isacharequal}{\kern0pt}{\isacharparenleft}{\kern0pt}{\isasymSum}x\ {\isasymin}\ UNIV{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}if\ x\ {\isacharequal}{\kern0pt}\ {\isadigit{0}}\ then\ prob{\isacharunderscore}{\kern0pt}ge{\isacharunderscore}{\kern0pt}inv\ {\isacharparenleft}{\kern0pt}a\ {\isacharplus}{\kern0pt}\ b{\isacharparenright}{\kern0pt}\ else\ if\ a\ {\isacharplus}{\kern0pt}\ b\ {\isacharless}{\kern0pt}\ x\ then\ {\isadigit{0}}\ else\ pmf\ demand{\isacharunderscore}{\kern0pt}prob\ {\isacharparenleft}{\kern0pt}a\ {\isacharplus}{\kern0pt}\ b\ {\isacharminus}{\kern0pt}\ x{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ sum{\isachardot}{\kern0pt}cong\ ennreal{\isacharunderscore}{\kern0pt}cong\ simp\ add{\isacharcolon}{\kern0pt}\ leD\ least{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}type{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isacharequal}{\kern0pt}\ prob{\isacharunderscore}{\kern0pt}ge{\isacharunderscore}{\kern0pt}inv\ {\isacharparenleft}{\kern0pt}a\ {\isacharplus}{\kern0pt}\ b{\isacharparenright}{\kern0pt}\ {\isacharplus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSum}x\ {\isasymin}\ UNIV{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}{\isadigit{0}}{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}if\ a\ {\isacharplus}{\kern0pt}\ b\ {\isacharless}{\kern0pt}\ x\ then\ {\isadigit{0}}\ else\ pmf\ demand{\isacharunderscore}{\kern0pt}prob\ {\isacharparenleft}{\kern0pt}a\ {\isacharplus}{\kern0pt}\ b\ {\isacharminus}{\kern0pt}\ x{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ sum{\isachardot}{\kern0pt}remove{\isacharbrackleft}{\kern0pt}of\ UNIV\ {\isadigit{0}}{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isacharequal}{\kern0pt}\ prob{\isacharunderscore}{\kern0pt}ge{\isacharunderscore}{\kern0pt}inv\ {\isacharparenleft}{\kern0pt}a\ {\isacharplus}{\kern0pt}\ b{\isacharparenright}{\kern0pt}\ {\isacharplus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSum}x{\isasymin}{\isacharbraceleft}{\kern0pt}{\isadigit{0}}{\isacharless}{\kern0pt}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}if\ a\ {\isacharplus}{\kern0pt}\ b\ {\isacharless}{\kern0pt}\ x\ then\ {\isadigit{0}}\ else\ pmf\ demand{\isacharunderscore}{\kern0pt}prob\ {\isacharparenleft}{\kern0pt}a\ {\isacharplus}{\kern0pt}\ b\ {\isacharminus}{\kern0pt}\ x{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp\ add{\isacharcolon}{\kern0pt}\ greaterThan{\isacharunderscore}{\kern0pt}def\ le{\isacharunderscore}{\kern0pt}neq{\isacharunderscore}{\kern0pt}trans\ least{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}type\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ cong{\isacharbrackleft}{\kern0pt}of\ {\isachardoublequoteopen}sum\ {\isacharunderscore}{\kern0pt}{\isachardoublequoteclose}{\isacharbrackright}{\kern0pt}\ ennreal{\isacharunderscore}{\kern0pt}cong{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isacharequal}{\kern0pt}\ prob{\isacharunderscore}{\kern0pt}ge{\isacharunderscore}{\kern0pt}inv\ {\isacharparenleft}{\kern0pt}a\ {\isacharplus}{\kern0pt}\ b{\isacharparenright}{\kern0pt}\ {\isacharplus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSum}x{\isasymin}{\isacharbraceleft}{\kern0pt}{\isadigit{0}}{\isacharless}{\kern0pt}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}a{\isacharplus}{\kern0pt}b{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}pmf\ demand{\isacharunderscore}{\kern0pt}prob\ {\isacharparenleft}{\kern0pt}a\ {\isacharplus}{\kern0pt}\ b\ {\isacharminus}{\kern0pt}\ x{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ atMost{\isacharunderscore}{\kern0pt}def\ greaterThan{\isacharunderscore}{\kern0pt}def\ greaterThanAtMost{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ Collect{\isacharunderscore}{\kern0pt}neg{\isacharunderscore}{\kern0pt}eq{\isacharbrackleft}{\kern0pt}symmetric{\isacharbrackright}{\kern0pt}\ not{\isacharunderscore}{\kern0pt}less\ sum{\isachardot}{\kern0pt}If{\isacharunderscore}{\kern0pt}cases{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isacharequal}{\kern0pt}\ \ {\isadigit{1}}\ {\isacharminus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSum}j{\isacharless}{\kern0pt}{\isacharparenleft}{\kern0pt}a\ {\isacharplus}{\kern0pt}\ b{\isacharparenright}{\kern0pt}{\isachardot}{\kern0pt}\ pmf\ demand{\isacharunderscore}{\kern0pt}prob\ j{\isacharparenright}{\kern0pt}\ {\isacharplus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSum}x{\isasymin}{\isacharbraceleft}{\kern0pt}{\isadigit{0}}{\isacharless}{\kern0pt}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}a{\isacharplus}{\kern0pt}b{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ pmf\ demand{\isacharunderscore}{\kern0pt}prob\ {\isacharparenleft}{\kern0pt}a\ {\isacharplus}{\kern0pt}\ b\ {\isacharminus}{\kern0pt}\ x{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ prob{\isacharunderscore}{\kern0pt}ge{\isacharunderscore}{\kern0pt}inv{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isacharequal}{\kern0pt}\ \ {\isadigit{1}}\ {\isacharminus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSum}j{\isacharless}{\kern0pt}{\isacharparenleft}{\kern0pt}a\ {\isacharplus}{\kern0pt}\ b{\isacharparenright}{\kern0pt}{\isachardot}{\kern0pt}\ pmf\ demand{\isacharunderscore}{\kern0pt}prob\ j{\isacharparenright}{\kern0pt}\ {\isacharplus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSum}x{\isasymin}{\isacharbraceleft}{\kern0pt}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharless}{\kern0pt}a{\isacharplus}{\kern0pt}b{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}pmf\ demand{\isacharunderscore}{\kern0pt}prob\ x{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isasymSum}x{\isasymin}{\isacharbraceleft}{\kern0pt}{\isadigit{0}}{\isacharless}{\kern0pt}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}a{\isacharplus}{\kern0pt}b{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ pmf\ demand{\isacharunderscore}{\kern0pt}prob\ {\isacharparenleft}{\kern0pt}a\ {\isacharplus}{\kern0pt}\ b\ {\isacharminus}{\kern0pt}\ x{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSum}x{\isasymin}{\isacharbraceleft}{\kern0pt}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharless}{\kern0pt}a{\isacharplus}{\kern0pt}b{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}pmf\ demand{\isacharunderscore}{\kern0pt}prob\ x{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}intro\ sum{\isachardot}{\kern0pt}reindex{\isacharunderscore}{\kern0pt}bij{\isacharunderscore}{\kern0pt}betw\ bij{\isacharunderscore}{\kern0pt}betw{\isacharunderscore}{\kern0pt}imageI{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}inj{\isacharunderscore}{\kern0pt}on\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}{\isacharminus}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}a\ {\isacharplus}{\kern0pt}\ b{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isadigit{0}}{\isacharless}{\kern0pt}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}a\ {\isacharplus}{\kern0pt}\ b{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ inj{\isacharunderscore}{\kern0pt}on{\isacharunderscore}{\kern0pt}def\ \isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ add{\isachardot}{\kern0pt}left{\isacharunderscore}{\kern0pt}cancel\ diff{\isacharunderscore}{\kern0pt}add{\isacharunderscore}{\kern0pt}cancel{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}x\ {\isacharplus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}a\ {\isacharplus}{\kern0pt}\ b{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ a\ {\isacharplus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}b\ {\isacharplus}{\kern0pt}\ x{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{for}\ x\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ add{\isachardot}{\kern0pt}assoc\ add{\isachardot}{\kern0pt}commute\ add{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}nat{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}x\ {\isacharless}{\kern0pt}\ a\ {\isacharplus}{\kern0pt}\ b\ {\isasymLongrightarrow}\ {\isadigit{0}}\ {\isacharless}{\kern0pt}\ a\ {\isacharplus}{\kern0pt}\ b\ {\isacharminus}{\kern0pt}\ x{\isachardoublequoteclose}\ \isakeyword{for}\ x\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ add{\isachardot}{\kern0pt}left{\isacharunderscore}{\kern0pt}neutral\ diff{\isacharunderscore}{\kern0pt}add{\isacharunderscore}{\kern0pt}cancel\ least{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}type\ less{\isacharunderscore}{\kern0pt}le{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}x\ {\isacharless}{\kern0pt}\ a\ {\isacharplus}{\kern0pt}\ b\ {\isasymLongrightarrow}\ a\ {\isacharplus}{\kern0pt}\ b\ {\isacharminus}{\kern0pt}\ x\ {\isasymle}\ a\ {\isacharplus}{\kern0pt}\ b{\isachardoublequoteclose}\ \isakeyword{for}\ x\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ diff{\isacharunderscore}{\kern0pt}{\isadigit{0}}{\isacharunderscore}{\kern0pt}right\ least{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}type\ less{\isacharunderscore}{\kern0pt}le\ mod{\isacharunderscore}{\kern0pt}less{\isacharunderscore}{\kern0pt}diff\ not{\isacharunderscore}{\kern0pt}less{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}x\ {\isacharless}{\kern0pt}\ a\ {\isacharplus}{\kern0pt}\ b\ {\isasymLongrightarrow}\ {\isasymexists}xa{\isachardot}{\kern0pt}\ x\ {\isacharequal}{\kern0pt}\ a\ {\isacharplus}{\kern0pt}\ b\ {\isacharminus}{\kern0pt}\ xa\ {\isasymand}\ {\isadigit{0}}\ {\isacharless}{\kern0pt}\ xa\ {\isasymand}\ xa\ {\isasymle}\ a\ {\isacharplus}{\kern0pt}\ b{\isachardoublequoteclose}\ \isakeyword{for}\ x\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ algebra{\isacharunderscore}{\kern0pt}simps\ intro{\isacharcolon}{\kern0pt}\ exI{\isacharbrackleft}{\kern0pt}of\ {\isacharunderscore}{\kern0pt}\ {\isachardoublequoteopen}a\ {\isacharplus}{\kern0pt}\ b\ {\isacharminus}{\kern0pt}\ x{\isachardoublequoteclose}{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isacharminus}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}a\ {\isacharplus}{\kern0pt}\ b{\isacharparenright}{\kern0pt}\ {\isacharbackquote}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isadigit{0}}{\isacharless}{\kern0pt}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}a\ {\isacharplus}{\kern0pt}\ b{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharless}{\kern0pt}a\ {\isacharplus}{\kern0pt}\ b{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\ \isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}fastforce\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ mod{\isacharunderscore}{\kern0pt}less{\isacharunderscore}{\kern0pt}diff{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\ \isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isacharequal}{\kern0pt}\ \ {\isadigit{1}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \isacommand{finally}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isacommand{{\isachardot}{\kern0pt}}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ nn{\isacharunderscore}{\kern0pt}integral{\isacharunderscore}{\kern0pt}count{\isacharunderscore}{\kern0pt}space{\isacharunderscore}{\kern0pt}finite{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ {\isachardoublequoteopen}A{\isacharunderscore}{\kern0pt}inv\ {\isacharparenleft}{\kern0pt}s{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}{\isacharprime}{\kern0pt}s{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}{\isacharprime}{\kern0pt}s{\isachardot}{\kern0pt}\ Rep\ s\ {\isacharplus}{\kern0pt}\ Rep\ a\ {\isacharless}{\kern0pt}\ CARD{\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}s{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{end}\isamarkupfalse%
-\isanewline
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ {\isachardoublequoteopen}var{\isacharunderscore}{\kern0pt}cost{\isacharunderscore}{\kern0pt}lin\ {\isacharparenleft}{\kern0pt}c{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}real{\isacharparenright}{\kern0pt}\ n\ {\isacharequal}{\kern0pt}\ c\ {\isacharasterisk}{\kern0pt}\ Rep\ n{\isachardoublequoteclose}\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ {\isachardoublequoteopen}inv{\isacharunderscore}{\kern0pt}cost{\isacharunderscore}{\kern0pt}lin\ {\isacharparenleft}{\kern0pt}c{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}real{\isacharparenright}{\kern0pt}\ n\ {\isacharequal}{\kern0pt}\ c\ {\isacharasterisk}{\kern0pt}\ Rep\ n{\isachardoublequoteclose}\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ {\isachardoublequoteopen}revenue{\isacharunderscore}{\kern0pt}lin\ {\isacharparenleft}{\kern0pt}c{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}real{\isacharparenright}{\kern0pt}\ n\ {\isacharequal}{\kern0pt}\ c\ {\isacharasterisk}{\kern0pt}\ Rep\ n{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{lift{\isacharunderscore}{\kern0pt}definition}\isamarkupfalse%
-\ demand{\isacharunderscore}{\kern0pt}unif\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}a{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}finite{\isacharparenright}{\kern0pt}\ pmf{\isachardoublequoteclose}\ \isakeyword{is}\ {\isachardoublequoteopen}{\isasymlambda}{\isacharunderscore}{\kern0pt}{\isachardot}{\kern0pt}\ {\isadigit{1}}\ {\isacharslash}{\kern0pt}\ card\ {\isacharparenleft}{\kern0pt}UNIV{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}{\isacharprime}{\kern0pt}a\ set{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ ennreal{\isacharunderscore}{\kern0pt}divide{\isacharunderscore}{\kern0pt}times\ divide{\isacharunderscore}{\kern0pt}ennreal{\isacharbrackleft}{\kern0pt}symmetric{\isacharbrackright}{\kern0pt}\ ennreal{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}nat{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}real{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}nat{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lift{\isacharunderscore}{\kern0pt}definition}\isamarkupfalse%
-\ demand{\isacharunderscore}{\kern0pt}three\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isadigit{3}}\ pmf{\isachardoublequoteclose}\ \isakeyword{is}\ {\isachardoublequoteopen}{\isasymlambda}i{\isachardot}{\kern0pt}\ if\ i\ {\isacharequal}{\kern0pt}\ {\isadigit{1}}\ then\ {\isadigit{1}}{\isacharslash}{\kern0pt}{\isadigit{4}}\ else\ if\ i\ {\isacharequal}{\kern0pt}\ {\isadigit{2}}\ then\ {\isadigit{1}}{\isacharslash}{\kern0pt}{\isadigit{2}}\ else\ {\isadigit{1}}{\isacharslash}{\kern0pt}{\isadigit{4}}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isacharasterisk}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}UNIV\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isadigit{3}}\ set{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isadigit{0}}{\isacharcomma}{\kern0pt}{\isadigit{1}}{\isacharcomma}{\kern0pt}{\isadigit{2}}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ exhaust{\isacharunderscore}{\kern0pt}{\isadigit{3}}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ fastforce\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{apply}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ nn{\isacharunderscore}{\kern0pt}integral{\isacharunderscore}{\kern0pt}count{\isacharunderscore}{\kern0pt}space{\isacharunderscore}{\kern0pt}finite{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ {\isacharasterisk}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{abbreviation}\isamarkupfalse%
-\ {\isachardoublequoteopen}fixed{\isacharunderscore}{\kern0pt}cost\ {\isasymequiv}\ {\isadigit{4}}{\isachardoublequoteclose}\isanewline
-\isacommand{abbreviation}\isamarkupfalse%
-\ {\isachardoublequoteopen}var{\isacharunderscore}{\kern0pt}cost\ {\isasymequiv}\ var{\isacharunderscore}{\kern0pt}cost{\isacharunderscore}{\kern0pt}lin\ {\isadigit{2}}{\isachardoublequoteclose}\isanewline
-\isacommand{abbreviation}\isamarkupfalse%
-\ {\isachardoublequoteopen}inv{\isacharunderscore}{\kern0pt}cost\ {\isasymequiv}\ inv{\isacharunderscore}{\kern0pt}cost{\isacharunderscore}{\kern0pt}lin\ {\isadigit{1}}{\isachardoublequoteclose}\isanewline
-\isacommand{abbreviation}\isamarkupfalse%
-\ {\isachardoublequoteopen}revenue\ {\isasymequiv}\ revenue{\isacharunderscore}{\kern0pt}lin\ {\isadigit{8}}{\isachardoublequoteclose}\isanewline
-\isacommand{abbreviation}\isamarkupfalse%
-\ {\isachardoublequoteopen}discount\ {\isasymequiv}\ {\isadigit{0}}{\isachardot}{\kern0pt}{\isadigit{9}}{\isadigit{9}}{\isachardoublequoteclose}\isanewline
-\isacommand{type{\isacharunderscore}{\kern0pt}synonym}\isamarkupfalse%
-\ capacity\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}{\isadigit{3}}{\isadigit{0}}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ card{\isacharunderscore}{\kern0pt}ge{\isacharunderscore}{\kern0pt}{\isadigit{2}}{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}ne{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}CARD{\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}a{\isacharparenright}{\kern0pt}\ {\isasymge}\ {\isadigit{2}}\ {\isasymLongrightarrow}\ {\isasymexists}{\isacharparenleft}{\kern0pt}x{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}{\isacharprime}{\kern0pt}a{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}finite{\isacharparenright}{\kern0pt}\ y{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}{\isacharprime}{\kern0pt}a{\isachardot}{\kern0pt}\ x\ {\isasymnoteq}\ y{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}meson\ card{\isacharunderscore}{\kern0pt}{\isadigit{2}}{\isacharunderscore}{\kern0pt}iff{\isacharprime}{\kern0pt}\ ex{\isacharunderscore}{\kern0pt}card{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\ \ \isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{global{\isacharunderscore}{\kern0pt}interpretation}\isamarkupfalse%
-\ inventory{\isacharunderscore}{\kern0pt}ex{\isacharcolon}{\kern0pt}\ inventory\ fixed{\isacharunderscore}{\kern0pt}cost\ {\isachardoublequoteopen}var{\isacharunderscore}{\kern0pt}cost{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ capacity\ {\isasymRightarrow}\ real{\isachardoublequoteclose}\ inv{\isacharunderscore}{\kern0pt}cost\ demand{\isacharunderscore}{\kern0pt}unif\ revenue\ discount\isanewline
-\ \ \isakeyword{defines}\ A{\isacharunderscore}{\kern0pt}inv\ {\isacharequal}{\kern0pt}\ inventory{\isacharunderscore}{\kern0pt}ex{\isachardot}{\kern0pt}A{\isacharunderscore}{\kern0pt}inv\isanewline
-\ \ \ \ \isakeyword{and}\ transition\ {\isacharequal}{\kern0pt}\ inventory{\isacharunderscore}{\kern0pt}ex{\isachardot}{\kern0pt}transition\isanewline
-\ \ \ \ \isakeyword{and}\ reward\ {\isacharequal}{\kern0pt}\ inventory{\isacharunderscore}{\kern0pt}ex{\isachardot}{\kern0pt}reward\isanewline
-\ \ \ \ \isakeyword{and}\ prob{\isacharunderscore}{\kern0pt}ge{\isacharunderscore}{\kern0pt}inv\ {\isacharequal}{\kern0pt}\ inventory{\isacharunderscore}{\kern0pt}ex{\isachardot}{\kern0pt}prob{\isacharunderscore}{\kern0pt}ge{\isacharunderscore}{\kern0pt}inv\isanewline
-\ \ \ \ \isakeyword{and}\ order{\isacharunderscore}{\kern0pt}cost\ {\isacharequal}{\kern0pt}\ inventory{\isacharunderscore}{\kern0pt}ex{\isachardot}{\kern0pt}order{\isacharunderscore}{\kern0pt}cost\isanewline
-\ \ \ \ \isakeyword{and}\ exp{\isacharunderscore}{\kern0pt}rev\ {\isacharequal}{\kern0pt}\ inventory{\isacharunderscore}{\kern0pt}ex{\isachardot}{\kern0pt}exp{\isacharunderscore}{\kern0pt}rev%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{{\isachardot}{\kern0pt}}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{abbreviation}\isamarkupfalse%
-\ {\isachardoublequoteopen}K\ {\isasymequiv}\ inventory{\isacharunderscore}{\kern0pt}ex{\isachardot}{\kern0pt}transition{\isachardoublequoteclose}\isanewline
-\isacommand{abbreviation}\isamarkupfalse%
-\ {\isachardoublequoteopen}A\ {\isasymequiv}\ inventory{\isacharunderscore}{\kern0pt}ex{\isachardot}{\kern0pt}A{\isacharunderscore}{\kern0pt}inv{\isachardoublequoteclose}\isanewline
-\isacommand{abbreviation}\isamarkupfalse%
-\ {\isachardoublequoteopen}r\ {\isasymequiv}\ inventory{\isacharunderscore}{\kern0pt}ex{\isachardot}{\kern0pt}reward{\isachardoublequoteclose}\isanewline
-\isacommand{abbreviation}\isamarkupfalse%
-\ {\isachardoublequoteopen}l\ {\isasymequiv}\ {\isadigit{0}}{\isachardot}{\kern0pt}{\isadigit{9}}{\isadigit{5}}{\isachardoublequoteclose}\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ {\isachardoublequoteopen}eps\ {\isacharequal}{\kern0pt}\ {\isadigit{0}}{\isachardot}{\kern0pt}{\isadigit{1}}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ {\isachardoublequoteopen}fun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}list\ f\ {\isacharequal}{\kern0pt}\ map\ f\ {\isacharparenleft}{\kern0pt}sorted{\isacharunderscore}{\kern0pt}list{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}set\ UNIV{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ {\isachardoublequoteopen}benchmark{\isacharunderscore}{\kern0pt}gs\ {\isacharparenleft}{\kern0pt}{\isacharunderscore}{\kern0pt}\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ unit{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ map\ Rep\ {\isacharparenleft}{\kern0pt}fun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}list\ {\isacharparenleft}{\kern0pt}vi{\isacharunderscore}{\kern0pt}policy{\isacharprime}{\kern0pt}\ K\ A\ r\ l\ eps\ {\isadigit{0}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ {\isachardoublequoteopen}benchmark{\isacharunderscore}{\kern0pt}vi\ {\isacharparenleft}{\kern0pt}{\isacharunderscore}{\kern0pt}\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ unit{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ map\ Rep\ {\isacharparenleft}{\kern0pt}fun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}list\ {\isacharparenleft}{\kern0pt}vi{\isacharunderscore}{\kern0pt}gs{\isacharunderscore}{\kern0pt}policy\ K\ A\ r\ l\ eps\ {\isadigit{0}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ {\isachardoublequoteopen}benchmark{\isacharunderscore}{\kern0pt}mpi\ {\isacharparenleft}{\kern0pt}{\isacharunderscore}{\kern0pt}\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ unit\ {\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ map\ Rep\ {\isacharparenleft}{\kern0pt}fun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}list\ {\isacharparenleft}{\kern0pt}fst\ {\isacharparenleft}{\kern0pt}mpi{\isacharunderscore}{\kern0pt}user\ K\ A\ r\ l\ eps\ {\isacharparenleft}{\kern0pt}{\isasymlambda}{\isacharunderscore}{\kern0pt}\ {\isacharunderscore}{\kern0pt}{\isachardot}{\kern0pt}\ {\isadigit{3}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ {\isachardoublequoteopen}benchmark{\isacharunderscore}{\kern0pt}pi\ {\isacharparenleft}{\kern0pt}{\isacharunderscore}{\kern0pt}\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ unit{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ map\ Rep\ {\isacharparenleft}{\kern0pt}fun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}list\ {\isacharparenleft}{\kern0pt}policy{\isacharunderscore}{\kern0pt}iteration\ K\ A\ r\ l\ {\isadigit{0}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{fun}\isamarkupfalse%
-\ vs{\isacharunderscore}{\kern0pt}n\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}vs{\isacharunderscore}{\kern0pt}n\ {\isadigit{0}}\ v\ {\isacharequal}{\kern0pt}\ v{\isachardoublequoteclose}\isanewline
-{\isacharbar}{\kern0pt}\ {\isachardoublequoteopen}vs{\isacharunderscore}{\kern0pt}n\ {\isacharparenleft}{\kern0pt}Suc\ n{\isacharparenright}{\kern0pt}\ v\ {\isacharequal}{\kern0pt}\ vs{\isacharunderscore}{\kern0pt}n\ n\ {\isacharparenleft}{\kern0pt}mod{\isacharunderscore}{\kern0pt}MDP{\isacharunderscore}{\kern0pt}{\isasymL}\isactrlsub b\ K\ A\ r\ l\ v{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ {\isachardoublequoteopen}vs{\isacharunderscore}{\kern0pt}n{\isacharprime}{\kern0pt}\ n\ {\isacharequal}{\kern0pt}\ vs{\isacharunderscore}{\kern0pt}n\ n\ {\isadigit{0}}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ {\isachardoublequoteopen}benchmark{\isacharunderscore}{\kern0pt}vi{\isacharunderscore}{\kern0pt}n\ n\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}fun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}list\ {\isacharparenleft}{\kern0pt}vs{\isacharunderscore}{\kern0pt}n\ n\ {\isadigit{0}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ {\isachardoublequoteopen}benchmark{\isacharunderscore}{\kern0pt}vi{\isacharunderscore}{\kern0pt}nopol\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}fun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}list\ {\isacharparenleft}{\kern0pt}mod{\isacharunderscore}{\kern0pt}MDP{\isacharunderscore}{\kern0pt}value{\isacharunderscore}{\kern0pt}iteration\ K\ A\ r\ l\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharslash}{\kern0pt}{\isadigit{1}}{\isadigit{0}}{\isacharparenright}{\kern0pt}\ {\isadigit{0}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isanewline
-\isanewline
-\isanewline
-\isacommand{export{\isacharunderscore}{\kern0pt}code}\isamarkupfalse%
-\ dist\ vs{\isacharunderscore}{\kern0pt}n{\isacharprime}{\kern0pt}\ benchmark{\isacharunderscore}{\kern0pt}vi{\isacharunderscore}{\kern0pt}nopol\ benchmark{\isacharunderscore}{\kern0pt}vi{\isacharunderscore}{\kern0pt}n\ nat{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}integer\ integer{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}int\ benchmark{\isacharunderscore}{\kern0pt}gs\ benchmark{\isacharunderscore}{\kern0pt}vi\ benchmark{\isacharunderscore}{\kern0pt}mpi\ benchmark{\isacharunderscore}{\kern0pt}pi\isanewline
-\ \ \isakeyword{in}\ Haskell\ \isakeyword{module{\isacharunderscore}{\kern0pt}name}\ DP\isanewline
-\isanewline
-\isacommand{export{\isacharunderscore}{\kern0pt}code}\isamarkupfalse%
-\ integer{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}int\ benchmark{\isacharunderscore}{\kern0pt}gs\ benchmark{\isacharunderscore}{\kern0pt}vi\ benchmark{\isacharunderscore}{\kern0pt}mpi\ benchmark{\isacharunderscore}{\kern0pt}pi\ \isakeyword{in}\ SML\ \isakeyword{module{\isacharunderscore}{\kern0pt}name}\ DP\isanewline
-%
-\isadelimtheory
-\isanewline
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{end}\isamarkupfalse%
-%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\end{isabellebody}%
-\endinput
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diff --git a/thys/MDP-Algorithms/output/document/Code_Mod.tex b/thys/MDP-Algorithms/output/document/Code_Mod.tex
deleted file mode 100644
--- a/thys/MDP-Algorithms/output/document/Code_Mod.tex
+++ /dev/null
@@ -1,303 +0,0 @@
-%
-\begin{isabellebody}%
-\setisabellecontext{Code{\isacharunderscore}{\kern0pt}Mod}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{theory}\isamarkupfalse%
-\ Code{\isacharunderscore}{\kern0pt}Mod\isanewline
-\ \ \isakeyword{imports}\ Code{\isacharunderscore}{\kern0pt}DP\isanewline
-\isakeyword{begin}%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsection{Code Generation for Concrete Finite MDPs%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{locale}\isamarkupfalse%
-\ mod{\isacharunderscore}{\kern0pt}MDP\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \isakeyword{fixes}\ transition\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}s{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}{\isacharbraceleft}{\kern0pt}enum{\isacharcomma}{\kern0pt}\ mod{\isacharunderscore}{\kern0pt}type{\isacharbraceright}{\kern0pt}\ {\isasymtimes}\ {\isacharprime}{\kern0pt}a{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}{\isacharbraceleft}{\kern0pt}enum{\isacharcomma}{\kern0pt}\ mod{\isacharunderscore}{\kern0pt}type{\isacharbraceright}{\kern0pt}\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}s\ pmf{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isakeyword{and}\ A\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}s\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isakeyword{and}\ reward\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}s\ {\isasymtimes}\ {\isacharprime}{\kern0pt}a\ {\isasymRightarrow}\ real{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isakeyword{and}\ discount\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}real{\isachardoublequoteclose}\isanewline
-\isakeyword{begin}\isanewline
-\isanewline
-\isacommand{sublocale}\isamarkupfalse%
-\ mdp{\isacharcolon}{\kern0pt}\ vi{\isacharunderscore}{\kern0pt}code\ \isanewline
-\ \ {\isachardoublequoteopen}{\isasymlambda}s{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}if\ Set{\isachardot}{\kern0pt}is{\isacharunderscore}{\kern0pt}empty\ {\isacharparenleft}{\kern0pt}A\ s{\isacharparenright}{\kern0pt}\ then\ UNIV\ else\ A\ s{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isanewline
-\ \ transition\ \isanewline
-\ \ reward\isanewline
-\ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}if\ {\isadigit{1}}\ {\isasymle}\ discount\ {\isasymor}\ discount\ {\isacharless}{\kern0pt}\ {\isadigit{0}}\ then\ {\isadigit{0}}\ else\ discount{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{defines}\ {\isasymL}\isactrlsub b\ {\isacharequal}{\kern0pt}\ mdp{\isachardot}{\kern0pt}{\isasymL}\isactrlsub b\isanewline
-\ \ \ \ \isakeyword{and}\ L{\isacharunderscore}{\kern0pt}det\ {\isacharequal}{\kern0pt}\ mdp{\isachardot}{\kern0pt}L{\isacharunderscore}{\kern0pt}det\isanewline
-\ \ \ \ \isakeyword{and}\ value{\isacharunderscore}{\kern0pt}iteration\ {\isacharequal}{\kern0pt}\ mdp{\isachardot}{\kern0pt}value{\isacharunderscore}{\kern0pt}iteration\isanewline
-\ \ \ \ \isakeyword{and}\ vi{\isacharunderscore}{\kern0pt}policy{\isacharprime}{\kern0pt}\ {\isacharequal}{\kern0pt}\ mdp{\isachardot}{\kern0pt}vi{\isacharunderscore}{\kern0pt}policy{\isacharprime}{\kern0pt}\isanewline
-\ \ \ \ \isakeyword{and}\ find{\isacharunderscore}{\kern0pt}policy{\isacharprime}{\kern0pt}\ {\isacharequal}{\kern0pt}\ mdp{\isachardot}{\kern0pt}find{\isacharunderscore}{\kern0pt}policy{\isacharprime}{\kern0pt}\isanewline
-\ \ \ \ \isakeyword{and}\ find{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}impl\ {\isacharequal}{\kern0pt}\ mdp{\isachardot}{\kern0pt}find{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}impl\isanewline
-\ \ \ \ \isakeyword{and}\ is{\isacharunderscore}{\kern0pt}opt{\isacharunderscore}{\kern0pt}act\ {\isacharequal}{\kern0pt}\ mdp{\isachardot}{\kern0pt}is{\isacharunderscore}{\kern0pt}opt{\isacharunderscore}{\kern0pt}act\isanewline
-\ \ \ \ \isakeyword{and}\ value{\isacharunderscore}{\kern0pt}iteration{\isacharunderscore}{\kern0pt}partial\ {\isacharequal}{\kern0pt}\ mdp{\isachardot}{\kern0pt}value{\isacharunderscore}{\kern0pt}iteration{\isacharunderscore}{\kern0pt}partial\isanewline
-\ \ \ \ \isakeyword{and}\ policy{\isacharunderscore}{\kern0pt}iteration\ {\isacharequal}{\kern0pt}\ mdp{\isachardot}{\kern0pt}policy{\isacharunderscore}{\kern0pt}iteration\isanewline
-\ \ \ \ \isakeyword{and}\ is{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\ {\isacharequal}{\kern0pt}\ mdp{\isachardot}{\kern0pt}is{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\isanewline
-\ \ \ \ \isakeyword{and}\ policy{\isacharunderscore}{\kern0pt}step\ {\isacharequal}{\kern0pt}\ mdp{\isachardot}{\kern0pt}policy{\isacharunderscore}{\kern0pt}step\isanewline
-\ \ \ \ \isakeyword{and}\ policy{\isacharunderscore}{\kern0pt}improvement\ {\isacharequal}{\kern0pt}\ mdp{\isachardot}{\kern0pt}policy{\isacharunderscore}{\kern0pt}improvement\isanewline
-\ \ \ \ \isakeyword{and}\ policy{\isacharunderscore}{\kern0pt}eval\ {\isacharequal}{\kern0pt}\ mdp{\isachardot}{\kern0pt}policy{\isacharunderscore}{\kern0pt}eval\isanewline
-\ \ \ \ \isakeyword{and}\ mk{\isacharunderscore}{\kern0pt}markovian\ {\isacharequal}{\kern0pt}\ mdp{\isachardot}{\kern0pt}mk{\isacharunderscore}{\kern0pt}markovian\isanewline
-\ \ \ \ \isakeyword{and}\ policy{\isacharunderscore}{\kern0pt}eval{\isacharprime}{\kern0pt}\ {\isacharequal}{\kern0pt}\ mdp{\isachardot}{\kern0pt}policy{\isacharunderscore}{\kern0pt}eval{\isacharprime}{\kern0pt}\isanewline
-\ \ \ \ \isakeyword{and}\ policy{\isacharunderscore}{\kern0pt}iteration{\isacharunderscore}{\kern0pt}partial{\isacharprime}{\kern0pt}\ {\isacharequal}{\kern0pt}\ mdp{\isachardot}{\kern0pt}policy{\isacharunderscore}{\kern0pt}iteration{\isacharunderscore}{\kern0pt}partial{\isacharprime}{\kern0pt}\isanewline
-\ \ \ \ \isakeyword{and}\ policy{\isacharunderscore}{\kern0pt}iteration{\isacharprime}{\kern0pt}\ {\isacharequal}{\kern0pt}\ mdp{\isachardot}{\kern0pt}policy{\isacharunderscore}{\kern0pt}iteration{\isacharprime}{\kern0pt}\isanewline
-\ \ \ \ \isakeyword{and}\ policy{\isacharunderscore}{\kern0pt}iteration{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}step{\isacharprime}{\kern0pt}\ {\isacharequal}{\kern0pt}\ mdp{\isachardot}{\kern0pt}policy{\isacharunderscore}{\kern0pt}step{\isacharprime}{\kern0pt}\isanewline
-\ \ \ \ \isakeyword{and}\ policy{\isacharunderscore}{\kern0pt}iteration{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}eval{\isacharprime}{\kern0pt}\ {\isacharequal}{\kern0pt}\ mdp{\isachardot}{\kern0pt}policy{\isacharunderscore}{\kern0pt}eval{\isacharprime}{\kern0pt}\isanewline
-\ \ \ \ \isakeyword{and}\ policy{\isacharunderscore}{\kern0pt}iteration{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}improvement{\isacharprime}{\kern0pt}\ {\isacharequal}{\kern0pt}\ mdp{\isachardot}{\kern0pt}policy{\isacharunderscore}{\kern0pt}improvement{\isacharprime}{\kern0pt}\isanewline
-\ \ \ \ \isakeyword{and}\ gs{\isacharunderscore}{\kern0pt}iteration\ {\isacharequal}{\kern0pt}\ mdp{\isachardot}{\kern0pt}gs{\isacharunderscore}{\kern0pt}iteration\isanewline
-\ \ \ \ \isakeyword{and}\ gs{\isacharunderscore}{\kern0pt}iteration{\isacharunderscore}{\kern0pt}partial\ {\isacharequal}{\kern0pt}\ mdp{\isachardot}{\kern0pt}gs{\isacharunderscore}{\kern0pt}iteration{\isacharunderscore}{\kern0pt}partial\isanewline
-\ \ \ \ \isakeyword{and}\ vi{\isacharunderscore}{\kern0pt}gs{\isacharunderscore}{\kern0pt}policy\ {\isacharequal}{\kern0pt}\ mdp{\isachardot}{\kern0pt}vi{\isacharunderscore}{\kern0pt}gs{\isacharunderscore}{\kern0pt}policy\isanewline
-\ \ \ \ \isakeyword{and}\ opt{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}gs\ {\isacharequal}{\kern0pt}\ mdp{\isachardot}{\kern0pt}opt{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}gs\isanewline
-\ \ \ \ \isakeyword{and}\ opt{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}gs{\isacharprime}{\kern0pt}{\isacharprime}{\kern0pt}\ {\isacharequal}{\kern0pt}\ mdp{\isachardot}{\kern0pt}opt{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}gs{\isacharprime}{\kern0pt}{\isacharprime}{\kern0pt}\isanewline
-\ \ \ \ \isakeyword{and}\ P{\isacharunderscore}{\kern0pt}mat\ {\isacharequal}{\kern0pt}\ mdp{\isachardot}{\kern0pt}P{\isacharunderscore}{\kern0pt}mat\isanewline
-\ \ \ \ \isakeyword{and}\ r{\isacharunderscore}{\kern0pt}vec{\isacharprime}{\kern0pt}\ {\isacharequal}{\kern0pt}\ mdp{\isachardot}{\kern0pt}r{\isacharunderscore}{\kern0pt}vec{\isacharprime}{\kern0pt}\isanewline
-\ \ \ \ \isakeyword{and}\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}fun\isactrlsub b\ {\isacharequal}{\kern0pt}\ mdp{\isachardot}{\kern0pt}GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}fun\isactrlsub b\isanewline
-\ \ \ \ \isakeyword{and}\ GS{\isacharunderscore}{\kern0pt}iter{\isacharunderscore}{\kern0pt}max\ {\isacharequal}{\kern0pt}\ mdp{\isachardot}{\kern0pt}GS{\isacharunderscore}{\kern0pt}iter{\isacharunderscore}{\kern0pt}max\isanewline
-\ \ \ \ \isakeyword{and}\ GS{\isacharunderscore}{\kern0pt}iter\ {\isacharequal}{\kern0pt}\ mdp{\isachardot}{\kern0pt}GS{\isacharunderscore}{\kern0pt}iter\isanewline
-\ \ \ \ \isakeyword{and}\ mpi{\isacharunderscore}{\kern0pt}user\ {\isacharequal}{\kern0pt}\ mdp{\isachardot}{\kern0pt}mpi{\isacharunderscore}{\kern0pt}user\isanewline
-\ \ \ \ \isakeyword{and}\ v{\isadigit{0}}{\isacharunderscore}{\kern0pt}mpi\isactrlsub b\ {\isacharequal}{\kern0pt}\ mdp{\isachardot}{\kern0pt}v{\isadigit{0}}{\isacharunderscore}{\kern0pt}mpi\isactrlsub b\isanewline
-\ \ \ \ \isakeyword{and}\ mpi{\isacharunderscore}{\kern0pt}partial{\isacharprime}{\kern0pt}\ {\isacharequal}{\kern0pt}\ mdp{\isachardot}{\kern0pt}mpi{\isacharunderscore}{\kern0pt}partial{\isacharprime}{\kern0pt}\isanewline
-\ \ \ \ \isakeyword{and}\ L{\isacharunderscore}{\kern0pt}pow\ {\isacharequal}{\kern0pt}\ mdp{\isachardot}{\kern0pt}L{\isacharunderscore}{\kern0pt}pow\isanewline
-\ \ \ \ \isakeyword{and}\ v{\isadigit{0}}{\isacharunderscore}{\kern0pt}mpi\ {\isacharequal}{\kern0pt}\ mdp{\isachardot}{\kern0pt}v{\isadigit{0}}{\isacharunderscore}{\kern0pt}mpi\isanewline
-\ \ \ \ \isakeyword{and}\ r{\isacharunderscore}{\kern0pt}min\ {\isacharequal}{\kern0pt}\ mdp{\isachardot}{\kern0pt}r{\isacharunderscore}{\kern0pt}min\isanewline
-\ \ \ \ \isakeyword{and}\ d{\isadigit{0}}\ {\isacharequal}{\kern0pt}\ mdp{\isachardot}{\kern0pt}d{\isadigit{0}}\isanewline
-\ \ \ \ \isakeyword{and}\ d{\isadigit{0}}{\isacharprime}{\kern0pt}\ {\isacharequal}{\kern0pt}\ mdp{\isachardot}{\kern0pt}d{\isadigit{0}}{\isacharprime}{\kern0pt}\isanewline
-\ \ \ \ \isakeyword{and}\ {\isasymnu}\isactrlsub b\ {\isacharequal}{\kern0pt}\ mdp{\isachardot}{\kern0pt}{\isasymnu}\isactrlsub b\isanewline
-\ \ \ \ \isakeyword{and}\ vi{\isacharunderscore}{\kern0pt}test\ {\isacharequal}{\kern0pt}\ mdp{\isachardot}{\kern0pt}vi{\isacharunderscore}{\kern0pt}test\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ unfold{\isacharunderscore}{\kern0pt}locales\ {\isacharparenleft}{\kern0pt}auto\ simp\ add{\isacharcolon}{\kern0pt}\ Set{\isachardot}{\kern0pt}is{\isacharunderscore}{\kern0pt}empty{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isacommand{end}\isamarkupfalse%
-\isanewline
-\isanewline
-\isacommand{global{\isacharunderscore}{\kern0pt}interpretation}\isamarkupfalse%
-\ mod{\isacharunderscore}{\kern0pt}MDP\ transition\ A\ reward\ discount\ \isanewline
-\ \ \isakeyword{for}\ transition\ A\ reward\ discount\ \isanewline
-\ \ \isakeyword{defines}\ mod{\isacharunderscore}{\kern0pt}MDP{\isacharunderscore}{\kern0pt}{\isasymL}\isactrlsub b\ {\isacharequal}{\kern0pt}\ mdp{\isachardot}{\kern0pt}{\isasymL}\isactrlsub b\isanewline
-\ \ \ \ \isakeyword{and}\ mod{\isacharunderscore}{\kern0pt}MDP{\isacharunderscore}{\kern0pt}{\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}L{\isacharunderscore}{\kern0pt}det\ {\isacharequal}{\kern0pt}\ mdp{\isachardot}{\kern0pt}L{\isacharunderscore}{\kern0pt}det\isanewline
-\ \ \ \ \isakeyword{and}\ mod{\isacharunderscore}{\kern0pt}MDP{\isacharunderscore}{\kern0pt}value{\isacharunderscore}{\kern0pt}iteration\ {\isacharequal}{\kern0pt}\ mdp{\isachardot}{\kern0pt}value{\isacharunderscore}{\kern0pt}iteration\isanewline
-\ \ \ \ \isakeyword{and}\ mod{\isacharunderscore}{\kern0pt}MDP{\isacharunderscore}{\kern0pt}vi{\isacharunderscore}{\kern0pt}policy{\isacharprime}{\kern0pt}\ {\isacharequal}{\kern0pt}\ mdp{\isachardot}{\kern0pt}vi{\isacharunderscore}{\kern0pt}policy{\isacharprime}{\kern0pt}\isanewline
-\ \ \ \ \isakeyword{and}\ mod{\isacharunderscore}{\kern0pt}MDP{\isacharunderscore}{\kern0pt}find{\isacharunderscore}{\kern0pt}policy{\isacharprime}{\kern0pt}\ {\isacharequal}{\kern0pt}\ mdp{\isachardot}{\kern0pt}find{\isacharunderscore}{\kern0pt}policy{\isacharprime}{\kern0pt}\isanewline
-\ \ \ \ \isakeyword{and}\ mod{\isacharunderscore}{\kern0pt}MDP{\isacharunderscore}{\kern0pt}find{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}impl\ {\isacharequal}{\kern0pt}\ mdp{\isachardot}{\kern0pt}find{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}impl\isanewline
-\ \ \ \ \isakeyword{and}\ mod{\isacharunderscore}{\kern0pt}MDP{\isacharunderscore}{\kern0pt}is{\isacharunderscore}{\kern0pt}opt{\isacharunderscore}{\kern0pt}act\ {\isacharequal}{\kern0pt}\ mdp{\isachardot}{\kern0pt}is{\isacharunderscore}{\kern0pt}opt{\isacharunderscore}{\kern0pt}act\isanewline
-\ \ \ \ \isakeyword{and}\ mod{\isacharunderscore}{\kern0pt}MDP{\isacharunderscore}{\kern0pt}value{\isacharunderscore}{\kern0pt}iteration{\isacharunderscore}{\kern0pt}partial\ {\isacharequal}{\kern0pt}\ mdp{\isachardot}{\kern0pt}value{\isacharunderscore}{\kern0pt}iteration{\isacharunderscore}{\kern0pt}partial\isanewline
-\ \ \ \ \isakeyword{and}\ mod{\isacharunderscore}{\kern0pt}MDP{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}iteration\ {\isacharequal}{\kern0pt}\ mdp{\isachardot}{\kern0pt}policy{\isacharunderscore}{\kern0pt}iteration\isanewline
-\ \ \ \ \isakeyword{and}\ mod{\isacharunderscore}{\kern0pt}MDP{\isacharunderscore}{\kern0pt}is{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\ {\isacharequal}{\kern0pt}\ mdp{\isachardot}{\kern0pt}is{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\isanewline
-\ \ \ \ \isakeyword{and}\ mod{\isacharunderscore}{\kern0pt}MDP{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}step\ {\isacharequal}{\kern0pt}\ mdp{\isachardot}{\kern0pt}policy{\isacharunderscore}{\kern0pt}step\isanewline
-\ \ \ \ \isakeyword{and}\ mod{\isacharunderscore}{\kern0pt}MDP{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}improvement\ {\isacharequal}{\kern0pt}\ mdp{\isachardot}{\kern0pt}policy{\isacharunderscore}{\kern0pt}improvement\isanewline
-\ \ \ \ \isakeyword{and}\ mod{\isacharunderscore}{\kern0pt}MDP{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}eval\ {\isacharequal}{\kern0pt}\ mdp{\isachardot}{\kern0pt}policy{\isacharunderscore}{\kern0pt}eval\isanewline
-\ \ \ \ \isakeyword{and}\ mod{\isacharunderscore}{\kern0pt}MDP{\isacharunderscore}{\kern0pt}mk{\isacharunderscore}{\kern0pt}markovian\ {\isacharequal}{\kern0pt}\ mdp{\isachardot}{\kern0pt}mk{\isacharunderscore}{\kern0pt}markovian\isanewline
-\ \ \ \ \isakeyword{and}\ mod{\isacharunderscore}{\kern0pt}MDP{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}eval{\isacharprime}{\kern0pt}\ {\isacharequal}{\kern0pt}\ mdp{\isachardot}{\kern0pt}policy{\isacharunderscore}{\kern0pt}eval{\isacharprime}{\kern0pt}\isanewline
-\ \ \ \ \isakeyword{and}\ mod{\isacharunderscore}{\kern0pt}MDP{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}iteration{\isacharunderscore}{\kern0pt}partial{\isacharprime}{\kern0pt}\ {\isacharequal}{\kern0pt}\ mdp{\isachardot}{\kern0pt}policy{\isacharunderscore}{\kern0pt}iteration{\isacharunderscore}{\kern0pt}partial{\isacharprime}{\kern0pt}\isanewline
-\ \ \ \ \isakeyword{and}\ mod{\isacharunderscore}{\kern0pt}MDP{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}iteration{\isacharprime}{\kern0pt}\ {\isacharequal}{\kern0pt}\ mdp{\isachardot}{\kern0pt}policy{\isacharunderscore}{\kern0pt}iteration{\isacharprime}{\kern0pt}\isanewline
-\ \ \ \ \isakeyword{and}\ mod{\isacharunderscore}{\kern0pt}MDP{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}iteration{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}step{\isacharprime}{\kern0pt}\ {\isacharequal}{\kern0pt}\ mdp{\isachardot}{\kern0pt}policy{\isacharunderscore}{\kern0pt}step{\isacharprime}{\kern0pt}\isanewline
-\ \ \ \ \isakeyword{and}\ mod{\isacharunderscore}{\kern0pt}MDP{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}iteration{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}eval{\isacharprime}{\kern0pt}\ {\isacharequal}{\kern0pt}\ mdp{\isachardot}{\kern0pt}policy{\isacharunderscore}{\kern0pt}eval{\isacharprime}{\kern0pt}\isanewline
-\ \ \ \ \isakeyword{and}\ mod{\isacharunderscore}{\kern0pt}MDP{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}iteration{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}improvement{\isacharprime}{\kern0pt}\ {\isacharequal}{\kern0pt}\ mdp{\isachardot}{\kern0pt}policy{\isacharunderscore}{\kern0pt}improvement{\isacharprime}{\kern0pt}\isanewline
-\ \ \ \ \isakeyword{and}\ mod{\isacharunderscore}{\kern0pt}MDP{\isacharunderscore}{\kern0pt}gs{\isacharunderscore}{\kern0pt}iteration\ {\isacharequal}{\kern0pt}\ mdp{\isachardot}{\kern0pt}gs{\isacharunderscore}{\kern0pt}iteration\isanewline
-\ \ \ \ \isakeyword{and}\ mod{\isacharunderscore}{\kern0pt}MDP{\isacharunderscore}{\kern0pt}gs{\isacharunderscore}{\kern0pt}iteration{\isacharunderscore}{\kern0pt}partial\ {\isacharequal}{\kern0pt}\ mdp{\isachardot}{\kern0pt}gs{\isacharunderscore}{\kern0pt}iteration{\isacharunderscore}{\kern0pt}partial\isanewline
-\ \ \ \ \isakeyword{and}\ mod{\isacharunderscore}{\kern0pt}MDP{\isacharunderscore}{\kern0pt}vi{\isacharunderscore}{\kern0pt}gs{\isacharunderscore}{\kern0pt}policy\ {\isacharequal}{\kern0pt}\ mdp{\isachardot}{\kern0pt}vi{\isacharunderscore}{\kern0pt}gs{\isacharunderscore}{\kern0pt}policy\isanewline
-\ \ \ \ \isakeyword{and}\ mod{\isacharunderscore}{\kern0pt}MDP{\isacharunderscore}{\kern0pt}opt{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}gs\ {\isacharequal}{\kern0pt}\ mdp{\isachardot}{\kern0pt}opt{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}gs\isanewline
-\ \ \ \ \isakeyword{and}\ mod{\isacharunderscore}{\kern0pt}MDP{\isacharunderscore}{\kern0pt}opt{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}gs{\isacharprime}{\kern0pt}{\isacharprime}{\kern0pt}\ {\isacharequal}{\kern0pt}\ mdp{\isachardot}{\kern0pt}opt{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}gs{\isacharprime}{\kern0pt}{\isacharprime}{\kern0pt}\isanewline
-\ \ \ \ \isakeyword{and}\ mod{\isacharunderscore}{\kern0pt}MDP{\isacharunderscore}{\kern0pt}P{\isacharunderscore}{\kern0pt}mat\ {\isacharequal}{\kern0pt}\ mdp{\isachardot}{\kern0pt}P{\isacharunderscore}{\kern0pt}mat\isanewline
-\ \ \ \ \isakeyword{and}\ mod{\isacharunderscore}{\kern0pt}MDP{\isacharunderscore}{\kern0pt}r{\isacharunderscore}{\kern0pt}vec{\isacharprime}{\kern0pt}\ {\isacharequal}{\kern0pt}\ mdp{\isachardot}{\kern0pt}r{\isacharunderscore}{\kern0pt}vec{\isacharprime}{\kern0pt}\isanewline
-\ \ \ \ \isakeyword{and}\ mod{\isacharunderscore}{\kern0pt}MDP{\isacharunderscore}{\kern0pt}GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}fun\isactrlsub b\ {\isacharequal}{\kern0pt}\ mdp{\isachardot}{\kern0pt}GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}fun\isactrlsub b\isanewline
-\ \ \ \ \isakeyword{and}\ mod{\isacharunderscore}{\kern0pt}MDP{\isacharunderscore}{\kern0pt}GS{\isacharunderscore}{\kern0pt}iter{\isacharunderscore}{\kern0pt}max\ {\isacharequal}{\kern0pt}\ mdp{\isachardot}{\kern0pt}GS{\isacharunderscore}{\kern0pt}iter{\isacharunderscore}{\kern0pt}max\isanewline
-\ \ \ \ \isakeyword{and}\ mod{\isacharunderscore}{\kern0pt}MDP{\isacharunderscore}{\kern0pt}GS{\isacharunderscore}{\kern0pt}iter\ {\isacharequal}{\kern0pt}\ mdp{\isachardot}{\kern0pt}GS{\isacharunderscore}{\kern0pt}iter\isanewline
-\ \ \ \ \isakeyword{and}\ mod{\isacharunderscore}{\kern0pt}MDP{\isacharunderscore}{\kern0pt}mpi{\isacharunderscore}{\kern0pt}user\ {\isacharequal}{\kern0pt}\ mdp{\isachardot}{\kern0pt}mpi{\isacharunderscore}{\kern0pt}user\isanewline
-\ \ \ \ \isakeyword{and}\ mod{\isacharunderscore}{\kern0pt}MDP{\isacharunderscore}{\kern0pt}v{\isadigit{0}}{\isacharunderscore}{\kern0pt}mpi\isactrlsub b\ {\isacharequal}{\kern0pt}\ mdp{\isachardot}{\kern0pt}v{\isadigit{0}}{\isacharunderscore}{\kern0pt}mpi\isactrlsub b\isanewline
-\ \ \ \ \isakeyword{and}\ mod{\isacharunderscore}{\kern0pt}MDP{\isacharunderscore}{\kern0pt}mpi{\isacharunderscore}{\kern0pt}partial{\isacharprime}{\kern0pt}\ {\isacharequal}{\kern0pt}\ mdp{\isachardot}{\kern0pt}mpi{\isacharunderscore}{\kern0pt}partial{\isacharprime}{\kern0pt}\isanewline
-\ \ \ \ \isakeyword{and}\ mod{\isacharunderscore}{\kern0pt}MDP{\isacharunderscore}{\kern0pt}L{\isacharunderscore}{\kern0pt}pow\ {\isacharequal}{\kern0pt}\ mdp{\isachardot}{\kern0pt}L{\isacharunderscore}{\kern0pt}pow\isanewline
-\ \ \ \ \isakeyword{and}\ mod{\isacharunderscore}{\kern0pt}MDP{\isacharunderscore}{\kern0pt}v{\isadigit{0}}{\isacharunderscore}{\kern0pt}mpi\ {\isacharequal}{\kern0pt}\ mdp{\isachardot}{\kern0pt}v{\isadigit{0}}{\isacharunderscore}{\kern0pt}mpi\isanewline
-\ \ \ \ \isakeyword{and}\ mod{\isacharunderscore}{\kern0pt}MDP{\isacharunderscore}{\kern0pt}r{\isacharunderscore}{\kern0pt}min\ {\isacharequal}{\kern0pt}\ mdp{\isachardot}{\kern0pt}r{\isacharunderscore}{\kern0pt}min\isanewline
-\ \ \ \ \isakeyword{and}\ mod{\isacharunderscore}{\kern0pt}MDP{\isacharunderscore}{\kern0pt}d{\isadigit{0}}\ {\isacharequal}{\kern0pt}\ mdp{\isachardot}{\kern0pt}d{\isadigit{0}}\isanewline
-\ \ \ \ \isakeyword{and}\ mod{\isacharunderscore}{\kern0pt}MDP{\isacharunderscore}{\kern0pt}d{\isadigit{0}}{\isacharprime}{\kern0pt}\ {\isacharequal}{\kern0pt}\ mdp{\isachardot}{\kern0pt}d{\isadigit{0}}{\isacharprime}{\kern0pt}\isanewline
-\ \ \ \ \isakeyword{and}\ mod{\isacharunderscore}{\kern0pt}MDP{\isacharunderscore}{\kern0pt}{\isasymnu}\isactrlsub b\ {\isacharequal}{\kern0pt}\ mdp{\isachardot}{\kern0pt}{\isasymnu}\isactrlsub b\isanewline
-\ \ \ \ \isakeyword{and}\ mod{\isacharunderscore}{\kern0pt}MDP{\isacharunderscore}{\kern0pt}vi{\isacharunderscore}{\kern0pt}test\ {\isacharequal}{\kern0pt}\ mdp{\isachardot}{\kern0pt}vi{\isacharunderscore}{\kern0pt}test\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{{\isachardot}{\kern0pt}}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-%
-\isadelimtheory
-\isanewline
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{end}\isamarkupfalse%
-%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\end{isabellebody}%
-\endinput
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diff --git a/thys/MDP-Algorithms/output/document/Code_Real_Approx_By_Float_Fix.tex b/thys/MDP-Algorithms/output/document/Code_Real_Approx_By_Float_Fix.tex
deleted file mode 100644
--- a/thys/MDP-Algorithms/output/document/Code_Real_Approx_By_Float_Fix.tex
+++ /dev/null
@@ -1,59 +0,0 @@
-%
-\begin{isabellebody}%
-\setisabellecontext{Code{\isacharunderscore}{\kern0pt}Real{\isacharunderscore}{\kern0pt}Approx{\isacharunderscore}{\kern0pt}By{\isacharunderscore}{\kern0pt}Float{\isacharunderscore}{\kern0pt}Fix}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{theory}\isamarkupfalse%
-\ Code{\isacharunderscore}{\kern0pt}Real{\isacharunderscore}{\kern0pt}Approx{\isacharunderscore}{\kern0pt}By{\isacharunderscore}{\kern0pt}Float{\isacharunderscore}{\kern0pt}Fix\isanewline
-\ \ \isakeyword{imports}\isanewline
-\ \ {\isachardoublequoteopen}HOL{\isacharminus}{\kern0pt}Library{\isachardot}{\kern0pt}Code{\isacharunderscore}{\kern0pt}Real{\isacharunderscore}{\kern0pt}Approx{\isacharunderscore}{\kern0pt}By{\isacharunderscore}{\kern0pt}Float{\isachardoublequoteclose}\isanewline
-\ \ {\isachardoublequoteopen}Gauss{\isacharunderscore}{\kern0pt}Jordan{\isachardot}{\kern0pt}Code{\isacharunderscore}{\kern0pt}Real{\isacharunderscore}{\kern0pt}Approx{\isacharunderscore}{\kern0pt}By{\isacharunderscore}{\kern0pt}Float{\isacharunderscore}{\kern0pt}Haskell{\isachardoublequoteclose}\isanewline
-\isakeyword{begin}%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{end}\isamarkupfalse%
-%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\end{isabellebody}%
-\endinput
-%:%file=Code_Real_Approx_By_Float_Fix.tex%:%
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diff --git a/thys/MDP-Algorithms/output/document/Examples.tex b/thys/MDP-Algorithms/output/document/Examples.tex
deleted file mode 100644
--- a/thys/MDP-Algorithms/output/document/Examples.tex
+++ /dev/null
@@ -1,38 +0,0 @@
-%
-\begin{isabellebody}%
-\setisabellecontext{Examples}%
-%
-\isadelimtheory
-\isanewline
-\isanewline
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{theory}\isamarkupfalse%
-\ Examples\isanewline
-\ \ \isakeyword{imports}\isanewline
-\ \ \ \ Code{\isacharunderscore}{\kern0pt}Inventory\isanewline
-\ \ \ \ Code{\isacharunderscore}{\kern0pt}Gridworld\isanewline
-\isakeyword{begin}\isanewline
-\isacommand{end}\isamarkupfalse%
-%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\end{isabellebody}%
-\endinput
-%:%file=Examples.tex%:%
-%:%6=1%:%
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diff --git a/thys/MDP-Algorithms/output/document/Matrix_Util.tex b/thys/MDP-Algorithms/output/document/Matrix_Util.tex
deleted file mode 100644
--- a/thys/MDP-Algorithms/output/document/Matrix_Util.tex
+++ /dev/null
@@ -1,1103 +0,0 @@
-%
-\begin{isabellebody}%
-\setisabellecontext{Matrix{\isacharunderscore}{\kern0pt}Util}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{theory}\isamarkupfalse%
-\ Matrix{\isacharunderscore}{\kern0pt}Util\isanewline
-\ \ \isakeyword{imports}\ {\isachardoublequoteopen}HOL{\isacharminus}{\kern0pt}Analysis{\isachardot}{\kern0pt}Analysis{\isachardoublequoteclose}\isanewline
-\isakeyword{begin}%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsection{Matrices%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{proposition}\isamarkupfalse%
-\ scalar{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}assoc{\isacharprime}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{fixes}\ C\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}b{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}real{\isacharunderscore}{\kern0pt}algebra{\isacharunderscore}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}{\isacharcircum}{\kern0pt}{\isacharprime}{\kern0pt}m{\isacharcircum}{\kern0pt}{\isacharprime}{\kern0pt}n{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}k\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isacharparenleft}{\kern0pt}C\ {\isacharasterisk}{\kern0pt}{\isacharasterisk}{\kern0pt}\ D{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ C\ {\isacharasterisk}{\kern0pt}{\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}k\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ D{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ matrix{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}mult{\isacharunderscore}{\kern0pt}def\ sum{\isacharunderscore}{\kern0pt}distrib{\isacharunderscore}{\kern0pt}left\ mult{\isacharunderscore}{\kern0pt}ac\ vec{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}iff\ scaleR{\isacharunderscore}{\kern0pt}sum{\isacharunderscore}{\kern0pt}right{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Nonnegative Matrices%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{lemma}\isamarkupfalse%
-\ nonneg{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}nonneg\ {\isacharbrackleft}{\kern0pt}dest{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isadigit{0}}\ {\isasymle}\ m\ {\isasymLongrightarrow}\ {\isadigit{0}}\ {\isasymle}\ m\ {\isachardollar}{\kern0pt}\ i\ {\isachardollar}{\kern0pt}\ j{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ Finite{\isacharunderscore}{\kern0pt}Cartesian{\isacharunderscore}{\kern0pt}Product{\isachardot}{\kern0pt}less{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}vec{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ matrix{\isacharunderscore}{\kern0pt}mult{\isacharunderscore}{\kern0pt}mono{\isacharcolon}{\kern0pt}\ \isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}{\isadigit{0}}\ {\isasymle}\ E{\isachardoublequoteclose}\ {\isachardoublequoteopen}{\isadigit{0}}\ {\isasymle}\ C{\isachardoublequoteclose}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}E\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ real{\isacharcircum}{\kern0pt}{\isacharprime}{\kern0pt}c{\isacharcircum}{\kern0pt}{\isacharprime}{\kern0pt}c{\isacharparenright}{\kern0pt}\ {\isasymle}\ B{\isachardoublequoteclose}\ {\isachardoublequoteopen}C\ {\isasymle}\ D{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}E\ {\isacharasterisk}{\kern0pt}{\isacharasterisk}{\kern0pt}\ C\ {\isasymle}\ B\ {\isacharasterisk}{\kern0pt}{\isacharasterisk}{\kern0pt}\ D{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ order{\isachardot}{\kern0pt}trans{\isacharbrackleft}{\kern0pt}OF\ assms{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}\ assms{\isacharparenleft}{\kern0pt}{\isadigit{3}}{\isacharparenright}{\kern0pt}{\isacharbrackright}{\kern0pt}\ assms\isanewline
-\ \ \isacommand{unfolding}\isamarkupfalse%
-\ Finite{\isacharunderscore}{\kern0pt}Cartesian{\isacharunderscore}{\kern0pt}Product{\isachardot}{\kern0pt}less{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}vec{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ sum{\isacharunderscore}{\kern0pt}mono\ mult{\isacharunderscore}{\kern0pt}mono\ simp{\isacharcolon}{\kern0pt}\ matrix{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}mult{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ nonneg{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}mult{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isadigit{0}}\ {\isasymle}\ {\isacharparenleft}{\kern0pt}C\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}b{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}{\isacharbraceleft}{\kern0pt}field{\isacharcomma}{\kern0pt}\ ordered{\isacharunderscore}{\kern0pt}ring{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharcircum}{\kern0pt}{\isacharunderscore}{\kern0pt}{\isacharcircum}{\kern0pt}{\isacharunderscore}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymLongrightarrow}\ {\isadigit{0}}\ {\isasymle}\ D\ {\isasymLongrightarrow}\ {\isadigit{0}}\ {\isasymle}\ C\ {\isacharasterisk}{\kern0pt}{\isacharasterisk}{\kern0pt}\ D{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ Finite{\isacharunderscore}{\kern0pt}Cartesian{\isacharunderscore}{\kern0pt}Product{\isachardot}{\kern0pt}less{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}vec{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ matrix{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}mult{\isacharunderscore}{\kern0pt}def\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ sum{\isacharunderscore}{\kern0pt}nonneg{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ zero{\isacharunderscore}{\kern0pt}le{\isacharunderscore}{\kern0pt}mat{\isacharunderscore}{\kern0pt}iff\ {\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isadigit{0}}\ {\isasymle}\ mat\ {\isacharparenleft}{\kern0pt}x\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isacharprime}{\kern0pt}c\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}zero{\isacharcomma}{\kern0pt}\ order{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymlongleftrightarrow}\ {\isadigit{0}}\ {\isasymle}\ x{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ Finite{\isacharunderscore}{\kern0pt}Cartesian{\isacharunderscore}{\kern0pt}Product{\isachardot}{\kern0pt}less{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}vec{\isacharunderscore}{\kern0pt}def\ mat{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ nonneg{\isacharunderscore}{\kern0pt}mat{\isacharunderscore}{\kern0pt}ge{\isacharunderscore}{\kern0pt}zero{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isadigit{0}}\ {\isasymle}\ Q\ {\isasymLongrightarrow}\ {\isadigit{0}}\ {\isasymle}\ v\ {\isasymLongrightarrow}\ {\isadigit{0}}\ {\isasymle}\ Q\ {\isacharasterisk}{\kern0pt}v\ {\isacharparenleft}{\kern0pt}v\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ real{\isacharcircum}{\kern0pt}{\isacharprime}{\kern0pt}c{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ Finite{\isacharunderscore}{\kern0pt}Cartesian{\isacharunderscore}{\kern0pt}Product{\isachardot}{\kern0pt}less{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}vec{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ sum{\isacharunderscore}{\kern0pt}nonneg\ simp{\isacharcolon}{\kern0pt}\ matrix{\isacharunderscore}{\kern0pt}vector{\isacharunderscore}{\kern0pt}mult{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ nonneg{\isacharunderscore}{\kern0pt}mat{\isacharunderscore}{\kern0pt}mono{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isadigit{0}}\ {\isasymle}\ Q\ {\isasymLongrightarrow}\ u\ {\isasymle}\ v\ {\isasymLongrightarrow}\ Q\ {\isacharasterisk}{\kern0pt}v\ u\ {\isasymle}\ Q\ {\isacharasterisk}{\kern0pt}v\ {\isacharparenleft}{\kern0pt}v\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ real{\isacharcircum}{\kern0pt}{\isacharprime}{\kern0pt}c{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ nonneg{\isacharunderscore}{\kern0pt}mat{\isacharunderscore}{\kern0pt}ge{\isacharunderscore}{\kern0pt}zero{\isacharbrackleft}{\kern0pt}of\ Q\ {\isachardoublequoteopen}v\ {\isacharminus}{\kern0pt}\ u{\isachardoublequoteclose}{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ vec{\isachardot}{\kern0pt}diff{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ nonneg{\isacharunderscore}{\kern0pt}mult{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}nonneg{\isacharunderscore}{\kern0pt}mat{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}{\isasymAnd}v{\isachardot}{\kern0pt}\ v\ {\isasymge}\ {\isadigit{0}}\ {\isasymLongrightarrow}\ X\ {\isacharasterisk}{\kern0pt}v\ v\ {\isasymge}\ {\isadigit{0}}{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}X\ {\isasymge}\ {\isacharparenleft}{\kern0pt}{\isadigit{0}}\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ real\ {\isacharcircum}{\kern0pt}\ {\isacharunderscore}{\kern0pt}\ {\isacharcircum}{\kern0pt}{\isacharunderscore}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{{\isacharbraceleft}{\kern0pt}}\isamarkupfalse%
-\ \isacommand{assume}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymnot}\ {\isacharparenleft}{\kern0pt}{\isadigit{0}}\ {\isasymle}\ X{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{then}\isamarkupfalse%
-\ \isacommand{obtain}\isamarkupfalse%
-\ i\ j\ \isakeyword{where}\ neg{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}X\ {\isachardollar}{\kern0pt}\ i\ {\isachardollar}{\kern0pt}\ j\ {\isacharless}{\kern0pt}\ {\isadigit{0}}{\isachardoublequoteclose}\ \isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ less{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}vec{\isacharunderscore}{\kern0pt}def\ not{\isacharunderscore}{\kern0pt}le\ zero{\isacharunderscore}{\kern0pt}index{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}v\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}{\isasymchi}\ k{\isachardot}{\kern0pt}\ if\ j\ {\isacharequal}{\kern0pt}\ k\ then\ {\isadigit{1}}{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}real\ else\ {\isadigit{0}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}X\ {\isacharasterisk}{\kern0pt}v\ {\isacharquery}{\kern0pt}v{\isacharparenright}{\kern0pt}\ {\isachardollar}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ {\isadigit{0}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ neg\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ matrix{\isacharunderscore}{\kern0pt}vector{\isacharunderscore}{\kern0pt}mult{\isacharunderscore}{\kern0pt}def\ if{\isacharunderscore}{\kern0pt}distrib\ cong{\isacharcolon}{\kern0pt}\ if{\isacharunderscore}{\kern0pt}cong{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharquery}{\kern0pt}v\ {\isasymge}\ {\isadigit{0}}\ {\isasymand}\ {\isasymnot}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}X\ {\isacharasterisk}{\kern0pt}v\ {\isacharquery}{\kern0pt}v{\isacharparenright}{\kern0pt}\ {\isasymge}\ {\isadigit{0}}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ less{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}vec{\isacharunderscore}{\kern0pt}def\ not{\isacharunderscore}{\kern0pt}le{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymexists}v{\isachardot}{\kern0pt}\ v\ {\isasymge}\ {\isadigit{0}}\ {\isasymand}\ {\isasymnot}\ X\ {\isacharasterisk}{\kern0pt}v\ v\ {\isasymge}\ {\isadigit{0}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{{\isacharbraceright}{\kern0pt}}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assms\ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ nonneg{\isacharunderscore}{\kern0pt}mat{\isacharunderscore}{\kern0pt}iff{\isacharcolon}{\kern0pt}\isanewline
-\ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}X\ {\isasymge}\ {\isacharparenleft}{\kern0pt}{\isadigit{0}}\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ real\ {\isacharcircum}{\kern0pt}\ {\isacharunderscore}{\kern0pt}\ {\isacharcircum}{\kern0pt}{\isacharunderscore}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymlongleftrightarrow}\ {\isacharparenleft}{\kern0pt}{\isasymforall}v{\isachardot}{\kern0pt}\ v\ {\isasymge}\ {\isadigit{0}}\ {\isasymlongrightarrow}\ X\ {\isacharasterisk}{\kern0pt}v\ v\ {\isasymge}\ {\isadigit{0}}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ nonneg{\isacharunderscore}{\kern0pt}mat{\isacharunderscore}{\kern0pt}ge{\isacharunderscore}{\kern0pt}zero\ nonneg{\isacharunderscore}{\kern0pt}mult{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}nonneg{\isacharunderscore}{\kern0pt}mat\ \isacommand{by}\isamarkupfalse%
-\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ mat{\isacharunderscore}{\kern0pt}le{\isacharunderscore}{\kern0pt}iff{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}X\ {\isasymle}\ Y{\isacharparenright}{\kern0pt}\ {\isasymlongleftrightarrow}\ {\isacharparenleft}{\kern0pt}{\isasymforall}x{\isasymge}{\isadigit{0}}{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}X{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}real{\isacharcircum}{\kern0pt}{\isacharunderscore}{\kern0pt}{\isacharcircum}{\kern0pt}{\isacharunderscore}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharasterisk}{\kern0pt}v\ x\ {\isasymle}\ Y\ {\isacharasterisk}{\kern0pt}v\ x{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ diff{\isacharunderscore}{\kern0pt}ge{\isacharunderscore}{\kern0pt}{\isadigit{0}}{\isacharunderscore}{\kern0pt}iff{\isacharunderscore}{\kern0pt}ge\ matrix{\isacharunderscore}{\kern0pt}vector{\isacharunderscore}{\kern0pt}mult{\isacharunderscore}{\kern0pt}diff{\isacharunderscore}{\kern0pt}rdistrib\ nonneg{\isacharunderscore}{\kern0pt}mat{\isacharunderscore}{\kern0pt}iff{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Matrix Powers%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{primrec}\isamarkupfalse%
-\ matpow\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}semiring{\isacharunderscore}{\kern0pt}{\isadigit{1}}{\isacharcircum}{\kern0pt}{\isacharprime}{\kern0pt}n{\isacharcircum}{\kern0pt}{\isacharprime}{\kern0pt}n\ {\isasymRightarrow}\ nat\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a{\isacharcircum}{\kern0pt}{\isacharprime}{\kern0pt}n{\isacharcircum}{\kern0pt}{\isacharprime}{\kern0pt}n{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ matpow{\isacharunderscore}{\kern0pt}{\isadigit{0}}{\isacharcolon}{\kern0pt}\ \ \ {\isachardoublequoteopen}matpow\ A\ {\isadigit{0}}\ {\isacharequal}{\kern0pt}\ mat\ {\isadigit{1}}{\isachardoublequoteclose}\ {\isacharbar}{\kern0pt}\isanewline
-\ \ matpow{\isacharunderscore}{\kern0pt}Suc{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}matpow\ A\ {\isacharparenleft}{\kern0pt}Suc\ n{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}matpow\ A\ n{\isacharparenright}{\kern0pt}\ {\isacharasterisk}{\kern0pt}{\isacharasterisk}{\kern0pt}\ A{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ nonneg{\isacharunderscore}{\kern0pt}matpow{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isadigit{0}}\ {\isasymle}\ X\ {\isasymLongrightarrow}\ {\isadigit{0}}\ {\isasymle}\ matpow\ {\isacharparenleft}{\kern0pt}X\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ real\ {\isacharcircum}{\kern0pt}\ {\isacharunderscore}{\kern0pt}\ {\isacharcircum}{\kern0pt}\ {\isacharunderscore}{\kern0pt}{\isacharparenright}{\kern0pt}\ i{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}induction\ i{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ nonneg{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}mult{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ matpow{\isacharunderscore}{\kern0pt}mono{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isadigit{0}}\ {\isasymle}\ C\ {\isasymLongrightarrow}\ C\ {\isasymle}\ D\ {\isasymLongrightarrow}\ matpow\ {\isacharparenleft}{\kern0pt}C\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ real{\isacharcircum}{\kern0pt}{\isacharunderscore}{\kern0pt}{\isacharcircum}{\kern0pt}{\isacharunderscore}{\kern0pt}{\isacharparenright}{\kern0pt}\ n\ {\isasymle}\ matpow\ D\ n{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}induction\ n{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}auto\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ matrix{\isacharunderscore}{\kern0pt}mult{\isacharunderscore}{\kern0pt}mono\ nonneg{\isacharunderscore}{\kern0pt}matpow{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ matpow{\isacharunderscore}{\kern0pt}scaleR{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}matpow\ {\isacharparenleft}{\kern0pt}c\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isacharparenleft}{\kern0pt}X\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isacharprime}{\kern0pt}b\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ real{\isacharunderscore}{\kern0pt}algebra{\isacharunderscore}{\kern0pt}{\isadigit{1}}{\isacharcircum}{\kern0pt}{\isacharunderscore}{\kern0pt}{\isacharcircum}{\kern0pt}{\isacharunderscore}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ n\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}c{\isacharcircum}{\kern0pt}n{\isacharparenright}{\kern0pt}\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isacharparenleft}{\kern0pt}matpow\ X{\isacharparenright}{\kern0pt}\ n{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}induction\ n\ arbitrary{\isacharcolon}{\kern0pt}\ X\ c{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{case}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}Suc\ n{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}matpow\ {\isacharparenleft}{\kern0pt}c\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ X{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}Suc\ n{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}c{\isacharcircum}{\kern0pt}n{\isacharparenright}{\kern0pt}{\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isacharparenleft}{\kern0pt}matpow\ X{\isacharparenright}{\kern0pt}\ n\ {\isacharasterisk}{\kern0pt}{\isacharasterisk}{\kern0pt}\ c\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ X{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ Suc\ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isacharequal}{\kern0pt}\ c\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}c{\isacharcircum}{\kern0pt}n{\isacharparenright}{\kern0pt}\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isacharparenleft}{\kern0pt}matpow\ X{\isacharparenright}{\kern0pt}\ n\ {\isacharasterisk}{\kern0pt}{\isacharasterisk}{\kern0pt}\ X{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ scalar{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}assoc{\isacharprime}{\kern0pt}\ \isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ scalar{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}assoc{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{finally}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}case\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ scalar{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}assoc{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ matrix{\isacharunderscore}{\kern0pt}vector{\isacharunderscore}{\kern0pt}mult{\isacharunderscore}{\kern0pt}code{\isacharprime}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}X\ {\isacharasterisk}{\kern0pt}v\ x{\isacharparenright}{\kern0pt}\ {\isachardollar}{\kern0pt}\ i\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSum}j{\isasymin}UNIV{\isachardot}{\kern0pt}\ X\ {\isachardollar}{\kern0pt}\ i\ {\isachardollar}{\kern0pt}\ j\ {\isacharasterisk}{\kern0pt}\ x\ {\isachardollar}{\kern0pt}\ j{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ matrix{\isacharunderscore}{\kern0pt}vector{\isacharunderscore}{\kern0pt}mult{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ matrix{\isacharunderscore}{\kern0pt}vector{\isacharunderscore}{\kern0pt}mult{\isacharunderscore}{\kern0pt}mono{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isadigit{0}}{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}real{\isacharcircum}{\kern0pt}{\isacharunderscore}{\kern0pt}{\isacharcircum}{\kern0pt}{\isacharunderscore}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymle}\ X\ {\isasymLongrightarrow}\ {\isadigit{0}}\ {\isasymle}\ v\ {\isasymLongrightarrow}\ X\ {\isasymle}\ Y\ {\isasymLongrightarrow}\ X\ {\isacharasterisk}{\kern0pt}v\ v\ {\isasymle}\ Y\ {\isacharasterisk}{\kern0pt}v\ v{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ diff{\isacharunderscore}{\kern0pt}ge{\isacharunderscore}{\kern0pt}{\isadigit{0}}{\isacharunderscore}{\kern0pt}iff{\isacharunderscore}{\kern0pt}ge\ matrix{\isacharunderscore}{\kern0pt}vector{\isacharunderscore}{\kern0pt}mult{\isacharunderscore}{\kern0pt}diff{\isacharunderscore}{\kern0pt}rdistrib\ nonneg{\isacharunderscore}{\kern0pt}mat{\isacharunderscore}{\kern0pt}iff{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Triangular Matrices%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{definition}\isamarkupfalse%
-\ {\isachardoublequoteopen}lower{\isacharunderscore}{\kern0pt}triangular{\isacharunderscore}{\kern0pt}mat\ X\ {\isasymlongleftrightarrow}\ {\isacharparenleft}{\kern0pt}{\isasymforall}i\ j{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}i\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isacharprime}{\kern0pt}b{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}{\isacharbraceleft}{\kern0pt}finite{\isacharcomma}{\kern0pt}\ linorder{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharless}{\kern0pt}\ j\ {\isasymlongrightarrow}\ X\ {\isachardollar}{\kern0pt}\ i\ {\isachardollar}{\kern0pt}\ j\ {\isacharequal}{\kern0pt}\ {\isadigit{0}}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ {\isachardoublequoteopen}strict{\isacharunderscore}{\kern0pt}lower{\isacharunderscore}{\kern0pt}triangular{\isacharunderscore}{\kern0pt}mat\ X\ {\isasymlongleftrightarrow}\ {\isacharparenleft}{\kern0pt}{\isasymforall}i\ j{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}i\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isacharprime}{\kern0pt}b{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}{\isacharbraceleft}{\kern0pt}finite{\isacharcomma}{\kern0pt}\ linorder{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymle}\ j\ {\isasymlongrightarrow}\ X\ {\isachardollar}{\kern0pt}\ i\ {\isachardollar}{\kern0pt}\ j\ {\isacharequal}{\kern0pt}\ {\isadigit{0}}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ {\isachardoublequoteopen}upper{\isacharunderscore}{\kern0pt}triangular{\isacharunderscore}{\kern0pt}mat\ X\ {\isasymlongleftrightarrow}\ {\isacharparenleft}{\kern0pt}{\isasymforall}i\ j{\isachardot}{\kern0pt}\ j\ {\isacharless}{\kern0pt}\ i\ {\isasymlongrightarrow}\ X\ {\isachardollar}{\kern0pt}\ i\ {\isachardollar}{\kern0pt}\ j\ {\isacharequal}{\kern0pt}\ {\isadigit{0}}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ stlI{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}strict{\isacharunderscore}{\kern0pt}lower{\isacharunderscore}{\kern0pt}triangular{\isacharunderscore}{\kern0pt}mat\ X\ {\isasymLongrightarrow}\ lower{\isacharunderscore}{\kern0pt}triangular{\isacharunderscore}{\kern0pt}mat\ X{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ strict{\isacharunderscore}{\kern0pt}lower{\isacharunderscore}{\kern0pt}triangular{\isacharunderscore}{\kern0pt}mat{\isacharunderscore}{\kern0pt}def\ lower{\isacharunderscore}{\kern0pt}triangular{\isacharunderscore}{\kern0pt}mat{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ lower{\isacharunderscore}{\kern0pt}triangular{\isacharunderscore}{\kern0pt}mat{\isacharunderscore}{\kern0pt}mat{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}lower{\isacharunderscore}{\kern0pt}triangular{\isacharunderscore}{\kern0pt}mat\ {\isacharparenleft}{\kern0pt}mat\ x{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ lower{\isacharunderscore}{\kern0pt}triangular{\isacharunderscore}{\kern0pt}mat{\isacharunderscore}{\kern0pt}def\ mat{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ lower{\isacharunderscore}{\kern0pt}triangular{\isacharunderscore}{\kern0pt}mult{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}lower{\isacharunderscore}{\kern0pt}triangular{\isacharunderscore}{\kern0pt}mat\ X{\isachardoublequoteclose}\ {\isachardoublequoteopen}lower{\isacharunderscore}{\kern0pt}triangular{\isacharunderscore}{\kern0pt}mat\ Y{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}lower{\isacharunderscore}{\kern0pt}triangular{\isacharunderscore}{\kern0pt}mat\ {\isacharparenleft}{\kern0pt}X\ {\isacharasterisk}{\kern0pt}{\isacharasterisk}{\kern0pt}\ Y{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ assms\ \isanewline
-\ \ \isacommand{unfolding}\isamarkupfalse%
-\ matrix{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}mult{\isacharunderscore}{\kern0pt}def\ lower{\isacharunderscore}{\kern0pt}triangular{\isacharunderscore}{\kern0pt}mat{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ sum{\isachardot}{\kern0pt}neutral{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}metis\ mult{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}zero\ neqE\ less{\isacharunderscore}{\kern0pt}trans{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ lower{\isacharunderscore}{\kern0pt}triangular{\isacharunderscore}{\kern0pt}pow{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}lower{\isacharunderscore}{\kern0pt}triangular{\isacharunderscore}{\kern0pt}mat\ X{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}lower{\isacharunderscore}{\kern0pt}triangular{\isacharunderscore}{\kern0pt}mat\ {\isacharparenleft}{\kern0pt}matpow\ X\ i{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ assms\ lower{\isacharunderscore}{\kern0pt}triangular{\isacharunderscore}{\kern0pt}mult\ lower{\isacharunderscore}{\kern0pt}triangular{\isacharunderscore}{\kern0pt}mat{\isacharunderscore}{\kern0pt}mat\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}induction\ i{\isacharparenright}{\kern0pt}\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ lower{\isacharunderscore}{\kern0pt}triangular{\isacharunderscore}{\kern0pt}suminf{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}{\isasymAnd}i{\isachardot}{\kern0pt}\ lower{\isacharunderscore}{\kern0pt}triangular{\isacharunderscore}{\kern0pt}mat\ {\isacharparenleft}{\kern0pt}f\ i{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ {\isachardoublequoteopen}summable\ {\isacharparenleft}{\kern0pt}f\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ nat\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}b{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}real{\isacharunderscore}{\kern0pt}normed{\isacharunderscore}{\kern0pt}vector{\isacharcircum}{\kern0pt}{\isacharunderscore}{\kern0pt}{\isacharcircum}{\kern0pt}{\isacharunderscore}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}lower{\isacharunderscore}{\kern0pt}triangular{\isacharunderscore}{\kern0pt}mat\ {\isacharparenleft}{\kern0pt}{\isasymSum}i{\isachardot}{\kern0pt}\ f\ i{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ assms\isanewline
-\ \ \isacommand{unfolding}\isamarkupfalse%
-\ lower{\isacharunderscore}{\kern0pt}triangular{\isacharunderscore}{\kern0pt}mat{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}subst\ bounded{\isacharunderscore}{\kern0pt}linear{\isachardot}{\kern0pt}suminf{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}auto\ intro{\isacharcolon}{\kern0pt}\ bounded{\isacharunderscore}{\kern0pt}linear{\isacharunderscore}{\kern0pt}compose{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ lower{\isacharunderscore}{\kern0pt}triangular{\isacharunderscore}{\kern0pt}pow{\isacharunderscore}{\kern0pt}eq{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}lower{\isacharunderscore}{\kern0pt}triangular{\isacharunderscore}{\kern0pt}mat\ X{\isachardoublequoteclose}\ {\isachardoublequoteopen}lower{\isacharunderscore}{\kern0pt}triangular{\isacharunderscore}{\kern0pt}mat\ Y{\isachardoublequoteclose}\ {\isachardoublequoteopen}{\isasymAnd}s{\isacharprime}{\kern0pt}{\isachardot}{\kern0pt}\ s{\isacharprime}{\kern0pt}\ {\isasymle}\ s\ {\isasymLongrightarrow}\ row\ s{\isacharprime}{\kern0pt}\ X\ {\isacharequal}{\kern0pt}\ row\ s{\isacharprime}{\kern0pt}\ Y{\isachardoublequoteclose}\ {\isachardoublequoteopen}s{\isacharprime}{\kern0pt}\ {\isasymle}\ s{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}row\ s{\isacharprime}{\kern0pt}\ {\isacharparenleft}{\kern0pt}matpow\ X\ i{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ row\ s{\isacharprime}{\kern0pt}\ {\isacharparenleft}{\kern0pt}matpow\ Y\ i{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ assms\isanewline
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}induction\ i{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{case}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}Suc\ i{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}case\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ ltX{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}lower{\isacharunderscore}{\kern0pt}triangular{\isacharunderscore}{\kern0pt}mat\ {\isacharparenleft}{\kern0pt}matpow\ X\ i{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ Suc{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}\ lower{\isacharunderscore}{\kern0pt}triangular{\isacharunderscore}{\kern0pt}pow{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ ltY{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}lower{\isacharunderscore}{\kern0pt}triangular{\isacharunderscore}{\kern0pt}mat\ {\isacharparenleft}{\kern0pt}matpow\ Y\ i{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ Suc{\isacharparenleft}{\kern0pt}{\isadigit{3}}{\isacharparenright}{\kern0pt}\ lower{\isacharunderscore}{\kern0pt}triangular{\isacharunderscore}{\kern0pt}pow{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}\ {\isacharparenleft}{\kern0pt}{\isasymSum}k{\isasymin}UNIV{\isachardot}{\kern0pt}\ matpow\ X\ i\ {\isachardollar}{\kern0pt}\ s{\isacharprime}{\kern0pt}\ {\isachardollar}{\kern0pt}\ k\ {\isacharasterisk}{\kern0pt}\ X\ {\isachardollar}{\kern0pt}\ k\ {\isachardollar}{\kern0pt}\ j{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSum}k{\isasymin}UNIV{\isachardot}{\kern0pt}\ matpow\ Y\ i\ {\isachardollar}{\kern0pt}\ s{\isacharprime}{\kern0pt}\ {\isachardollar}{\kern0pt}\ k\ {\isacharasterisk}{\kern0pt}\ Y\ {\isachardollar}{\kern0pt}\ k\ {\isachardollar}{\kern0pt}\ j{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{for}\ j\isanewline
-\ \ \ \ \isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isasymSum}k{\isasymin}UNIV{\isachardot}{\kern0pt}\ matpow\ X\ i\ {\isachardollar}{\kern0pt}\ s{\isacharprime}{\kern0pt}\ {\isachardollar}{\kern0pt}\ k\ {\isacharasterisk}{\kern0pt}\ X\ {\isachardollar}{\kern0pt}\ k\ {\isachardollar}{\kern0pt}\ j{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSum}k{\isasymin}UNIV{\isachardot}{\kern0pt}\ if\ s{\isacharprime}{\kern0pt}\ {\isacharless}{\kern0pt}\ k\ then\ {\isadigit{0}}\ else\ matpow\ Y\ i\ {\isachardollar}{\kern0pt}\ s{\isacharprime}{\kern0pt}\ {\isachardollar}{\kern0pt}\ k\ {\isacharasterisk}{\kern0pt}\ X\ {\isachardollar}{\kern0pt}\ k\ {\isachardollar}{\kern0pt}\ j{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ Suc\ ltY\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ row{\isacharunderscore}{\kern0pt}def\ lower{\isacharunderscore}{\kern0pt}triangular{\isacharunderscore}{\kern0pt}mat{\isacharunderscore}{\kern0pt}def\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ sum{\isachardot}{\kern0pt}cong{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSum}k\ {\isasymin}\ UNIV\ {\isachardot}{\kern0pt}\ matpow\ Y\ i\ {\isachardollar}{\kern0pt}\ s{\isacharprime}{\kern0pt}\ {\isachardollar}{\kern0pt}\ k\ {\isacharasterisk}{\kern0pt}\ Y\ {\isachardollar}{\kern0pt}\ k\ {\isachardollar}{\kern0pt}\ j{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ Suc\ ltY\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ row{\isacharunderscore}{\kern0pt}def\ lower{\isacharunderscore}{\kern0pt}triangular{\isacharunderscore}{\kern0pt}mat{\isacharunderscore}{\kern0pt}def\ cong{\isacharcolon}{\kern0pt}\ if{\isacharunderscore}{\kern0pt}cong\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ sum{\isachardot}{\kern0pt}cong{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{finally}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isacommand{{\isachardot}{\kern0pt}}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ row{\isacharunderscore}{\kern0pt}def\ matrix{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}mult{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\isacommand{qed}\isamarkupfalse%
-\ simp%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ lower{\isacharunderscore}{\kern0pt}triangular{\isacharunderscore}{\kern0pt}mat{\isacharunderscore}{\kern0pt}mult{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}lower{\isacharunderscore}{\kern0pt}triangular{\isacharunderscore}{\kern0pt}mat\ M{\isachardoublequoteclose}\ {\isachardoublequoteopen}{\isasymAnd}i{\isachardot}{\kern0pt}\ i\ {\isasymle}\ j\ {\isasymLongrightarrow}\ v\ {\isachardollar}{\kern0pt}\ i\ {\isacharequal}{\kern0pt}\ v{\isacharprime}{\kern0pt}\ {\isachardollar}{\kern0pt}\ i{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}M\ {\isacharasterisk}{\kern0pt}v\ v{\isacharparenright}{\kern0pt}\ {\isachardollar}{\kern0pt}\ j\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}M\ {\isacharasterisk}{\kern0pt}v\ v{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isachardollar}{\kern0pt}\ j{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}M\ {\isacharasterisk}{\kern0pt}v\ v{\isacharparenright}{\kern0pt}\ {\isachardollar}{\kern0pt}\ j\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSum}i{\isasymin}UNIV{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}if\ j\ {\isacharless}{\kern0pt}\ i\ then\ {\isadigit{0}}\ else\ \ M\ {\isachardollar}{\kern0pt}\ j\ {\isachardollar}{\kern0pt}\ i\ {\isacharasterisk}{\kern0pt}\ v\ {\isachardollar}{\kern0pt}\ i{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assms\ \isacommand{unfolding}\isamarkupfalse%
-\ lower{\isacharunderscore}{\kern0pt}triangular{\isacharunderscore}{\kern0pt}mat{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ matrix{\isacharunderscore}{\kern0pt}vector{\isacharunderscore}{\kern0pt}mult{\isacharunderscore}{\kern0pt}def\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ sum{\isachardot}{\kern0pt}cong{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSum}i{\isasymin}UNIV{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}if\ j\ {\isacharless}{\kern0pt}\ i\ then\ {\isadigit{0}}\ else\ \ M\ {\isachardollar}{\kern0pt}\ j\ {\isachardollar}{\kern0pt}\ i\ {\isacharasterisk}{\kern0pt}\ v{\isacharprime}{\kern0pt}\ {\isachardollar}{\kern0pt}\ i{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assms\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ sum{\isachardot}{\kern0pt}cong{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}M\ {\isacharasterisk}{\kern0pt}v\ v{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isachardollar}{\kern0pt}\ j{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assms\ \isacommand{unfolding}\isamarkupfalse%
-\ lower{\isacharunderscore}{\kern0pt}triangular{\isacharunderscore}{\kern0pt}mat{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ matrix{\isacharunderscore}{\kern0pt}vector{\isacharunderscore}{\kern0pt}mult{\isacharunderscore}{\kern0pt}def\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ sum{\isachardot}{\kern0pt}cong{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{finally}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isacommand{{\isachardot}{\kern0pt}}\isamarkupfalse%
-\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Inverses%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{lemma}\isamarkupfalse%
-\ matrix{\isacharunderscore}{\kern0pt}inv{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}invertible\ M{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ matrix{\isacharunderscore}{\kern0pt}inv{\isacharunderscore}{\kern0pt}left{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}matrix{\isacharunderscore}{\kern0pt}inv\ M\ {\isacharasterisk}{\kern0pt}{\isacharasterisk}{\kern0pt}\ M\ {\isacharequal}{\kern0pt}\ mat\ {\isadigit{1}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isakeyword{and}\ matrix{\isacharunderscore}{\kern0pt}inv{\isacharunderscore}{\kern0pt}right{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}M\ {\isacharasterisk}{\kern0pt}{\isacharasterisk}{\kern0pt}\ matrix{\isacharunderscore}{\kern0pt}inv\ M\ {\isacharequal}{\kern0pt}\ mat\ {\isadigit{1}}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ {\isacartoucheopen}invertible\ M{\isacartoucheclose}\ \isakeyword{and}\ someI{\isacharunderscore}{\kern0pt}ex\ {\isacharbrackleft}{\kern0pt}of\ {\isachardoublequoteopen}{\isasymlambda}\ N{\isachardot}{\kern0pt}\ M\ {\isacharasterisk}{\kern0pt}{\isacharasterisk}{\kern0pt}\ N\ {\isacharequal}{\kern0pt}\ mat\ {\isadigit{1}}\ {\isasymand}\ N\ {\isacharasterisk}{\kern0pt}{\isacharasterisk}{\kern0pt}\ M\ {\isacharequal}{\kern0pt}\ mat\ {\isadigit{1}}{\isachardoublequoteclose}{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \isacommand{unfolding}\isamarkupfalse%
-\ invertible{\isacharunderscore}{\kern0pt}def\ \isakeyword{and}\ matrix{\isacharunderscore}{\kern0pt}inv{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ simp{\isacharunderscore}{\kern0pt}all%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ matrix{\isacharunderscore}{\kern0pt}inv{\isacharunderscore}{\kern0pt}unique{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{fixes}\ A{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}{\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}{\isacharbraceleft}{\kern0pt}semiring{\isacharunderscore}{\kern0pt}{\isadigit{1}}{\isacharbraceright}{\kern0pt}{\isacharcircum}{\kern0pt}{\isacharprime}{\kern0pt}n{\isacharcircum}{\kern0pt}{\isacharprime}{\kern0pt}n{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{assumes}\ AB{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}A\ {\isacharasterisk}{\kern0pt}{\isacharasterisk}{\kern0pt}\ B\ {\isacharequal}{\kern0pt}\ mat\ {\isadigit{1}}{\isachardoublequoteclose}\ \isakeyword{and}\ BA{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}B\ {\isacharasterisk}{\kern0pt}{\isacharasterisk}{\kern0pt}\ A\ {\isacharequal}{\kern0pt}\ mat\ {\isadigit{1}}{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}matrix{\isacharunderscore}{\kern0pt}inv\ A\ {\isacharequal}{\kern0pt}\ B{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ AB\ BA\ invertible{\isacharunderscore}{\kern0pt}def\ matrix{\isacharunderscore}{\kern0pt}inv{\isacharunderscore}{\kern0pt}right\ matrix{\isacharunderscore}{\kern0pt}mul{\isacharunderscore}{\kern0pt}assoc\ matrix{\isacharunderscore}{\kern0pt}mul{\isacharunderscore}{\kern0pt}lid{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\ \isanewline
-%
-\endisadelimproof
-%
-\isadelimtheory
-\isanewline
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{end}\isamarkupfalse%
-%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\end{isabellebody}%
-\endinput
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diff --git a/thys/MDP-Algorithms/output/document/Modified_Policy_Iteration.tex b/thys/MDP-Algorithms/output/document/Modified_Policy_Iteration.tex
deleted file mode 100644
--- a/thys/MDP-Algorithms/output/document/Modified_Policy_Iteration.tex
+++ /dev/null
@@ -1,3615 +0,0 @@
-%
-\begin{isabellebody}%
-\setisabellecontext{Modified{\isacharunderscore}{\kern0pt}Policy{\isacharunderscore}{\kern0pt}Iteration}%
-%
-\isadelimtheory
-\isanewline
-\isanewline
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{theory}\isamarkupfalse%
-\ Modified{\isacharunderscore}{\kern0pt}Policy{\isacharunderscore}{\kern0pt}Iteration\isanewline
-\ \ \isakeyword{imports}\ \isanewline
-\ \ \ \ Policy{\isacharunderscore}{\kern0pt}Iteration\isanewline
-\ \ \ \ Value{\isacharunderscore}{\kern0pt}Iteration\isanewline
-\isakeyword{begin}%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsection{Modified Policy Iteration%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{locale}\isamarkupfalse%
-\ MDP{\isacharunderscore}{\kern0pt}MPI\ {\isacharequal}{\kern0pt}\ MDP{\isacharunderscore}{\kern0pt}finite{\isacharunderscore}{\kern0pt}type\ A\ K\ r\ l\ {\isacharplus}{\kern0pt}\ MDP{\isacharunderscore}{\kern0pt}act\ A\ K\ r\ l\ arb{\isacharunderscore}{\kern0pt}act\isanewline
-\ \ \isakeyword{for}\ A\ \isakeyword{and}\ K\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}s\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ finite\ {\isasymtimes}\ {\isacharprime}{\kern0pt}a\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ finite\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}s\ pmf{\isachardoublequoteclose}\ \isakeyword{and}\ r\ l\ arb{\isacharunderscore}{\kern0pt}act\isanewline
-\isakeyword{begin}%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{The Advantage Function \isa{B}%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{definition}\isamarkupfalse%
-\ {\isachardoublequoteopen}B\ v\ s\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSqunion}d\ {\isasymin}\ D\isactrlsub R{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}r{\isacharunderscore}{\kern0pt}dec\ d\ s\ {\isacharplus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isasymP}\isactrlsub {\isadigit{1}}\ d\ {\isacharminus}{\kern0pt}\ id{\isacharunderscore}{\kern0pt}blinfun{\isacharparenright}{\kern0pt}\ v\ s{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}%
-\begin{isamarkuptext}%
-The function \isa{B} denotes the advantage of choosing the optimal action vs.
-  the current value estimate%
-\end{isamarkuptext}\isamarkuptrue%
-\isacommand{lemma}\isamarkupfalse%
-\ B{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}{\isasymL}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}B\ v\ s\ {\isacharequal}{\kern0pt}\ {\isasymL}\ v\ s\ {\isacharminus}{\kern0pt}\ v\ s{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isacharasterisk}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}B\ v\ s\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSqunion}d\ {\isasymin}\ D\isactrlsub R{\isachardot}{\kern0pt}\ L\ d\ v\ s\ {\isacharminus}{\kern0pt}\ v\ s{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ B{\isacharunderscore}{\kern0pt}def\ L{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp\ add{\isacharcolon}{\kern0pt}\ blinfun{\isachardot}{\kern0pt}bilinear{\isacharunderscore}{\kern0pt}simps\ add{\isacharunderscore}{\kern0pt}diff{\isacharunderscore}{\kern0pt}eq{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ {\isacharasterisk}{\kern0pt}\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}rule\ antisym{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isasymSqunion}d{\isasymin}D\isactrlsub R{\isachardot}{\kern0pt}\ L\ d\ v\ s\ {\isacharminus}{\kern0pt}\ v\ s{\isacharparenright}{\kern0pt}\ {\isasymle}\ {\isasymL}\ v\ s\ {\isacharminus}{\kern0pt}\ v\ s{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ {\isasymL}{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ ex{\isacharunderscore}{\kern0pt}dec\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}fastforce\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ cSUP{\isacharunderscore}{\kern0pt}upper\ cSUP{\isacharunderscore}{\kern0pt}least{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}bdd{\isacharunderscore}{\kern0pt}above\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}{\isasymlambda}d{\isachardot}{\kern0pt}\ L\ d\ v\ s\ {\isacharminus}{\kern0pt}\ v\ s{\isacharparenright}{\kern0pt}\ {\isacharbackquote}{\kern0pt}\ D\isactrlsub R{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ bounded{\isacharunderscore}{\kern0pt}const\ bounded{\isacharunderscore}{\kern0pt}minus{\isacharunderscore}{\kern0pt}comp\ bounded{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}bdd{\isacharunderscore}{\kern0pt}above{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymL}\ v\ s\ {\isacharminus}{\kern0pt}\ v\ s\ {\isasymle}\ {\isacharparenleft}{\kern0pt}{\isasymSqunion}d\ {\isasymin}\ D\isactrlsub R{\isachardot}{\kern0pt}\ L\ d\ v\ s\ {\isacharminus}{\kern0pt}\ v\ s{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ {\isasymL}{\isacharunderscore}{\kern0pt}def\ diff{\isacharunderscore}{\kern0pt}le{\isacharunderscore}{\kern0pt}eq\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}intro\ cSUP{\isacharunderscore}{\kern0pt}least{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}auto\ intro{\isacharcolon}{\kern0pt}\ cSUP{\isacharunderscore}{\kern0pt}upper{\isadigit{2}}\ simp{\isacharcolon}{\kern0pt}\ diff{\isacharunderscore}{\kern0pt}le{\isacharunderscore}{\kern0pt}eq{\isacharbrackleft}{\kern0pt}symmetric{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\begin{isamarkuptext}%
-\isa{B} is a bounded function.%
-\end{isamarkuptext}\isamarkuptrue%
-\isacommand{lift{\isacharunderscore}{\kern0pt}definition}\isamarkupfalse%
-\ B\isactrlsub b\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}s\ {\isasymRightarrow}\isactrlsub b\ real{\isacharparenright}{\kern0pt}\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}s\ {\isasymRightarrow}\isactrlsub b\ real{\isachardoublequoteclose}\ \isakeyword{is}\ {\isachardoublequoteopen}B{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ {\isasymL}\isactrlsub b{\isachardot}{\kern0pt}rep{\isacharunderscore}{\kern0pt}eq{\isacharbrackleft}{\kern0pt}symmetric{\isacharbrackright}{\kern0pt}\ B{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}{\isasymL}\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ bfun{\isacharunderscore}{\kern0pt}normI\ order{\isachardot}{\kern0pt}trans{\isacharbrackleft}{\kern0pt}OF\ abs{\isacharunderscore}{\kern0pt}triangle{\isacharunderscore}{\kern0pt}ineq{\isadigit{4}}{\isacharbrackright}{\kern0pt}\ add{\isacharunderscore}{\kern0pt}mono\ abs{\isacharunderscore}{\kern0pt}le{\isacharunderscore}{\kern0pt}norm{\isacharunderscore}{\kern0pt}bfun{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ B\isactrlsub b{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}{\isasymL}\isactrlsub b{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}B\isactrlsub b\ v\ {\isacharequal}{\kern0pt}\ {\isasymL}\isactrlsub b\ v\ {\isacharminus}{\kern0pt}\ v{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ {\isasymL}\isactrlsub b{\isachardot}{\kern0pt}rep{\isacharunderscore}{\kern0pt}eq\ B\isactrlsub b{\isachardot}{\kern0pt}rep{\isacharunderscore}{\kern0pt}eq\ B{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}{\isasymL}{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ {\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}SUP{\isacharunderscore}{\kern0pt}L\isactrlsub a{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymL}\isactrlsub b\ v\ s\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSqunion}a\ {\isasymin}\ A\ s{\isachardot}{\kern0pt}\ L\isactrlsub a\ a\ v\ s{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ L{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}L\isactrlsub a{\isacharunderscore}{\kern0pt}det\ {\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}SUP{\isacharunderscore}{\kern0pt}det\ SUP{\isacharunderscore}{\kern0pt}step{\isacharunderscore}{\kern0pt}det{\isacharunderscore}{\kern0pt}eq\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Optimization of the Value Function over Multiple Steps%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{definition}\isamarkupfalse%
-\ {\isachardoublequoteopen}U\ m\ v\ s\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSqunion}d\ {\isasymin}\ D\isactrlsub R{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}fin\ {\isacharparenleft}{\kern0pt}mk{\isacharunderscore}{\kern0pt}stationary\ d{\isacharparenright}{\kern0pt}\ m\ {\isacharplus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isasymP}\isactrlsub {\isadigit{1}}\ d{\isacharparenright}{\kern0pt}{\isacharcircum}{\kern0pt}{\isacharcircum}{\kern0pt}m{\isacharparenright}{\kern0pt}\ v{\isacharparenright}{\kern0pt}\ s{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}%
-\begin{isamarkuptext}%
-\isa{U} expresses the value estimate obtained by optimizing the first \isa{m} steps and 
-  afterwards using the current estimate.%
-\end{isamarkuptext}\isamarkuptrue%
-\isacommand{lemma}\isamarkupfalse%
-\ U{\isacharunderscore}{\kern0pt}zero\ {\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}U\ {\isadigit{0}}\ v\ {\isacharequal}{\kern0pt}\ v{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ U{\isacharunderscore}{\kern0pt}def\ {\isasymL}{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}fin{\isachardot}{\kern0pt}rep{\isacharunderscore}{\kern0pt}eq{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ U{\isacharunderscore}{\kern0pt}one{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}{\isasymL}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}U\ {\isadigit{1}}\ v\ s\ {\isacharequal}{\kern0pt}\ {\isasymL}\ v\ s{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ U{\isacharunderscore}{\kern0pt}def\ {\isasymL}{\isacharunderscore}{\kern0pt}def\ \isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}fin{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}{\isasymP}\isactrlsub X\ L{\isacharunderscore}{\kern0pt}def\ blinfun{\isachardot}{\kern0pt}bilinear{\isacharunderscore}{\kern0pt}simps{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lift{\isacharunderscore}{\kern0pt}definition}\isamarkupfalse%
-\ U\isactrlsub b\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}nat\ {\isasymRightarrow}\ {\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}s\ {\isasymRightarrow}\isactrlsub b\ real{\isacharparenright}{\kern0pt}\ {\isasymRightarrow}\ {\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}s\ {\isasymRightarrow}\isactrlsub b\ real{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{is}\ U\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\ n\ v\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}norm\ {\isacharparenleft}{\kern0pt}{\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}fin\ {\isacharparenleft}{\kern0pt}mk{\isacharunderscore}{\kern0pt}stationary\ d{\isacharparenright}{\kern0pt}\ m{\isacharparenright}{\kern0pt}\ {\isasymle}\ {\isacharparenleft}{\kern0pt}{\isasymSum}i{\isacharless}{\kern0pt}m{\isachardot}{\kern0pt}\ l\ {\isacharcircum}{\kern0pt}\ i\ {\isacharasterisk}{\kern0pt}\ r\isactrlsub M{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{for}\ d\ m\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ abs{\isacharunderscore}{\kern0pt}{\isasymnu}{\isacharunderscore}{\kern0pt}fin{\isacharunderscore}{\kern0pt}le\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}fin{\isachardot}{\kern0pt}rep{\isacharunderscore}{\kern0pt}eq\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ norm{\isacharunderscore}{\kern0pt}bound{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}norm\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isasymP}\isactrlsub {\isadigit{1}}\ d{\isacharparenright}{\kern0pt}{\isacharcircum}{\kern0pt}{\isacharcircum}{\kern0pt}m{\isacharparenright}{\kern0pt}\ v{\isacharparenright}{\kern0pt}\ {\isasymle}\ l\ {\isacharcircum}{\kern0pt}\ m\ {\isacharasterisk}{\kern0pt}\ norm\ v{\isachardoublequoteclose}\ \isakeyword{for}\ d\ m\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ {\isasymP}\isactrlsub X{\isacharunderscore}{\kern0pt}const{\isacharbrackleft}{\kern0pt}symmetric{\isacharbrackright}{\kern0pt}\ blinfun{\isachardot}{\kern0pt}bilinear{\isacharunderscore}{\kern0pt}simps\ blincomp{\isacharunderscore}{\kern0pt}scaleR{\isacharunderscore}{\kern0pt}right\ simp\ del{\isacharcolon}{\kern0pt}\ {\isasymP}\isactrlsub X{\isacharunderscore}{\kern0pt}sconst\ \isanewline
-\ \ \ \ \ \ \ \ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ boundedI\ order{\isachardot}{\kern0pt}trans{\isacharbrackleft}{\kern0pt}OF\ abs{\isacharunderscore}{\kern0pt}le{\isacharunderscore}{\kern0pt}norm{\isacharunderscore}{\kern0pt}bfun{\isacharbrackright}{\kern0pt}\ mult{\isacharunderscore}{\kern0pt}left{\isacharunderscore}{\kern0pt}mono{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isacharasterisk}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}norm\ {\isacharparenleft}{\kern0pt}{\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}fin\ {\isacharparenleft}{\kern0pt}mk{\isacharunderscore}{\kern0pt}stationary\ d{\isacharparenright}{\kern0pt}\ m\ {\isacharplus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isasymP}\isactrlsub {\isadigit{1}}\ d{\isacharparenright}{\kern0pt}{\isacharcircum}{\kern0pt}{\isacharcircum}{\kern0pt}m{\isacharparenright}{\kern0pt}\ v{\isacharparenright}{\kern0pt}\ {\isasymle}\ {\isacharparenleft}{\kern0pt}{\isasymSum}i{\isacharless}{\kern0pt}m{\isachardot}{\kern0pt}\ l\ {\isacharcircum}{\kern0pt}\ i\ {\isacharasterisk}{\kern0pt}\ r\isactrlsub M{\isacharparenright}{\kern0pt}\ {\isacharplus}{\kern0pt}\ \ l\ {\isacharcircum}{\kern0pt}\ m\ {\isacharasterisk}{\kern0pt}\ norm\ v{\isachardoublequoteclose}\ \isakeyword{for}\ d\ m\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ norm{\isacharunderscore}{\kern0pt}triangle{\isacharunderscore}{\kern0pt}mono\ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}U\ n\ v\ {\isasymin}\ bfun{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ ex{\isacharunderscore}{\kern0pt}dec\ order{\isachardot}{\kern0pt}trans{\isacharbrackleft}{\kern0pt}OF\ abs{\isacharunderscore}{\kern0pt}le{\isacharunderscore}{\kern0pt}norm{\isacharunderscore}{\kern0pt}bfun\ {\isacharasterisk}{\kern0pt}{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}fastforce\ simp{\isacharcolon}{\kern0pt}\ U{\isacharunderscore}{\kern0pt}def\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ bfun{\isacharunderscore}{\kern0pt}normI\ cSup{\isacharunderscore}{\kern0pt}abs{\isacharunderscore}{\kern0pt}le{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ U\isactrlsub b{\isacharunderscore}{\kern0pt}contraction{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}dist\ {\isacharparenleft}{\kern0pt}U\isactrlsub b\ m\ v{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}U\isactrlsub b\ m\ u{\isacharparenright}{\kern0pt}\ {\isasymle}\ l\ {\isacharcircum}{\kern0pt}\ m\ {\isacharasterisk}{\kern0pt}\ dist\ v\ u{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ aux{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}dist\ {\isacharparenleft}{\kern0pt}U\isactrlsub b\ m\ v\ s{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}U\isactrlsub b\ m\ u\ s{\isacharparenright}{\kern0pt}\ {\isasymle}\ l\ {\isacharcircum}{\kern0pt}\ m\ {\isacharasterisk}{\kern0pt}\ dist\ v\ u{\isachardoublequoteclose}\ \isakeyword{if}\ le{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}U\isactrlsub b\ m\ u\ s\ {\isasymle}\ U\isactrlsub b\ m\ v\ s{\isachardoublequoteclose}\ \isakeyword{for}\ s\ v\ u\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}U\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}{\isasymlambda}m\ v\ d{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}fin\ {\isacharparenleft}{\kern0pt}mk{\isacharunderscore}{\kern0pt}stationary\ d{\isacharparenright}{\kern0pt}\ m\ {\isacharplus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isasymP}\isactrlsub {\isadigit{1}}\ d{\isacharparenright}{\kern0pt}\ {\isacharcircum}{\kern0pt}{\isacharcircum}{\kern0pt}\ m{\isacharparenright}{\kern0pt}\ v{\isacharparenright}{\kern0pt}\ s{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}U\isactrlsub b\ m\ v\ s\ {\isacharminus}{\kern0pt}\ U\isactrlsub b\ m\ u\ s\ {\isasymle}\ {\isacharparenleft}{\kern0pt}{\isasymSqunion}d\ {\isasymin}\ D\isactrlsub R{\isachardot}{\kern0pt}\ {\isacharquery}{\kern0pt}U\ m\ v\ d\ {\isacharminus}{\kern0pt}\ {\isacharquery}{\kern0pt}U\ m\ u\ d{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ bounded{\isacharunderscore}{\kern0pt}stationary{\isacharunderscore}{\kern0pt}{\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}fin\ bounded{\isacharunderscore}{\kern0pt}disc{\isacharunderscore}{\kern0pt}{\isasymP}\isactrlsub {\isadigit{1}}\ le\isanewline
-\ \ \ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ U\isactrlsub b{\isachardot}{\kern0pt}rep{\isacharunderscore}{\kern0pt}eq\ U{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}intro\ le{\isacharunderscore}{\kern0pt}SUP{\isacharunderscore}{\kern0pt}diff{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}auto\ intro{\isacharcolon}{\kern0pt}\ bounded{\isacharunderscore}{\kern0pt}plus{\isacharunderscore}{\kern0pt}comp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSqunion}d\ {\isasymin}\ D\isactrlsub R{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isasymP}\isactrlsub {\isadigit{1}}\ d{\isacharparenright}{\kern0pt}\ {\isacharcircum}{\kern0pt}{\isacharcircum}{\kern0pt}\ m{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}v\ {\isacharminus}{\kern0pt}\ u{\isacharparenright}{\kern0pt}\ s{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ L{\isacharunderscore}{\kern0pt}def\ scale{\isacharunderscore}{\kern0pt}right{\isacharunderscore}{\kern0pt}diff{\isacharunderscore}{\kern0pt}distrib\ blinfun{\isachardot}{\kern0pt}bilinear{\isacharunderscore}{\kern0pt}simps{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSqunion}d\ {\isasymin}\ D\isactrlsub R{\isachardot}{\kern0pt}\ l{\isacharcircum}{\kern0pt}m\ {\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}{\isasymP}\isactrlsub {\isadigit{1}}\ d\ {\isacharcircum}{\kern0pt}{\isacharcircum}{\kern0pt}\ m{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}v\ {\isacharminus}{\kern0pt}\ u{\isacharparenright}{\kern0pt}\ s{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ blincomp{\isacharunderscore}{\kern0pt}scaleR{\isacharunderscore}{\kern0pt}right\ blinfun{\isachardot}{\kern0pt}scaleR{\isacharunderscore}{\kern0pt}left{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isacharequal}{\kern0pt}\ l{\isacharcircum}{\kern0pt}m\ {\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSqunion}d\ {\isasymin}\ D\isactrlsub R{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}{\isasymP}\isactrlsub {\isadigit{1}}\ d\ {\isacharcircum}{\kern0pt}{\isacharcircum}{\kern0pt}\ m{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}v\ {\isacharminus}{\kern0pt}\ u{\isacharparenright}{\kern0pt}\ s{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ D\isactrlsub R{\isacharunderscore}{\kern0pt}ne\ bounded{\isacharunderscore}{\kern0pt}P\ bounded{\isacharunderscore}{\kern0pt}disc{\isacharunderscore}{\kern0pt}{\isasymP}\isactrlsub {\isadigit{1}}{\isacharprime}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ intro{\isacharcolon}{\kern0pt}\ bounded{\isacharunderscore}{\kern0pt}SUP{\isacharunderscore}{\kern0pt}mul{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isasymle}\ l{\isacharcircum}{\kern0pt}m\ {\isacharasterisk}{\kern0pt}\ norm\ {\isacharparenleft}{\kern0pt}{\isasymSqunion}d\ {\isasymin}\ D\isactrlsub R{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}{\isasymP}\isactrlsub {\isadigit{1}}\ d\ {\isacharcircum}{\kern0pt}{\isacharcircum}{\kern0pt}\ m{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}v\ {\isacharminus}{\kern0pt}\ u{\isacharparenright}{\kern0pt}\ s{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ mult{\isacharunderscore}{\kern0pt}left{\isacharunderscore}{\kern0pt}mono{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isasymle}\ l{\isacharcircum}{\kern0pt}m\ {\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSqunion}d\ {\isasymin}\ D\isactrlsub R{\isachardot}{\kern0pt}\ norm\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}{\isasymP}\isactrlsub {\isadigit{1}}\ d\ {\isacharcircum}{\kern0pt}{\isacharcircum}{\kern0pt}\ m{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}v\ {\isacharminus}{\kern0pt}\ u{\isacharparenright}{\kern0pt}\ s{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ D\isactrlsub R{\isacharunderscore}{\kern0pt}ne\ ex{\isacharunderscore}{\kern0pt}dec\ bounded{\isacharunderscore}{\kern0pt}norm{\isacharunderscore}{\kern0pt}comp\ bounded{\isacharunderscore}{\kern0pt}disc{\isacharunderscore}{\kern0pt}{\isasymP}\isactrlsub {\isadigit{1}}{\isacharprime}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}fastforce\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ mult{\isacharunderscore}{\kern0pt}left{\isacharunderscore}{\kern0pt}mono{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isasymle}\ l{\isacharcircum}{\kern0pt}m\ {\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSqunion}d\ {\isasymin}\ D\isactrlsub R{\isachardot}{\kern0pt}\ norm\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}{\isasymP}\isactrlsub {\isadigit{1}}\ d\ {\isacharcircum}{\kern0pt}{\isacharcircum}{\kern0pt}\ m{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}v\ {\isacharminus}{\kern0pt}\ u{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ ex{\isacharunderscore}{\kern0pt}dec\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}fastforce\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ order{\isachardot}{\kern0pt}trans{\isacharbrackleft}{\kern0pt}OF\ norm{\isacharunderscore}{\kern0pt}blinfun{\isacharbrackright}{\kern0pt}\ abs{\isacharunderscore}{\kern0pt}le{\isacharunderscore}{\kern0pt}norm{\isacharunderscore}{\kern0pt}bfun\ mult{\isacharunderscore}{\kern0pt}left{\isacharunderscore}{\kern0pt}mono\ cSUP{\isacharunderscore}{\kern0pt}mono{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isasymle}\ l{\isacharcircum}{\kern0pt}m\ {\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSqunion}d\ {\isasymin}\ D\isactrlsub R{\isachardot}{\kern0pt}\ norm\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}v\ {\isacharminus}{\kern0pt}\ u{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ norm{\isacharunderscore}{\kern0pt}{\isasymP}\isactrlsub X{\isacharunderscore}{\kern0pt}apply\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ {\isasymP}\isactrlsub X{\isacharunderscore}{\kern0pt}const{\isacharbrackleft}{\kern0pt}symmetric{\isacharbrackright}{\kern0pt}\ cSUP{\isacharunderscore}{\kern0pt}least\ mult{\isacharunderscore}{\kern0pt}left{\isacharunderscore}{\kern0pt}mono{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isacharequal}{\kern0pt}\ l\ {\isacharcircum}{\kern0pt}m\ {\isacharasterisk}{\kern0pt}\ dist\ v\ u{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ dist{\isacharunderscore}{\kern0pt}norm{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{finally}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}U\isactrlsub b\ m\ v\ s\ {\isacharminus}{\kern0pt}\ U\isactrlsub b\ m\ u\ s\ {\isasymle}\ l{\isacharcircum}{\kern0pt}m\ {\isacharasterisk}{\kern0pt}\ dist\ v\ u{\isachardoublequoteclose}\ \isacommand{{\isachardot}{\kern0pt}}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ dist{\isacharunderscore}{\kern0pt}real{\isacharunderscore}{\kern0pt}def\ le{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}U\isactrlsub b\ m\ v\ s\ {\isasymle}\ U\isactrlsub b\ m\ u\ s\ {\isasymLongrightarrow}\ dist\ {\isacharparenleft}{\kern0pt}U\isactrlsub b\ m\ v\ s{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}U\isactrlsub b\ m\ u\ s{\isacharparenright}{\kern0pt}\ {\isasymle}\ l{\isacharcircum}{\kern0pt}m\ {\isacharasterisk}{\kern0pt}\ dist\ v\ u{\isachardoublequoteclose}\ \isakeyword{for}\ u\ v\ s\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ aux\ dist{\isacharunderscore}{\kern0pt}commute{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}dist\ {\isacharparenleft}{\kern0pt}U\isactrlsub b\ m\ v\ s{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}U\isactrlsub b\ m\ u\ s{\isacharparenright}{\kern0pt}\ {\isasymle}\ l{\isacharcircum}{\kern0pt}m\ {\isacharasterisk}{\kern0pt}\ dist\ v\ u{\isachardoublequoteclose}\ \isakeyword{for}\ u\ v\ s\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ linear\ \isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}dist\ {\isacharparenleft}{\kern0pt}U\isactrlsub b\ m\ v{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}U\isactrlsub b\ m\ u{\isacharparenright}{\kern0pt}\ {\isasymle}\ l{\isacharcircum}{\kern0pt}m\ {\isacharasterisk}{\kern0pt}\ dist\ v\ u{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ dist{\isacharunderscore}{\kern0pt}bound{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ U\isactrlsub b{\isacharunderscore}{\kern0pt}conv{\isacharcolon}{\kern0pt}\isanewline
-\ \ {\isachardoublequoteopen}{\isasymexists}{\isacharbang}{\kern0pt}v{\isachardot}{\kern0pt}\ U\isactrlsub b\ {\isacharparenleft}{\kern0pt}Suc\ m{\isacharparenright}{\kern0pt}\ v\ {\isacharequal}{\kern0pt}\ v{\isachardoublequoteclose}\ \isanewline
-\ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isasymlambda}n{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}U\isactrlsub b\ {\isacharparenleft}{\kern0pt}Suc\ m{\isacharparenright}{\kern0pt}\ {\isacharcircum}{\kern0pt}{\isacharcircum}{\kern0pt}\ n{\isacharparenright}{\kern0pt}\ v{\isacharparenright}{\kern0pt}\ {\isasymlonglonglongrightarrow}\ {\isacharparenleft}{\kern0pt}THE\ v{\isachardot}{\kern0pt}\ U\isactrlsub b\ {\isacharparenleft}{\kern0pt}Suc\ m{\isacharparenright}{\kern0pt}\ v\ {\isacharequal}{\kern0pt}\ v{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isacharasterisk}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}is{\isacharunderscore}{\kern0pt}contraction\ {\isacharparenleft}{\kern0pt}U\isactrlsub b\ {\isacharparenleft}{\kern0pt}Suc\ m{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ is{\isacharunderscore}{\kern0pt}contraction{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ U\isactrlsub b{\isacharunderscore}{\kern0pt}contraction{\isacharbrackleft}{\kern0pt}of\ {\isachardoublequoteopen}Suc\ m{\isachardoublequoteclose}{\isacharbrackright}{\kern0pt}\ le{\isacharunderscore}{\kern0pt}neq{\isacharunderscore}{\kern0pt}trans{\isacharbrackleft}{\kern0pt}OF\ zero{\isacharunderscore}{\kern0pt}le{\isacharunderscore}{\kern0pt}disc{\isacharbrackright}{\kern0pt}\ \isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}cases\ {\isachardoublequoteopen}l\ {\isacharequal}{\kern0pt}\ {\isadigit{0}}{\isachardoublequoteclose}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}auto\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ power{\isacharunderscore}{\kern0pt}Suc{\isacharunderscore}{\kern0pt}less{\isacharunderscore}{\kern0pt}one\ intro{\isacharcolon}{\kern0pt}\ exI{\isacharbrackleft}{\kern0pt}of\ {\isacharunderscore}{\kern0pt}\ {\isachardoublequoteopen}l{\isacharcircum}{\kern0pt}{\isacharparenleft}{\kern0pt}Suc\ m{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymexists}{\isacharbang}{\kern0pt}v{\isachardot}{\kern0pt}\ U\isactrlsub b\ {\isacharparenleft}{\kern0pt}Suc\ m{\isacharparenright}{\kern0pt}\ v\ {\isacharequal}{\kern0pt}\ v{\isachardoublequoteclose}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isasymlambda}n{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}U\isactrlsub b\ {\isacharparenleft}{\kern0pt}Suc\ m{\isacharparenright}{\kern0pt}\ {\isacharcircum}{\kern0pt}{\isacharcircum}{\kern0pt}\ n{\isacharparenright}{\kern0pt}\ v{\isacharparenright}{\kern0pt}\ {\isasymlonglonglongrightarrow}\ {\isacharparenleft}{\kern0pt}THE\ v{\isachardot}{\kern0pt}\ U\isactrlsub b\ {\isacharparenleft}{\kern0pt}Suc\ m{\isacharparenright}{\kern0pt}\ v\ {\isacharequal}{\kern0pt}\ v{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ banach{\isacharprime}{\kern0pt}{\isacharbrackleft}{\kern0pt}OF\ {\isacharasterisk}{\kern0pt}{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ U\isactrlsub b{\isacharunderscore}{\kern0pt}convergent{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}convergent\ {\isacharparenleft}{\kern0pt}{\isasymlambda}n{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}U\isactrlsub b\ {\isacharparenleft}{\kern0pt}Suc\ m{\isacharparenright}{\kern0pt}\ {\isacharcircum}{\kern0pt}{\isacharcircum}{\kern0pt}\ n{\isacharparenright}{\kern0pt}\ v{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}intro\ convergentI{\isacharbrackleft}{\kern0pt}OF\ U\isactrlsub b{\isacharunderscore}{\kern0pt}conv{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ U\isactrlsub b{\isacharunderscore}{\kern0pt}mono{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}v\ {\isasymle}\ u{\isachardoublequoteclose}\ \isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}U\isactrlsub b\ m\ v\ {\isasymle}\ U\isactrlsub b\ m\ u{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ \ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}U\isactrlsub b\ m\ v\ s\ {\isasymle}\ U\isactrlsub b\ m\ u\ s{\isachardoublequoteclose}\ \isakeyword{for}\ s\isanewline
-\ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ U\isactrlsub b{\isachardot}{\kern0pt}rep{\isacharunderscore}{\kern0pt}eq\ U{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}intro\ cSUP{\isacharunderscore}{\kern0pt}mono{\isacharcomma}{\kern0pt}\ goal{\isacharunderscore}{\kern0pt}cases{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{case}\isamarkupfalse%
-\ {\isadigit{2}}\isanewline
-\ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}case\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ bounded{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}bdd{\isacharunderscore}{\kern0pt}above\ bounded{\isacharunderscore}{\kern0pt}disc{\isacharunderscore}{\kern0pt}{\isasymP}\isactrlsub {\isadigit{1}}\ bounded{\isacharunderscore}{\kern0pt}plus{\isacharunderscore}{\kern0pt}comp\ bounded{\isacharunderscore}{\kern0pt}stationary{\isacharunderscore}{\kern0pt}{\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}fin{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{case}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}{\isadigit{3}}\ n{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}case\ \isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ less{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}bfunD{\isacharbrackleft}{\kern0pt}OF\ {\isasymP}\isactrlsub X{\isacharunderscore}{\kern0pt}mono{\isacharbrackleft}{\kern0pt}OF\ assms{\isacharbrackright}{\kern0pt}{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ {\isasymP}\isactrlsub X{\isacharunderscore}{\kern0pt}const{\isacharbrackleft}{\kern0pt}symmetric{\isacharbrackright}{\kern0pt}\ blincomp{\isacharunderscore}{\kern0pt}scaleR{\isacharunderscore}{\kern0pt}right\ blinfun{\isachardot}{\kern0pt}scaleR{\isacharunderscore}{\kern0pt}left\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ mult{\isacharunderscore}{\kern0pt}left{\isacharunderscore}{\kern0pt}mono\ exI{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assms\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ U\isactrlsub b{\isacharunderscore}{\kern0pt}le{\isacharunderscore}{\kern0pt}{\isasymL}\isactrlsub b{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}U\isactrlsub b\ m\ v\ {\isasymle}\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ {\isacharcircum}{\kern0pt}{\isacharcircum}{\kern0pt}\ m{\isacharparenright}{\kern0pt}\ v{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}U\isactrlsub b\ m\ v\ s\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSqunion}d\ {\isasymin}\ D\isactrlsub R{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}L\ d{\isacharcircum}{\kern0pt}{\isacharcircum}{\kern0pt}\ m{\isacharparenright}{\kern0pt}\ v\ s{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{for}\ m\ v\ s\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ L{\isacharunderscore}{\kern0pt}iter\ U\isactrlsub b{\isachardot}{\kern0pt}rep{\isacharunderscore}{\kern0pt}eq\ {\isasymL}\isactrlsub b{\isachardot}{\kern0pt}rep{\isacharunderscore}{\kern0pt}eq\ U{\isacharunderscore}{\kern0pt}def\ {\isasymL}{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ L{\isacharunderscore}{\kern0pt}iter{\isacharunderscore}{\kern0pt}le{\isacharunderscore}{\kern0pt}{\isasymL}\isactrlsub b\ ex{\isacharunderscore}{\kern0pt}dec\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}fastforce\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ cSUP{\isacharunderscore}{\kern0pt}least{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ L{\isacharunderscore}{\kern0pt}iter{\isacharunderscore}{\kern0pt}le{\isacharunderscore}{\kern0pt}U\isactrlsub b{\isacharcolon}{\kern0pt}\ \isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}d\ {\isasymin}\ D\isactrlsub R{\isachardoublequoteclose}\ \isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}L\ d{\isacharcircum}{\kern0pt}{\isacharcircum}{\kern0pt}m{\isacharparenright}{\kern0pt}\ v\ {\isasymle}\ U\isactrlsub b\ m\ v{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ assms\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}fastforce\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ cSUP{\isacharunderscore}{\kern0pt}upper\ bounded{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}bdd{\isacharunderscore}{\kern0pt}above\isanewline
-\ \ \ \ \ \ simp{\isacharcolon}{\kern0pt}\ L{\isacharunderscore}{\kern0pt}iter\ U\isactrlsub b{\isachardot}{\kern0pt}rep{\isacharunderscore}{\kern0pt}eq\ U{\isacharunderscore}{\kern0pt}def\ bounded{\isacharunderscore}{\kern0pt}disc{\isacharunderscore}{\kern0pt}{\isasymP}\isactrlsub {\isadigit{1}}\ bounded{\isacharunderscore}{\kern0pt}plus{\isacharunderscore}{\kern0pt}comp\ bounded{\isacharunderscore}{\kern0pt}stationary{\isacharunderscore}{\kern0pt}{\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}fin{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ lim{\isacharunderscore}{\kern0pt}U\isactrlsub b{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}lim\ {\isacharparenleft}{\kern0pt}{\isasymlambda}n{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}U\isactrlsub b\ {\isacharparenleft}{\kern0pt}Suc\ m{\isacharparenright}{\kern0pt}\ {\isacharcircum}{\kern0pt}{\isacharcircum}{\kern0pt}\ n{\isacharparenright}{\kern0pt}\ v{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ le{\isacharunderscore}{\kern0pt}U{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt\ {\isasymle}\ U\isactrlsub b\ m\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt{\isachardoublequoteclose}\ \isakeyword{for}\ m\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{obtain}\isamarkupfalse%
-\ d\ \isakeyword{where}\ d{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymnu}{\isacharunderscore}{\kern0pt}improving\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt\ {\isacharparenleft}{\kern0pt}mk{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\ d{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ {\isachardoublequoteopen}d\ {\isasymin}\ D\isactrlsub D{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ ex{\isacharunderscore}{\kern0pt}improving{\isacharunderscore}{\kern0pt}det\ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}L\ {\isacharparenleft}{\kern0pt}mk{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\ d{\isacharparenright}{\kern0pt}\ {\isacharcircum}{\kern0pt}{\isacharcircum}{\kern0pt}\ m{\isacharparenright}{\kern0pt}\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}induction\ m{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}metis\ L{\isacharunderscore}{\kern0pt}{\isasymnu}{\isacharunderscore}{\kern0pt}fix{\isacharunderscore}{\kern0pt}iff\ {\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}opt\ {\isasymnu}{\isacharunderscore}{\kern0pt}improving{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}{\isasymL}\isactrlsub b\ d{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}\ funpow{\isacharunderscore}{\kern0pt}swap{\isadigit{1}}{\isacharparenright}{\kern0pt}{\isacharplus}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isacartoucheopen}d\ {\isasymin}\ D\isactrlsub D{\isacartoucheclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ order{\isachardot}{\kern0pt}trans{\isacharbrackleft}{\kern0pt}OF\ {\isacharunderscore}{\kern0pt}\ L{\isacharunderscore}{\kern0pt}iter{\isacharunderscore}{\kern0pt}le{\isacharunderscore}{\kern0pt}U\isactrlsub b{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}U\isactrlsub b\ m\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt\ {\isasymle}\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt{\isachardoublequoteclose}\ \isakeyword{for}\ m\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isasymL}{\isacharunderscore}{\kern0pt}inc{\isacharunderscore}{\kern0pt}le{\isacharunderscore}{\kern0pt}opt\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ order{\isachardot}{\kern0pt}trans{\isacharbrackleft}{\kern0pt}OF\ U\isactrlsub b{\isacharunderscore}{\kern0pt}le{\isacharunderscore}{\kern0pt}{\isasymL}\isactrlsub b{\isacharbrackright}{\kern0pt}\ simp{\isacharcolon}{\kern0pt}\ funpow{\isacharunderscore}{\kern0pt}swap{\isadigit{1}}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}U\isactrlsub b\ {\isacharparenleft}{\kern0pt}Suc\ m{\isacharparenright}{\kern0pt}\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt\ {\isacharequal}{\kern0pt}\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ le{\isacharunderscore}{\kern0pt}U\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ antisym{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}lim\ {\isacharparenleft}{\kern0pt}{\isasymlambda}n{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}U\isactrlsub b\ {\isacharparenleft}{\kern0pt}Suc\ m{\isacharparenright}{\kern0pt}\ {\isacharcircum}{\kern0pt}{\isacharcircum}{\kern0pt}n{\isacharparenright}{\kern0pt}\ v{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ U\isactrlsub b\ {\isacharparenleft}{\kern0pt}Suc\ m{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}lim\ {\isacharparenleft}{\kern0pt}{\isasymlambda}n{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}U\isactrlsub b\ {\isacharparenleft}{\kern0pt}Suc\ m{\isacharparenright}{\kern0pt}\ {\isacharcircum}{\kern0pt}{\isacharcircum}{\kern0pt}n{\isacharparenright}{\kern0pt}\ v{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ limI{\isacharbrackleft}{\kern0pt}OF\ U\isactrlsub b{\isacharunderscore}{\kern0pt}conv{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isacharbrackright}{\kern0pt}\ theI{\isacharprime}{\kern0pt}{\isacharbrackleft}{\kern0pt}OF\ U\isactrlsub b{\isacharunderscore}{\kern0pt}conv{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ U\isactrlsub b{\isacharunderscore}{\kern0pt}conv{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ U\isactrlsub b{\isacharunderscore}{\kern0pt}tendsto{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isasymlambda}n{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}U\isactrlsub b\ {\isacharparenleft}{\kern0pt}Suc\ m{\isacharparenright}{\kern0pt}\ {\isacharcircum}{\kern0pt}{\isacharcircum}{\kern0pt}\ n{\isacharparenright}{\kern0pt}\ v{\isacharparenright}{\kern0pt}\ {\isasymlonglonglongrightarrow}\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ lim{\isacharunderscore}{\kern0pt}U\isactrlsub b\ U\isactrlsub b{\isacharunderscore}{\kern0pt}convergent\ convergent{\isacharunderscore}{\kern0pt}LIMSEQ{\isacharunderscore}{\kern0pt}iff\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ metis%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ U\isactrlsub b{\isacharunderscore}{\kern0pt}fix{\isacharunderscore}{\kern0pt}unique{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}U\isactrlsub b\ {\isacharparenleft}{\kern0pt}Suc\ m{\isacharparenright}{\kern0pt}\ v\ {\isacharequal}{\kern0pt}\ v\ {\isasymlongleftrightarrow}\ v\ {\isacharequal}{\kern0pt}\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt{\isachardoublequoteclose}\ \isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ theI{\isacharprime}{\kern0pt}{\isacharbrackleft}{\kern0pt}OF\ U\isactrlsub b{\isacharunderscore}{\kern0pt}conv{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}{\isacharbrackright}{\kern0pt}\ U\isactrlsub b{\isacharunderscore}{\kern0pt}conv{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ LIMSEQ{\isacharunderscore}{\kern0pt}unique{\isacharbrackleft}{\kern0pt}OF\ U\isactrlsub b{\isacharunderscore}{\kern0pt}tendsto\ U\isactrlsub b{\isacharunderscore}{\kern0pt}conv{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isacharbrackleft}{\kern0pt}of\ m{\isacharbrackright}{\kern0pt}{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ dist{\isacharunderscore}{\kern0pt}U\isactrlsub b{\isacharunderscore}{\kern0pt}opt{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}dist\ {\isacharparenleft}{\kern0pt}U\isactrlsub b\ m\ v{\isacharparenright}{\kern0pt}\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt\ {\isasymle}\ l{\isacharcircum}{\kern0pt}m\ {\isacharasterisk}{\kern0pt}\ dist\ v\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}dist\ {\isacharparenleft}{\kern0pt}U\isactrlsub b\ m\ v{\isacharparenright}{\kern0pt}\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt\ {\isacharequal}{\kern0pt}\ dist\ {\isacharparenleft}{\kern0pt}U\isactrlsub b\ m\ v{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}U\isactrlsub b\ m\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ U\isactrlsub b{\isachardot}{\kern0pt}abs{\isacharunderscore}{\kern0pt}eq\ U\isactrlsub b{\isacharunderscore}{\kern0pt}fix{\isacharunderscore}{\kern0pt}unique\ U{\isacharunderscore}{\kern0pt}zero\ apply{\isacharunderscore}{\kern0pt}bfun{\isacharunderscore}{\kern0pt}inverse\ not{\isadigit{0}}{\isacharunderscore}{\kern0pt}implies{\isacharunderscore}{\kern0pt}Suc{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isasymle}\ l{\isacharcircum}{\kern0pt}m\ {\isacharasterisk}{\kern0pt}\ dist\ v\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}meson\ U\isactrlsub b{\isacharunderscore}{\kern0pt}contraction{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{finally}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\ \isacommand{{\isachardot}{\kern0pt}}\isamarkupfalse%
-\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Expressing a Single Step of Modified Policy Iteration%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\begin{isamarkuptext}%
-The function \isa{W} equals the value computed by the Modified Policy Iteration Algorithm
-  in a single iteration.
-  The right hand addend in the definition describes the advantage of using the optimal action for 
-  the first m steps.%
-\end{isamarkuptext}\isamarkuptrue%
-\isacommand{definition}\isamarkupfalse%
-\ {\isachardoublequoteopen}W\ d\ m\ v\ {\isacharequal}{\kern0pt}\ v\ {\isacharplus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSum}i\ {\isacharless}{\kern0pt}\ m{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isasymP}\isactrlsub {\isadigit{1}}\ d{\isacharparenright}{\kern0pt}{\isacharcircum}{\kern0pt}{\isacharcircum}{\kern0pt}i{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}B\isactrlsub b\ v{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ W{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}L{\isacharunderscore}{\kern0pt}iter{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}{\isasymnu}{\isacharunderscore}{\kern0pt}improving\ v\ d{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}W\ d\ m\ v\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}L\ d{\isacharcircum}{\kern0pt}{\isacharcircum}{\kern0pt}m{\isacharparenright}{\kern0pt}\ v{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isasymSum}i{\isacharless}{\kern0pt}m{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isasymP}\isactrlsub {\isadigit{1}}\ d{\isacharparenright}{\kern0pt}{\isacharcircum}{\kern0pt}{\isacharcircum}{\kern0pt}i{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ v{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSum}i{\isacharless}{\kern0pt}m{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isasymP}\isactrlsub {\isadigit{1}}\ d{\isacharparenright}{\kern0pt}{\isacharcircum}{\kern0pt}{\isacharcircum}{\kern0pt}i{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}L\ d\ v{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isasymnu}{\isacharunderscore}{\kern0pt}improving{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}{\isasymL}\isactrlsub b\ assms\ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}W\ d\ m\ v\ {\isacharequal}{\kern0pt}\ \ v\ {\isacharplus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}{\isasymSum}i{\isacharless}{\kern0pt}m{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isasymP}\isactrlsub {\isadigit{1}}\ d{\isacharparenright}{\kern0pt}{\isacharcircum}{\kern0pt}{\isacharcircum}{\kern0pt}i{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}L\ d\ v{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharminus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSum}i{\isacharless}{\kern0pt}m{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isasymP}\isactrlsub {\isadigit{1}}\ d{\isacharparenright}{\kern0pt}{\isacharcircum}{\kern0pt}{\isacharcircum}{\kern0pt}i{\isacharparenright}{\kern0pt}\ v{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ W{\isacharunderscore}{\kern0pt}def\ B\isactrlsub b{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}{\isasymL}\isactrlsub b\ blinfun{\isachardot}{\kern0pt}bilinear{\isacharunderscore}{\kern0pt}simps\ algebra{\isacharunderscore}{\kern0pt}simps\ {\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isacharequal}{\kern0pt}\ v\ {\isacharplus}{\kern0pt}\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}fin\ {\isacharparenleft}{\kern0pt}mk{\isacharunderscore}{\kern0pt}stationary\ d{\isacharparenright}{\kern0pt}\ m\ {\isacharplus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSum}i{\isacharless}{\kern0pt}m{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isasymP}\isactrlsub {\isadigit{1}}\ d{\isacharparenright}{\kern0pt}{\isacharcircum}{\kern0pt}{\isacharcircum}{\kern0pt}i{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isasymP}\isactrlsub {\isadigit{1}}\ d{\isacharparenright}{\kern0pt}\ v{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharminus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSum}i{\isacharless}{\kern0pt}m{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isasymP}\isactrlsub {\isadigit{1}}\ d{\isacharparenright}{\kern0pt}{\isacharcircum}{\kern0pt}{\isacharcircum}{\kern0pt}i{\isacharparenright}{\kern0pt}\ v{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ L{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}fin{\isacharunderscore}{\kern0pt}eq\ blinfun{\isachardot}{\kern0pt}bilinear{\isacharunderscore}{\kern0pt}simps\ blinfun{\isachardot}{\kern0pt}sum{\isacharunderscore}{\kern0pt}left\ scaleR{\isacharunderscore}{\kern0pt}right{\isachardot}{\kern0pt}sum{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isacharequal}{\kern0pt}\ v\ {\isacharplus}{\kern0pt}\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}fin\ {\isacharparenleft}{\kern0pt}mk{\isacharunderscore}{\kern0pt}stationary\ d{\isacharparenright}{\kern0pt}\ m\ {\isacharplus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSum}i{\isacharless}{\kern0pt}m{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isasymP}\isactrlsub {\isadigit{1}}\ d{\isacharparenright}{\kern0pt}{\isacharcircum}{\kern0pt}{\isacharcircum}{\kern0pt}Suc\ i{\isacharparenright}{\kern0pt}\ v{\isacharparenright}{\kern0pt}\ {\isacharminus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSum}i{\isacharless}{\kern0pt}m{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isasymP}\isactrlsub {\isadigit{1}}\ d{\isacharparenright}{\kern0pt}{\isacharcircum}{\kern0pt}{\isacharcircum}{\kern0pt}i{\isacharparenright}{\kern0pt}\ v{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp\ del{\isacharcolon}{\kern0pt}\ blinfunpow{\isachardot}{\kern0pt}simps\ simp{\isacharcolon}{\kern0pt}\ blinfunpow{\isacharunderscore}{\kern0pt}assoc{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isacharequal}{\kern0pt}\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}fin\ {\isacharparenleft}{\kern0pt}mk{\isacharunderscore}{\kern0pt}stationary\ d{\isacharparenright}{\kern0pt}\ m\ {\isacharplus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSum}i{\isacharless}{\kern0pt}Suc\ m{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isasymP}\isactrlsub {\isadigit{1}}\ d{\isacharparenright}{\kern0pt}{\isacharcircum}{\kern0pt}{\isacharcircum}{\kern0pt}\ i{\isacharparenright}{\kern0pt}\ v{\isacharparenright}{\kern0pt}\ \ {\isacharminus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSum}i{\isacharless}{\kern0pt}m{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isasymP}\isactrlsub {\isadigit{1}}\ d{\isacharparenright}{\kern0pt}{\isacharcircum}{\kern0pt}{\isacharcircum}{\kern0pt}\ i{\isacharparenright}{\kern0pt}\ v{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}subst\ sum{\isachardot}{\kern0pt}lessThan{\isacharunderscore}{\kern0pt}Suc{\isacharunderscore}{\kern0pt}shift{\isacharparenright}{\kern0pt}\ auto\isanewline
-\ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isacharequal}{\kern0pt}\ \ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}fin\ {\isacharparenleft}{\kern0pt}mk{\isacharunderscore}{\kern0pt}stationary\ d{\isacharparenright}{\kern0pt}\ m\ {\isacharplus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isasymP}\isactrlsub {\isadigit{1}}\ d{\isacharparenright}{\kern0pt}{\isacharcircum}{\kern0pt}{\isacharcircum}{\kern0pt}m{\isacharparenright}{\kern0pt}\ v{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ blinfun{\isachardot}{\kern0pt}sum{\isacharunderscore}{\kern0pt}left{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}L\ d\ {\isacharcircum}{\kern0pt}{\isacharcircum}{\kern0pt}\ m{\isacharparenright}{\kern0pt}\ v{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ L{\isacharunderscore}{\kern0pt}iter\ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{finally}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\ \isacommand{{\isachardot}{\kern0pt}}\isamarkupfalse%
-\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ W{\isacharunderscore}{\kern0pt}le{\isacharunderscore}{\kern0pt}U\isactrlsub b{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}v\ {\isasymle}\ u{\isachardoublequoteclose}\ {\isachardoublequoteopen}{\isasymnu}{\isacharunderscore}{\kern0pt}improving\ v\ d{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}W\ d\ m\ v\ {\isasymle}\ U\isactrlsub b\ m\ u{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}U\isactrlsub b\ m\ u\ {\isacharminus}{\kern0pt}\ W\ d\ m\ v\ {\isasymge}\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}fin\ {\isacharparenleft}{\kern0pt}mk{\isacharunderscore}{\kern0pt}stationary\ d{\isacharparenright}{\kern0pt}\ m\ {\isacharplus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isasymP}\isactrlsub {\isadigit{1}}\ d{\isacharparenright}{\kern0pt}\ {\isacharcircum}{\kern0pt}{\isacharcircum}{\kern0pt}\ m{\isacharparenright}{\kern0pt}\ u\ {\isacharminus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}fin\ {\isacharparenleft}{\kern0pt}mk{\isacharunderscore}{\kern0pt}stationary\ d{\isacharparenright}{\kern0pt}\ m\ {\isacharplus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isasymP}\isactrlsub {\isadigit{1}}\ d{\isacharparenright}{\kern0pt}{\isacharcircum}{\kern0pt}{\isacharcircum}{\kern0pt}m{\isacharparenright}{\kern0pt}\ v{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isasymnu}{\isacharunderscore}{\kern0pt}improving{\isacharunderscore}{\kern0pt}D{\isacharunderscore}{\kern0pt}MR\ assms{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}\ bounded{\isacharunderscore}{\kern0pt}stationary{\isacharunderscore}{\kern0pt}{\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}fin\ bounded{\isacharunderscore}{\kern0pt}disc{\isacharunderscore}{\kern0pt}{\isasymP}\isactrlsub {\isadigit{1}}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}fastforce\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ diff{\isacharunderscore}{\kern0pt}mono\ bounded{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}bdd{\isacharunderscore}{\kern0pt}above\ cSUP{\isacharunderscore}{\kern0pt}upper\ bounded{\isacharunderscore}{\kern0pt}plus{\isacharunderscore}{\kern0pt}comp\ simp{\isacharcolon}{\kern0pt}\ U\isactrlsub b{\isachardot}{\kern0pt}rep{\isacharunderscore}{\kern0pt}eq\ U{\isacharunderscore}{\kern0pt}def\ L{\isacharunderscore}{\kern0pt}iter\ W{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}L{\isacharunderscore}{\kern0pt}iter{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isacharasterisk}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}U\isactrlsub b\ m\ u\ {\isacharminus}{\kern0pt}\ W\ d\ m\ v\ {\isasymge}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isasymP}\isactrlsub {\isadigit{1}}\ d{\isacharparenright}{\kern0pt}\ {\isacharcircum}{\kern0pt}{\isacharcircum}{\kern0pt}\ m{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}u\ {\isacharminus}{\kern0pt}\ v{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ blinfun{\isachardot}{\kern0pt}diff{\isacharunderscore}{\kern0pt}right{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}W\ d\ m\ v\ {\isasymle}\ U\isactrlsub b\ m\ u{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ order{\isachardot}{\kern0pt}trans{\isacharbrackleft}{\kern0pt}OF\ {\isasymP}\isactrlsub {\isadigit{1}}{\isacharunderscore}{\kern0pt}n{\isacharunderscore}{\kern0pt}disc{\isacharunderscore}{\kern0pt}pos{\isacharbrackleft}{\kern0pt}unfolded\ blincomp{\isacharunderscore}{\kern0pt}scaleR{\isacharunderscore}{\kern0pt}right{\isacharbrackleft}{\kern0pt}symmetric{\isacharbrackright}{\kern0pt}{\isacharbrackright}{\kern0pt}\ {\isacharasterisk}{\kern0pt}{\isacharbrackright}{\kern0pt}\ assms\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ W{\isacharunderscore}{\kern0pt}ge{\isacharunderscore}{\kern0pt}{\isasymL}\isactrlsub b{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}v\ {\isasymle}\ u{\isachardoublequoteclose}\ {\isachardoublequoteopen}{\isadigit{0}}\ {\isasymle}\ B\isactrlsub b\ u{\isachardoublequoteclose}\ {\isachardoublequoteopen}{\isasymnu}{\isacharunderscore}{\kern0pt}improving\ u\ d{\isacharprime}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}{\isasymL}\isactrlsub b\ v\ {\isasymle}\ W\ d{\isacharprime}{\kern0pt}\ {\isacharparenleft}{\kern0pt}Suc\ m{\isacharparenright}{\kern0pt}\ u{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymL}\isactrlsub b\ v\ {\isasymle}\ u\ {\isacharplus}{\kern0pt}\ B\isactrlsub b\ u{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assms{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}mono\ B\isactrlsub b{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}{\isasymL}\isactrlsub b\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isasymle}\ W\ d{\isacharprime}{\kern0pt}\ {\isacharparenleft}{\kern0pt}Suc\ m{\isacharparenright}{\kern0pt}\ u{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ L{\isacharunderscore}{\kern0pt}mono\ {\isasymnu}{\isacharunderscore}{\kern0pt}improving{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}{\isasymL}\isactrlsub b\ assms{\isacharparenleft}{\kern0pt}{\isadigit{3}}{\isacharparenright}{\kern0pt}\ assms\ \isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}induction\ m{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ W{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}L{\isacharunderscore}{\kern0pt}iter\ B\isactrlsub b{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}{\isasymL}\isactrlsub b{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{finally}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\ \isacommand{{\isachardot}{\kern0pt}}\isamarkupfalse%
-\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ B\isactrlsub b{\isacharunderscore}{\kern0pt}le{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}{\isasymnu}{\isacharunderscore}{\kern0pt}improving\ v\ d{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}B\isactrlsub b\ v\ {\isacharplus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isasymP}\isactrlsub {\isadigit{1}}\ d\ {\isacharminus}{\kern0pt}\ id{\isacharunderscore}{\kern0pt}blinfun{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}u\ {\isacharminus}{\kern0pt}\ v{\isacharparenright}{\kern0pt}\ {\isasymle}\ B\isactrlsub b\ u{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}r{\isacharunderscore}{\kern0pt}dec\isactrlsub b\ d\ {\isacharplus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isasymP}\isactrlsub {\isadigit{1}}\ d\ {\isacharminus}{\kern0pt}\ id{\isacharunderscore}{\kern0pt}blinfun{\isacharparenright}{\kern0pt}\ u\ {\isasymle}\ B\isactrlsub b\ u{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ L{\isacharunderscore}{\kern0pt}def\ L{\isacharunderscore}{\kern0pt}le{\isacharunderscore}{\kern0pt}{\isasymL}\isactrlsub b\ assms\ \ \ \ \ \isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ B\isactrlsub b{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}{\isasymL}\isactrlsub b\ {\isasymL}\isactrlsub b{\isachardot}{\kern0pt}rep{\isacharunderscore}{\kern0pt}eq\ {\isasymL}{\isacharunderscore}{\kern0pt}def\ blinfun{\isachardot}{\kern0pt}bilinear{\isacharunderscore}{\kern0pt}simps{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}B\isactrlsub b\ v\ {\isacharequal}{\kern0pt}\ r{\isacharunderscore}{\kern0pt}dec\isactrlsub b\ d\ {\isacharplus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isasymP}\isactrlsub {\isadigit{1}}\ d\ {\isacharminus}{\kern0pt}\ id{\isacharunderscore}{\kern0pt}blinfun{\isacharparenright}{\kern0pt}\ v{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assms\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ B\isactrlsub b{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}{\isasymL}\isactrlsub b\ {\isasymnu}{\isacharunderscore}{\kern0pt}improving{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}{\isasymL}\isactrlsub b{\isacharbrackleft}{\kern0pt}of\ {\isacharunderscore}{\kern0pt}\ d{\isacharbrackright}{\kern0pt}\ L{\isacharunderscore}{\kern0pt}def\ blinfun{\isachardot}{\kern0pt}bilinear{\isacharunderscore}{\kern0pt}simps{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ blinfun{\isachardot}{\kern0pt}diff{\isacharunderscore}{\kern0pt}right{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ {\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}W{\isacharunderscore}{\kern0pt}ge{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}u\ {\isasymle}\ {\isasymL}\isactrlsub b\ u{\isachardoublequoteclose}\ {\isachardoublequoteopen}{\isasymnu}{\isacharunderscore}{\kern0pt}improving\ u\ d{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}W\ d\ m\ u\ {\isasymle}\ {\isasymL}\isactrlsub b\ {\isacharparenleft}{\kern0pt}W\ d\ m\ u{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isadigit{0}}\ {\isasymle}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isasymP}\isactrlsub {\isadigit{1}}\ d{\isacharparenright}{\kern0pt}\ {\isacharcircum}{\kern0pt}{\isacharcircum}{\kern0pt}\ m{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}B\isactrlsub b\ u{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ B\isactrlsub b{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}{\isasymL}\isactrlsub b\ {\isasymP}\isactrlsub {\isadigit{1}}{\isacharunderscore}{\kern0pt}n{\isacharunderscore}{\kern0pt}disc{\isacharunderscore}{\kern0pt}pos\ assms{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}\ blincomp{\isacharunderscore}{\kern0pt}scaleR{\isacharunderscore}{\kern0pt}right\ diff{\isacharunderscore}{\kern0pt}ge{\isacharunderscore}{\kern0pt}{\isadigit{0}}{\isacharunderscore}{\kern0pt}iff{\isacharunderscore}{\kern0pt}ge{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isasymP}\isactrlsub {\isadigit{1}}\ d{\isacharparenright}{\kern0pt}{\isacharcircum}{\kern0pt}{\isacharcircum}{\kern0pt}{\isadigit{0}}\ {\isacharplus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSum}i\ {\isacharless}{\kern0pt}\ m{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isasymP}\isactrlsub {\isadigit{1}}\ d{\isacharparenright}{\kern0pt}{\isacharcircum}{\kern0pt}{\isacharcircum}{\kern0pt}{\isacharparenleft}{\kern0pt}Suc\ i{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}B\isactrlsub b\ u{\isacharparenright}{\kern0pt}\ {\isacharminus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSum}i\ {\isacharless}{\kern0pt}\ m{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isasymP}\isactrlsub {\isadigit{1}}\ d{\isacharparenright}{\kern0pt}{\isacharcircum}{\kern0pt}{\isacharcircum}{\kern0pt}\ i{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}B\isactrlsub b\ u{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}subst\ sum{\isachardot}{\kern0pt}lessThan{\isacharunderscore}{\kern0pt}Suc{\isacharunderscore}{\kern0pt}shift{\isacharbrackleft}{\kern0pt}symmetric{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ blinfun{\isachardot}{\kern0pt}diff{\isacharunderscore}{\kern0pt}left{\isacharbrackleft}{\kern0pt}symmetric{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isacharequal}{\kern0pt}\ B\isactrlsub b\ u\ {\isacharplus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isasymP}\isactrlsub {\isadigit{1}}\ d\ {\isacharminus}{\kern0pt}\ id{\isacharunderscore}{\kern0pt}blinfun{\isacharparenright}{\kern0pt}\ o\isactrlsub L\ {\isacharparenleft}{\kern0pt}{\isasymSum}i\ {\isacharless}{\kern0pt}\ m{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isasymP}\isactrlsub {\isadigit{1}}\ d{\isacharparenright}{\kern0pt}{\isacharcircum}{\kern0pt}{\isacharcircum}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}B\isactrlsub b\ u{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ blinfun{\isachardot}{\kern0pt}bilinear{\isacharunderscore}{\kern0pt}simps\ sum{\isacharunderscore}{\kern0pt}subtractf{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isacharequal}{\kern0pt}\ B\isactrlsub b\ u\ {\isacharplus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isasymP}\isactrlsub {\isadigit{1}}\ d\ {\isacharminus}{\kern0pt}\ id{\isacharunderscore}{\kern0pt}blinfun{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}W\ d\ m\ u\ {\isacharminus}{\kern0pt}\ u{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ W{\isacharunderscore}{\kern0pt}def\ sum{\isachardot}{\kern0pt}lessThan{\isacharunderscore}{\kern0pt}Suc{\isacharbrackleft}{\kern0pt}unfolded\ lessThan{\isacharunderscore}{\kern0pt}Suc{\isacharunderscore}{\kern0pt}atMost{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isasymle}\ B\isactrlsub b\ {\isacharparenleft}{\kern0pt}W\ d\ m\ u{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ B\isactrlsub b{\isacharunderscore}{\kern0pt}le\ assms{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}\ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{finally}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isadigit{0}}\ {\isasymle}\ B\isactrlsub b\ {\isacharparenleft}{\kern0pt}W\ d\ m\ u{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isacommand{{\isachardot}{\kern0pt}}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\ \isacommand{using}\isamarkupfalse%
-\ B\isactrlsub b{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}{\isasymL}\isactrlsub b\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Computing the Bellman Operator over Multiple Steps%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{definition}\isamarkupfalse%
-\ L{\isacharunderscore}{\kern0pt}pow\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}s\ {\isasymRightarrow}\isactrlsub b\ real{\isacharparenright}{\kern0pt}\ {\isasymRightarrow}\ {\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}s\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a{\isacharparenright}{\kern0pt}\ {\isasymRightarrow}\ nat\ {\isasymRightarrow}\ {\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}s\ {\isasymRightarrow}\isactrlsub b\ real{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}L{\isacharunderscore}{\kern0pt}pow\ v\ d\ m\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}L\ {\isacharparenleft}{\kern0pt}mk{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\ d{\isacharparenright}{\kern0pt}\ {\isacharcircum}{\kern0pt}{\isacharcircum}{\kern0pt}\ Suc\ m{\isacharparenright}{\kern0pt}\ v{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ sum{\isacharunderscore}{\kern0pt}telescope{\isacharprime}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isasymSum}i{\isasymle}k{\isachardot}{\kern0pt}\ f\ {\isacharparenleft}{\kern0pt}Suc\ i{\isacharparenright}{\kern0pt}\ {\isacharminus}{\kern0pt}\ f\ i\ {\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ f\ {\isacharparenleft}{\kern0pt}Suc\ k{\isacharparenright}{\kern0pt}\ {\isacharminus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}f\ {\isadigit{0}}\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isacharprime}{\kern0pt}c\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ ab{\isacharunderscore}{\kern0pt}group{\isacharunderscore}{\kern0pt}add{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ sum{\isacharunderscore}{\kern0pt}telescope{\isacharbrackleft}{\kern0pt}of\ {\isachardoublequoteopen}{\isacharminus}{\kern0pt}f{\isachardoublequoteclose}\ k{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ L{\isacharunderscore}{\kern0pt}pow{\isacharunderscore}{\kern0pt}eq{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}{\isasymnu}{\isacharunderscore}{\kern0pt}improving\ v\ {\isacharparenleft}{\kern0pt}mk{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\ d{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}L{\isacharunderscore}{\kern0pt}pow\ v\ d\ m\ {\isacharequal}{\kern0pt}\ v\ {\isacharplus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSum}i\ {\isasymle}\ m{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isasymP}\isactrlsub {\isadigit{1}}\ {\isacharparenleft}{\kern0pt}mk{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\ d{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharcircum}{\kern0pt}{\isacharcircum}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}B\isactrlsub b\ v{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}d\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}mk{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\ d{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isasymSum}i\ {\isasymle}\ m{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isasymP}\isactrlsub {\isadigit{1}}\ {\isacharquery}{\kern0pt}d{\isacharparenright}{\kern0pt}{\isacharcircum}{\kern0pt}{\isacharcircum}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}B\isactrlsub b\ v{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSum}i\ {\isasymle}\ m{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isasymP}\isactrlsub {\isadigit{1}}\ {\isacharquery}{\kern0pt}d{\isacharparenright}{\kern0pt}{\isacharcircum}{\kern0pt}{\isacharcircum}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}L\ {\isacharquery}{\kern0pt}d\ v{\isacharparenright}{\kern0pt}\ {\isacharminus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSum}i\ {\isasymle}\ m{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isasymP}\isactrlsub {\isadigit{1}}\ {\isacharquery}{\kern0pt}d{\isacharparenright}{\kern0pt}{\isacharcircum}{\kern0pt}{\isacharcircum}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ v{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assms\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ B\isactrlsub b{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}{\isasymL}\isactrlsub b\ blinfun{\isachardot}{\kern0pt}bilinear{\isacharunderscore}{\kern0pt}simps\ {\isasymnu}{\isacharunderscore}{\kern0pt}improving{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}{\isasymL}\isactrlsub b{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSum}i\ {\isasymle}\ m{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isasymP}\isactrlsub {\isadigit{1}}\ {\isacharquery}{\kern0pt}d{\isacharparenright}{\kern0pt}{\isacharcircum}{\kern0pt}{\isacharcircum}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}r{\isacharunderscore}{\kern0pt}dec\isactrlsub b\ {\isacharquery}{\kern0pt}d{\isacharparenright}{\kern0pt}\ {\isacharplus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSum}i\ {\isasymle}\ m{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isasymP}\isactrlsub {\isadigit{1}}\ {\isacharquery}{\kern0pt}d{\isacharparenright}{\kern0pt}{\isacharcircum}{\kern0pt}{\isacharcircum}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isasymP}\isactrlsub {\isadigit{1}}\ {\isacharquery}{\kern0pt}d{\isacharparenright}{\kern0pt}\ v{\isacharparenright}{\kern0pt}\ {\isacharminus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSum}i\ {\isasymle}\ m{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isasymP}\isactrlsub {\isadigit{1}}\ {\isacharquery}{\kern0pt}d{\isacharparenright}{\kern0pt}{\isacharcircum}{\kern0pt}{\isacharcircum}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ v{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ L{\isacharunderscore}{\kern0pt}def\ blinfun{\isachardot}{\kern0pt}bilinear{\isacharunderscore}{\kern0pt}simps{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSum}i\ {\isasymle}\ m{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isasymP}\isactrlsub {\isadigit{1}}\ {\isacharquery}{\kern0pt}d{\isacharparenright}{\kern0pt}{\isacharcircum}{\kern0pt}{\isacharcircum}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}r{\isacharunderscore}{\kern0pt}dec\isactrlsub b\ {\isacharquery}{\kern0pt}d{\isacharparenright}{\kern0pt}\ {\isacharplus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSum}i\ {\isasymle}\ m{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isasymP}\isactrlsub {\isadigit{1}}\ {\isacharquery}{\kern0pt}d{\isacharparenright}{\kern0pt}{\isacharcircum}{\kern0pt}{\isacharcircum}{\kern0pt}Suc\ i{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ v\ {\isacharminus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSum}i\ {\isasymle}\ m{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isasymP}\isactrlsub {\isadigit{1}}\ {\isacharquery}{\kern0pt}d{\isacharparenright}{\kern0pt}{\isacharcircum}{\kern0pt}{\isacharcircum}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ v{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ blinfun{\isachardot}{\kern0pt}sum{\isacharunderscore}{\kern0pt}left\ blinfunpow{\isacharunderscore}{\kern0pt}assoc\ simp\ del{\isacharcolon}{\kern0pt}\ blinfunpow{\isachardot}{\kern0pt}simps{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSum}i\ {\isasymle}\ m{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isasymP}\isactrlsub {\isadigit{1}}\ {\isacharquery}{\kern0pt}d{\isacharparenright}{\kern0pt}{\isacharcircum}{\kern0pt}{\isacharcircum}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}r{\isacharunderscore}{\kern0pt}dec\isactrlsub b\ {\isacharquery}{\kern0pt}d{\isacharparenright}{\kern0pt}\ {\isacharplus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSum}i\ {\isasymle}\ m{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isasymP}\isactrlsub {\isadigit{1}}\ {\isacharquery}{\kern0pt}d{\isacharparenright}{\kern0pt}{\isacharcircum}{\kern0pt}{\isacharcircum}{\kern0pt}Suc\ i{\isacharparenright}{\kern0pt}\ {\isacharminus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isasymP}\isactrlsub {\isadigit{1}}\ {\isacharquery}{\kern0pt}d{\isacharparenright}{\kern0pt}{\isacharcircum}{\kern0pt}{\isacharcircum}{\kern0pt}i{\isacharparenright}{\kern0pt}\ v{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ blinfun{\isachardot}{\kern0pt}diff{\isacharunderscore}{\kern0pt}left\ sum{\isacharunderscore}{\kern0pt}subtractf{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSum}i\ {\isasymle}\ m{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isasymP}\isactrlsub {\isadigit{1}}\ {\isacharquery}{\kern0pt}d{\isacharparenright}{\kern0pt}{\isacharcircum}{\kern0pt}{\isacharcircum}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}r{\isacharunderscore}{\kern0pt}dec\isactrlsub b\ {\isacharquery}{\kern0pt}d{\isacharparenright}{\kern0pt}\ {\isacharplus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isasymP}\isactrlsub {\isadigit{1}}\ {\isacharquery}{\kern0pt}d{\isacharparenright}{\kern0pt}{\isacharcircum}{\kern0pt}{\isacharcircum}{\kern0pt}Suc\ m{\isacharparenright}{\kern0pt}\ v\ {\isacharminus}{\kern0pt}\ v{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}subst\ sum{\isacharunderscore}{\kern0pt}telescope{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ blinfun{\isachardot}{\kern0pt}bilinear{\isacharunderscore}{\kern0pt}simps{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{finally}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isasymSum}i\ {\isasymle}\ m{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isasymP}\isactrlsub {\isadigit{1}}\ {\isacharquery}{\kern0pt}d{\isacharparenright}{\kern0pt}{\isacharcircum}{\kern0pt}{\isacharcircum}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}B\isactrlsub b\ v{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSum}i\ {\isasymle}\ m{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isasymP}\isactrlsub {\isadigit{1}}\ {\isacharquery}{\kern0pt}d{\isacharparenright}{\kern0pt}{\isacharcircum}{\kern0pt}{\isacharcircum}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}r{\isacharunderscore}{\kern0pt}dec\isactrlsub b\ {\isacharquery}{\kern0pt}d{\isacharparenright}{\kern0pt}\ {\isacharplus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isasymP}\isactrlsub {\isadigit{1}}\ {\isacharquery}{\kern0pt}d{\isacharparenright}{\kern0pt}{\isacharcircum}{\kern0pt}{\isacharcircum}{\kern0pt}Suc\ m{\isacharparenright}{\kern0pt}\ v\ {\isacharminus}{\kern0pt}\ v{\isachardoublequoteclose}\ \isacommand{{\isachardot}{\kern0pt}}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}L{\isacharunderscore}{\kern0pt}pow\ v\ d\ m\ {\isacharequal}{\kern0pt}\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}fin\ {\isacharparenleft}{\kern0pt}mk{\isacharunderscore}{\kern0pt}stationary{\isacharunderscore}{\kern0pt}det\ d{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}Suc\ m{\isacharparenright}{\kern0pt}\ {\isacharplus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isasymP}\isactrlsub {\isadigit{1}}\ {\isacharquery}{\kern0pt}d{\isacharparenright}{\kern0pt}{\isacharcircum}{\kern0pt}{\isacharcircum}{\kern0pt}Suc\ m{\isacharparenright}{\kern0pt}\ v{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ only{\isacharcolon}{\kern0pt}\ L{\isacharunderscore}{\kern0pt}pow{\isacharunderscore}{\kern0pt}def\ L{\isacharunderscore}{\kern0pt}iter\ lessThan{\isacharunderscore}{\kern0pt}Suc{\isacharunderscore}{\kern0pt}atMost{\isacharbrackleft}{\kern0pt}symmetric{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}fin{\isacharunderscore}{\kern0pt}eq\ lessThan{\isacharunderscore}{\kern0pt}Suc{\isacharunderscore}{\kern0pt}atMost{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ L{\isacharunderscore}{\kern0pt}pow{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}W{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}d\ {\isasymin}\ D\isactrlsub D{\isachardoublequoteclose}\ \isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}L{\isacharunderscore}{\kern0pt}pow\ v\ {\isacharparenleft}{\kern0pt}policy{\isacharunderscore}{\kern0pt}improvement\ d\ v{\isacharparenright}{\kern0pt}\ m\ {\isacharequal}{\kern0pt}\ W\ {\isacharparenleft}{\kern0pt}mk{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\ {\isacharparenleft}{\kern0pt}policy{\isacharunderscore}{\kern0pt}improvement\ d\ v{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}Suc\ m{\isacharparenright}{\kern0pt}\ v{\isachardoublequoteclose}\ \isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ assms\ policy{\isacharunderscore}{\kern0pt}improvement{\isacharunderscore}{\kern0pt}improving\ \isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ W{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}L{\isacharunderscore}{\kern0pt}iter\ L{\isacharunderscore}{\kern0pt}pow{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ L{\isacharunderscore}{\kern0pt}pow{\isacharunderscore}{\kern0pt}{\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}mono{\isacharunderscore}{\kern0pt}inv{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}d\ {\isasymin}\ D\isactrlsub D{\isachardoublequoteclose}\ {\isachardoublequoteopen}v\ {\isasymle}\ {\isasymL}\isactrlsub b\ v{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}L{\isacharunderscore}{\kern0pt}pow\ v\ {\isacharparenleft}{\kern0pt}policy{\isacharunderscore}{\kern0pt}improvement\ d\ v{\isacharparenright}{\kern0pt}\ m\ {\isasymle}\ {\isasymL}\isactrlsub b\ {\isacharparenleft}{\kern0pt}L{\isacharunderscore}{\kern0pt}pow\ v\ {\isacharparenleft}{\kern0pt}policy{\isacharunderscore}{\kern0pt}improvement\ d\ v{\isacharparenright}{\kern0pt}\ m{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ assms\ L{\isacharunderscore}{\kern0pt}pow{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}W\ {\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}W{\isacharunderscore}{\kern0pt}ge\ policy{\isacharunderscore}{\kern0pt}improvement{\isacharunderscore}{\kern0pt}improving\ \isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{The Modified Policy Iteration Algorithm%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{context}\isamarkupfalse%
-\isanewline
-\ \ \isakeyword{fixes}\ d{\isadigit{0}}\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}s\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{fixes}\ v{\isadigit{0}}\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}s\ {\isasymRightarrow}\isactrlsub b\ real{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{fixes}\ m\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}nat\ {\isasymRightarrow}\ {\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}s\ {\isasymRightarrow}\isactrlsub b\ real{\isacharparenright}{\kern0pt}\ {\isasymRightarrow}\ nat{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{assumes}\ d{\isadigit{0}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}d{\isadigit{0}}\ {\isasymin}\ D\isactrlsub D{\isachardoublequoteclose}\isanewline
-\isakeyword{begin}%
-\begin{isamarkuptext}%
-We first define a function that executes the algorithm for n steps.%
-\end{isamarkuptext}\isamarkuptrue%
-\isacommand{fun}\isamarkupfalse%
-\ mpi\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}nat\ {\isasymRightarrow}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}s\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a{\isacharparenright}{\kern0pt}\ {\isasymtimes}\ {\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}s\ {\isasymRightarrow}\isactrlsub b\ real{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}mpi\ {\isadigit{0}}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}policy{\isacharunderscore}{\kern0pt}improvement\ d{\isadigit{0}}\ v{\isadigit{0}}{\isacharcomma}{\kern0pt}\ v{\isadigit{0}}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ {\isacharbar}{\kern0pt}\isanewline
-\ \ {\isachardoublequoteopen}mpi\ {\isacharparenleft}{\kern0pt}Suc\ n{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\isanewline
-\ \ {\isacharparenleft}{\kern0pt}let\ {\isacharparenleft}{\kern0pt}d{\isacharcomma}{\kern0pt}\ v{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ mpi\ n{\isacharsemicolon}{\kern0pt}\ v{\isacharprime}{\kern0pt}\ {\isacharequal}{\kern0pt}\ L{\isacharunderscore}{\kern0pt}pow\ v\ d\ {\isacharparenleft}{\kern0pt}m\ n\ v{\isacharparenright}{\kern0pt}\ in\isanewline
-\ \ {\isacharparenleft}{\kern0pt}policy{\isacharunderscore}{\kern0pt}improvement\ d\ v{\isacharprime}{\kern0pt}{\isacharcomma}{\kern0pt}\ v{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ {\isachardoublequoteopen}mpi{\isacharunderscore}{\kern0pt}val\ n\ {\isacharequal}{\kern0pt}\ snd\ {\isacharparenleft}{\kern0pt}mpi\ n{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ {\isachardoublequoteopen}mpi{\isacharunderscore}{\kern0pt}pol\ n\ {\isacharequal}{\kern0pt}\ fst\ {\isacharparenleft}{\kern0pt}mpi\ n{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ mpi{\isacharunderscore}{\kern0pt}pol{\isacharunderscore}{\kern0pt}zero{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}mpi{\isacharunderscore}{\kern0pt}pol\ {\isadigit{0}}\ {\isacharequal}{\kern0pt}\ policy{\isacharunderscore}{\kern0pt}improvement\ d{\isadigit{0}}\ v{\isadigit{0}}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ mpi{\isacharunderscore}{\kern0pt}pol{\isacharunderscore}{\kern0pt}def\ \isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ mpi{\isacharunderscore}{\kern0pt}pol{\isacharunderscore}{\kern0pt}Suc{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}mpi{\isacharunderscore}{\kern0pt}pol\ {\isacharparenleft}{\kern0pt}Suc\ n{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ policy{\isacharunderscore}{\kern0pt}improvement\ {\isacharparenleft}{\kern0pt}mpi{\isacharunderscore}{\kern0pt}pol\ n{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}mpi{\isacharunderscore}{\kern0pt}val\ {\isacharparenleft}{\kern0pt}Suc\ n{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ case{\isacharunderscore}{\kern0pt}prod{\isacharunderscore}{\kern0pt}beta{\isacharprime}{\kern0pt}\ Let{\isacharunderscore}{\kern0pt}def\ mpi{\isacharunderscore}{\kern0pt}pol{\isacharunderscore}{\kern0pt}def\ mpi{\isacharunderscore}{\kern0pt}val{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ mpi{\isacharunderscore}{\kern0pt}pol{\isacharunderscore}{\kern0pt}is{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}mpi{\isacharunderscore}{\kern0pt}pol\ n\ {\isasymin}\ D\isactrlsub D{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ mpi{\isacharunderscore}{\kern0pt}pol{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \isacommand{using}\isamarkupfalse%
-\ policy{\isacharunderscore}{\kern0pt}improvement{\isacharunderscore}{\kern0pt}is{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\ d{\isadigit{0}}\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}induction\ n{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ Let{\isacharunderscore}{\kern0pt}def\ split{\isacharcolon}{\kern0pt}\ prod{\isachardot}{\kern0pt}splits{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ {\isasymnu}{\isacharunderscore}{\kern0pt}improving{\isacharunderscore}{\kern0pt}mpi{\isacharunderscore}{\kern0pt}pol{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymnu}{\isacharunderscore}{\kern0pt}improving\ {\isacharparenleft}{\kern0pt}mpi{\isacharunderscore}{\kern0pt}val\ n{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}mk{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\ {\isacharparenleft}{\kern0pt}mpi{\isacharunderscore}{\kern0pt}pol\ n{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ d{\isadigit{0}}\ policy{\isacharunderscore}{\kern0pt}improvement{\isacharunderscore}{\kern0pt}improving\ mpi{\isacharunderscore}{\kern0pt}pol{\isacharunderscore}{\kern0pt}is{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\ mpi{\isacharunderscore}{\kern0pt}pol{\isacharunderscore}{\kern0pt}Suc\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}cases\ n{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ mpi{\isacharunderscore}{\kern0pt}pol{\isacharunderscore}{\kern0pt}def\ mpi{\isacharunderscore}{\kern0pt}val{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ mpi{\isacharunderscore}{\kern0pt}val{\isacharunderscore}{\kern0pt}zero{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}mpi{\isacharunderscore}{\kern0pt}val\ {\isadigit{0}}\ {\isacharequal}{\kern0pt}\ v{\isadigit{0}}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ mpi{\isacharunderscore}{\kern0pt}val{\isacharunderscore}{\kern0pt}def\ \isacommand{by}\isamarkupfalse%
-\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ mpi{\isacharunderscore}{\kern0pt}val{\isacharunderscore}{\kern0pt}Suc{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}mpi{\isacharunderscore}{\kern0pt}val\ {\isacharparenleft}{\kern0pt}Suc\ n{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ L{\isacharunderscore}{\kern0pt}pow\ {\isacharparenleft}{\kern0pt}mpi{\isacharunderscore}{\kern0pt}val\ n{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}mpi{\isacharunderscore}{\kern0pt}pol\ n{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}m\ n\ {\isacharparenleft}{\kern0pt}mpi{\isacharunderscore}{\kern0pt}val\ n{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ mpi{\isacharunderscore}{\kern0pt}val{\isacharunderscore}{\kern0pt}def\ mpi{\isacharunderscore}{\kern0pt}pol{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ case{\isacharunderscore}{\kern0pt}prod{\isacharunderscore}{\kern0pt}beta{\isacharprime}{\kern0pt}\ Let{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ mpi{\isacharunderscore}{\kern0pt}val{\isacharunderscore}{\kern0pt}eq{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}mpi{\isacharunderscore}{\kern0pt}val\ {\isacharparenleft}{\kern0pt}Suc\ n{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ \isanewline
-\ \ mpi{\isacharunderscore}{\kern0pt}val\ n\ {\isacharplus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSum}i\ {\isasymle}\ m\ n\ {\isacharparenleft}{\kern0pt}mpi{\isacharunderscore}{\kern0pt}val\ n{\isacharparenright}{\kern0pt}{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isasymP}\isactrlsub {\isadigit{1}}\ {\isacharparenleft}{\kern0pt}mk{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\ {\isacharparenleft}{\kern0pt}mpi{\isacharunderscore}{\kern0pt}pol\ n{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharcircum}{\kern0pt}{\isacharcircum}{\kern0pt}\ i{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}B\isactrlsub b\ {\isacharparenleft}{\kern0pt}mpi{\isacharunderscore}{\kern0pt}val\ n{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \ \isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ L{\isacharunderscore}{\kern0pt}pow{\isacharunderscore}{\kern0pt}eq{\isacharbrackleft}{\kern0pt}OF\ {\isasymnu}{\isacharunderscore}{\kern0pt}improving{\isacharunderscore}{\kern0pt}mpi{\isacharunderscore}{\kern0pt}pol{\isacharbrackright}{\kern0pt}\ mpi{\isacharunderscore}{\kern0pt}val{\isacharunderscore}{\kern0pt}Suc\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\begin{isamarkuptext}%
-Value Iteration is a special case of MPI where \isa{{\isasymforall}n\ v{\isachardot}{\kern0pt}\ m\ n\ v\ {\isacharequal}{\kern0pt}\ {\isadigit{0}}}.%
-\end{isamarkuptext}\isamarkuptrue%
-\isacommand{lemma}\isamarkupfalse%
-\ mpi{\isacharunderscore}{\kern0pt}includes{\isacharunderscore}{\kern0pt}value{\isacharunderscore}{\kern0pt}it{\isacharcolon}{\kern0pt}\ \isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}{\isasymforall}n\ v{\isachardot}{\kern0pt}\ m\ n\ v\ {\isacharequal}{\kern0pt}\ {\isadigit{0}}{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}mpi{\isacharunderscore}{\kern0pt}val\ {\isacharparenleft}{\kern0pt}Suc\ n{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isasymL}\isactrlsub b\ {\isacharparenleft}{\kern0pt}mpi{\isacharunderscore}{\kern0pt}val\ n{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ assms\ B\isactrlsub b{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}{\isasymL}\isactrlsub b\ mpi{\isacharunderscore}{\kern0pt}val{\isacharunderscore}{\kern0pt}eq\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Convergence Proof%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\begin{isamarkuptext}%
-We define the sequence \isa{w} as an upper bound for the values of MPI.%
-\end{isamarkuptext}\isamarkuptrue%
-\isacommand{fun}\isamarkupfalse%
-\ w\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}w\ {\isadigit{0}}\ {\isacharequal}{\kern0pt}\ v{\isadigit{0}}{\isachardoublequoteclose}\ {\isacharbar}{\kern0pt}\isanewline
-\ \ {\isachardoublequoteopen}w\ {\isacharparenleft}{\kern0pt}Suc\ n{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ U\isactrlsub b\ {\isacharparenleft}{\kern0pt}Suc\ {\isacharparenleft}{\kern0pt}m\ n\ {\isacharparenleft}{\kern0pt}mpi{\isacharunderscore}{\kern0pt}val\ n{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}w\ n{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ dist{\isacharunderscore}{\kern0pt}{\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}dist\ {\isacharparenleft}{\kern0pt}w\ {\isacharparenleft}{\kern0pt}Suc\ n{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt\ {\isasymle}\ l\ {\isacharasterisk}{\kern0pt}\ dist\ {\isacharparenleft}{\kern0pt}w\ n{\isacharparenright}{\kern0pt}\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}fastforce\ simp{\isacharcolon}{\kern0pt}\ algebra{\isacharunderscore}{\kern0pt}simps\ intro{\isacharcolon}{\kern0pt}\ order{\isachardot}{\kern0pt}trans{\isacharbrackleft}{\kern0pt}OF\ dist{\isacharunderscore}{\kern0pt}U\isactrlsub b{\isacharunderscore}{\kern0pt}opt{\isacharbrackright}{\kern0pt}\ mult{\isacharunderscore}{\kern0pt}left{\isacharunderscore}{\kern0pt}mono\ power{\isacharunderscore}{\kern0pt}le{\isacharunderscore}{\kern0pt}one\isanewline
-\ \ \ \ \ \ mult{\isacharunderscore}{\kern0pt}left{\isacharunderscore}{\kern0pt}le{\isacharunderscore}{\kern0pt}one{\isacharunderscore}{\kern0pt}le\ order{\isachardot}{\kern0pt}strict{\isacharunderscore}{\kern0pt}implies{\isacharunderscore}{\kern0pt}order{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ dist{\isacharunderscore}{\kern0pt}{\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt{\isacharunderscore}{\kern0pt}n{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}dist\ {\isacharparenleft}{\kern0pt}w\ n{\isacharparenright}{\kern0pt}\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt\ {\isasymle}\ l{\isacharcircum}{\kern0pt}n\ {\isacharasterisk}{\kern0pt}\ dist\ v{\isadigit{0}}\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}induction\ n{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}fastforce\ simp{\isacharcolon}{\kern0pt}\ algebra{\isacharunderscore}{\kern0pt}simps\ intro{\isacharcolon}{\kern0pt}\ order{\isachardot}{\kern0pt}trans{\isacharbrackleft}{\kern0pt}OF\ dist{\isacharunderscore}{\kern0pt}{\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt{\isacharbrackright}{\kern0pt}\ mult{\isacharunderscore}{\kern0pt}left{\isacharunderscore}{\kern0pt}mono{\isacharparenright}{\kern0pt}{\isacharplus}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ w{\isacharunderscore}{\kern0pt}conv{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}w\ {\isasymlonglonglongrightarrow}\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isasymlambda}n{\isachardot}{\kern0pt}\ l{\isacharcircum}{\kern0pt}n\ {\isacharasterisk}{\kern0pt}\ dist\ v{\isadigit{0}}\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt{\isacharparenright}{\kern0pt}\ {\isasymlonglonglongrightarrow}\ {\isadigit{0}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ LIMSEQ{\isacharunderscore}{\kern0pt}realpow{\isacharunderscore}{\kern0pt}zero\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}cases\ {\isachardoublequoteopen}v{\isadigit{0}}\ {\isacharequal}{\kern0pt}\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt{\isachardoublequoteclose}{\isacharparenright}{\kern0pt}\ auto\isanewline
-\ \ \isacommand{then}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}fastforce\ intro{\isacharcolon}{\kern0pt}\ metric{\isacharunderscore}{\kern0pt}LIMSEQ{\isacharunderscore}{\kern0pt}I\ order{\isachardot}{\kern0pt}strict{\isacharunderscore}{\kern0pt}trans{\isadigit{1}}{\isacharbrackleft}{\kern0pt}OF\ dist{\isacharunderscore}{\kern0pt}{\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt{\isacharunderscore}{\kern0pt}n{\isacharbrackright}{\kern0pt}\ simp{\isacharcolon}{\kern0pt}\ LIMSEQ{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\begin{isamarkuptext}%
-MPI converges monotonically to the optimal value from below. 
-  The iterates are sandwiched between \isa{{\isasymL}\isactrlsub b} from below and \isa{U\isactrlsub b} from above.%
-\end{isamarkuptext}\isamarkuptrue%
-\isacommand{theorem}\isamarkupfalse%
-\ mpi{\isacharunderscore}{\kern0pt}conv{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}v{\isadigit{0}}\ {\isasymle}\ {\isasymL}\isactrlsub b\ v{\isadigit{0}}{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}mpi{\isacharunderscore}{\kern0pt}val\ {\isasymlonglonglongrightarrow}\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt{\isachardoublequoteclose}\ \isakeyword{and}\ {\isachardoublequoteopen}{\isasymAnd}n{\isachardot}{\kern0pt}\ mpi{\isacharunderscore}{\kern0pt}val\ n\ {\isasymle}\ mpi{\isacharunderscore}{\kern0pt}val\ {\isacharparenleft}{\kern0pt}Suc\ n{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{define}\isamarkupfalse%
-\ y\ \isakeyword{where}\ {\isachardoublequoteopen}y\ n\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b{\isacharcircum}{\kern0pt}{\isacharcircum}{\kern0pt}n{\isacharparenright}{\kern0pt}\ v{\isadigit{0}}{\isachardoublequoteclose}\ \isakeyword{for}\ n\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ aux{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}mpi{\isacharunderscore}{\kern0pt}val\ n\ {\isasymle}\ {\isasymL}\isactrlsub b\ {\isacharparenleft}{\kern0pt}mpi{\isacharunderscore}{\kern0pt}val\ n{\isacharparenright}{\kern0pt}\ {\isasymand}\ mpi{\isacharunderscore}{\kern0pt}val\ n\ {\isasymle}\ mpi{\isacharunderscore}{\kern0pt}val\ {\isacharparenleft}{\kern0pt}Suc\ n{\isacharparenright}{\kern0pt}\ {\isasymand}\ y\ n\ {\isasymle}\ mpi{\isacharunderscore}{\kern0pt}val\ n\ {\isasymand}\ mpi{\isacharunderscore}{\kern0pt}val\ n\ {\isasymle}\ w\ n{\isachardoublequoteclose}\ \isakeyword{for}\ n\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}induction\ n{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{case}\isamarkupfalse%
-\ {\isadigit{0}}\isanewline
-\ \ \ \ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}case\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assms\ B\isactrlsub b{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}{\isasymL}\isactrlsub b\isanewline
-\ \ \ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ y{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ mpi{\isacharunderscore}{\kern0pt}val{\isacharunderscore}{\kern0pt}eq\ blinfun{\isachardot}{\kern0pt}sum{\isacharunderscore}{\kern0pt}left\ {\isasymP}\isactrlsub {\isadigit{1}}{\isacharunderscore}{\kern0pt}n{\isacharunderscore}{\kern0pt}disc{\isacharunderscore}{\kern0pt}pos\ blincomp{\isacharunderscore}{\kern0pt}scaleR{\isacharunderscore}{\kern0pt}right\ sum{\isacharunderscore}{\kern0pt}nonneg{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{case}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}Suc\ n{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ val{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}W{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}mpi{\isacharunderscore}{\kern0pt}val\ {\isacharparenleft}{\kern0pt}Suc\ n{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ W\ {\isacharparenleft}{\kern0pt}mk{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\ {\isacharparenleft}{\kern0pt}mpi{\isacharunderscore}{\kern0pt}pol\ n{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}Suc\ {\isacharparenleft}{\kern0pt}m\ n\ {\isacharparenleft}{\kern0pt}mpi{\isacharunderscore}{\kern0pt}val\ n{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}mpi{\isacharunderscore}{\kern0pt}val\ n{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isasymnu}{\isacharunderscore}{\kern0pt}improving{\isacharunderscore}{\kern0pt}mpi{\isacharunderscore}{\kern0pt}pol\ mpi{\isacharunderscore}{\kern0pt}val{\isacharunderscore}{\kern0pt}Suc\ W{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}L{\isacharunderscore}{\kern0pt}iter\ L{\isacharunderscore}{\kern0pt}pow{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ {\isacharasterisk}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}mpi{\isacharunderscore}{\kern0pt}val\ {\isacharparenleft}{\kern0pt}Suc\ n{\isacharparenright}{\kern0pt}\ {\isasymle}\ {\isasymL}\isactrlsub b\ {\isacharparenleft}{\kern0pt}mpi{\isacharunderscore}{\kern0pt}val\ {\isacharparenleft}{\kern0pt}Suc\ n{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ Suc{\isachardot}{\kern0pt}IH\ {\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}W{\isacharunderscore}{\kern0pt}ge\ {\isasymnu}{\isacharunderscore}{\kern0pt}improving{\isacharunderscore}{\kern0pt}mpi{\isacharunderscore}{\kern0pt}pol\ \isacommand{by}\isamarkupfalse%
-\ presburger\isanewline
-\ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}mpi{\isacharunderscore}{\kern0pt}val\ {\isacharparenleft}{\kern0pt}Suc\ n{\isacharparenright}{\kern0pt}\ {\isasymle}\ mpi{\isacharunderscore}{\kern0pt}val\ {\isacharparenleft}{\kern0pt}Suc\ {\isacharparenleft}{\kern0pt}Suc\ n{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isacharasterisk}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ B\isactrlsub b{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}{\isasymL}\isactrlsub b\ mpi{\isacharunderscore}{\kern0pt}val{\isacharunderscore}{\kern0pt}eq\ {\isasymP}\isactrlsub {\isadigit{1}}{\isacharunderscore}{\kern0pt}n{\isacharunderscore}{\kern0pt}disc{\isacharunderscore}{\kern0pt}pos\ blincomp{\isacharunderscore}{\kern0pt}scaleR{\isacharunderscore}{\kern0pt}right\ blinfun{\isachardot}{\kern0pt}sum{\isacharunderscore}{\kern0pt}left\ sum{\isacharunderscore}{\kern0pt}nonneg{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}mpi{\isacharunderscore}{\kern0pt}val\ {\isacharparenleft}{\kern0pt}Suc\ n{\isacharparenright}{\kern0pt}\ {\isasymle}\ w\ {\isacharparenleft}{\kern0pt}Suc\ n{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ Suc{\isachardot}{\kern0pt}IH\ {\isasymnu}{\isacharunderscore}{\kern0pt}improving{\isacharunderscore}{\kern0pt}mpi{\isacharunderscore}{\kern0pt}pol\ \isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ val{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}W\ intro{\isacharcolon}{\kern0pt}\ order{\isachardot}{\kern0pt}trans{\isacharbrackleft}{\kern0pt}OF\ {\isacharunderscore}{\kern0pt}\ W{\isacharunderscore}{\kern0pt}le{\isacharunderscore}{\kern0pt}U\isactrlsub b{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}y\ {\isacharparenleft}{\kern0pt}Suc\ n{\isacharparenright}{\kern0pt}\ {\isasymle}\ mpi{\isacharunderscore}{\kern0pt}val\ {\isacharparenleft}{\kern0pt}Suc\ n{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ Suc{\isachardot}{\kern0pt}IH\ {\isasymnu}{\isacharunderscore}{\kern0pt}improving{\isacharunderscore}{\kern0pt}mpi{\isacharunderscore}{\kern0pt}pol\ W{\isacharunderscore}{\kern0pt}ge{\isacharunderscore}{\kern0pt}{\isasymL}\isactrlsub b\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ y{\isacharunderscore}{\kern0pt}def\ B\isactrlsub b{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}{\isasymL}\isactrlsub b\ val{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}W{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}case\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}mpi{\isacharunderscore}{\kern0pt}val\ n\ {\isasymle}\ mpi{\isacharunderscore}{\kern0pt}val\ {\isacharparenleft}{\kern0pt}Suc\ n{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{for}\ n\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}y\ {\isasymlonglonglongrightarrow}\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}lim\ y{\isacharunderscore}{\kern0pt}def\ \isacommand{by}\isamarkupfalse%
-\ presburger\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}mpi{\isacharunderscore}{\kern0pt}val\ {\isasymlonglonglongrightarrow}\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ aux\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ intro{\isacharcolon}{\kern0pt}\ tendsto{\isacharunderscore}{\kern0pt}bfun{\isacharunderscore}{\kern0pt}sandwich{\isacharbrackleft}{\kern0pt}OF\ {\isacharunderscore}{\kern0pt}\ w{\isacharunderscore}{\kern0pt}conv{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{$\epsilon$-Optimality%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\begin{isamarkuptext}%
-This gives an upper bound on the error of MPI.%
-\end{isamarkuptext}\isamarkuptrue%
-\isacommand{lemma}\isamarkupfalse%
-\ mpi{\isacharunderscore}{\kern0pt}pol{\isacharunderscore}{\kern0pt}eps{\isacharunderscore}{\kern0pt}opt{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}{\isadigit{2}}\ {\isacharasterisk}{\kern0pt}\ l\ {\isacharasterisk}{\kern0pt}\ dist\ {\isacharparenleft}{\kern0pt}mpi{\isacharunderscore}{\kern0pt}val\ n{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ {\isacharparenleft}{\kern0pt}mpi{\isacharunderscore}{\kern0pt}val\ n{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharless}{\kern0pt}\ eps\ {\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}\ {\isacharminus}{\kern0pt}\ l{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ {\isachardoublequoteopen}eps\ {\isachargreater}{\kern0pt}\ {\isadigit{0}}{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}dist\ {\isacharparenleft}{\kern0pt}{\isasymnu}\isactrlsub b\ {\isacharparenleft}{\kern0pt}mk{\isacharunderscore}{\kern0pt}stationary{\isacharunderscore}{\kern0pt}det\ {\isacharparenleft}{\kern0pt}mpi{\isacharunderscore}{\kern0pt}pol\ n{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ {\isacharparenleft}{\kern0pt}mpi{\isacharunderscore}{\kern0pt}val\ n{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymle}\ eps\ {\isacharslash}{\kern0pt}\ {\isadigit{2}}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}p\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}mk{\isacharunderscore}{\kern0pt}stationary{\isacharunderscore}{\kern0pt}det\ {\isacharparenleft}{\kern0pt}mpi{\isacharunderscore}{\kern0pt}pol\ n{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}d\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}mk{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\ {\isacharparenleft}{\kern0pt}mpi{\isacharunderscore}{\kern0pt}pol\ n{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}v\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}mpi{\isacharunderscore}{\kern0pt}val\ n{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}dist\ {\isacharparenleft}{\kern0pt}{\isasymnu}\isactrlsub b\ {\isacharquery}{\kern0pt}p{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ {\isacharquery}{\kern0pt}v{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ dist\ {\isacharparenleft}{\kern0pt}L\ {\isacharquery}{\kern0pt}d\ {\isacharparenleft}{\kern0pt}{\isasymnu}\isactrlsub b\ {\isacharquery}{\kern0pt}p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ {\isacharquery}{\kern0pt}v{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ L{\isacharunderscore}{\kern0pt}{\isasymnu}{\isacharunderscore}{\kern0pt}fix\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ force\isanewline
-\ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isacharequal}{\kern0pt}\ dist\ {\isacharparenleft}{\kern0pt}L\ {\isacharquery}{\kern0pt}d\ {\isacharparenleft}{\kern0pt}{\isasymnu}\isactrlsub b\ {\isacharquery}{\kern0pt}p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}L\ {\isacharquery}{\kern0pt}d\ {\isacharquery}{\kern0pt}v{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isasymnu}{\isacharunderscore}{\kern0pt}improving{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}{\isasymL}\isactrlsub b\ {\isasymnu}{\isacharunderscore}{\kern0pt}improving{\isacharunderscore}{\kern0pt}mpi{\isacharunderscore}{\kern0pt}pol{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isasymle}\ dist\ {\isacharparenleft}{\kern0pt}L\ {\isacharquery}{\kern0pt}d\ {\isacharparenleft}{\kern0pt}{\isasymnu}\isactrlsub b\ {\isacharquery}{\kern0pt}p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}L\ {\isacharquery}{\kern0pt}d\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ {\isacharquery}{\kern0pt}v{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharplus}{\kern0pt}\ dist\ {\isacharparenleft}{\kern0pt}L\ {\isacharquery}{\kern0pt}d\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ {\isacharquery}{\kern0pt}v{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}L\ {\isacharquery}{\kern0pt}d\ {\isacharquery}{\kern0pt}v{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ dist{\isacharunderscore}{\kern0pt}triangle\ \isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isasymle}\ l\ {\isacharasterisk}{\kern0pt}\ dist\ {\isacharparenleft}{\kern0pt}{\isasymnu}\isactrlsub b\ {\isacharquery}{\kern0pt}p{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ {\isacharquery}{\kern0pt}v{\isacharparenright}{\kern0pt}\ {\isacharplus}{\kern0pt}\ dist\ {\isacharparenleft}{\kern0pt}L\ {\isacharquery}{\kern0pt}d\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ {\isacharquery}{\kern0pt}v{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}L\ {\isacharquery}{\kern0pt}d\ {\isacharquery}{\kern0pt}v{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ contraction{\isacharunderscore}{\kern0pt}L\ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isasymle}\ l\ {\isacharasterisk}{\kern0pt}\ dist\ {\isacharparenleft}{\kern0pt}{\isasymnu}\isactrlsub b\ {\isacharquery}{\kern0pt}p{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ {\isacharquery}{\kern0pt}v{\isacharparenright}{\kern0pt}\ {\isacharplus}{\kern0pt}\ l\ {\isacharasterisk}{\kern0pt}\ dist\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ {\isacharquery}{\kern0pt}v{\isacharparenright}{\kern0pt}\ {\isacharquery}{\kern0pt}v{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ contraction{\isacharunderscore}{\kern0pt}L\ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{finally}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}dist\ {\isacharparenleft}{\kern0pt}{\isasymnu}\isactrlsub b\ {\isacharquery}{\kern0pt}p{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ {\isacharquery}{\kern0pt}v{\isacharparenright}{\kern0pt}\ {\isasymle}\ l\ {\isacharasterisk}{\kern0pt}\ dist\ {\isacharparenleft}{\kern0pt}{\isasymnu}\isactrlsub b\ {\isacharquery}{\kern0pt}p{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ {\isacharquery}{\kern0pt}v{\isacharparenright}{\kern0pt}\ {\isacharplus}{\kern0pt}\ l\ {\isacharasterisk}{\kern0pt}\ dist\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ {\isacharquery}{\kern0pt}v{\isacharparenright}{\kern0pt}\ {\isacharquery}{\kern0pt}v{\isachardoublequoteclose}\isacommand{{\isachardot}{\kern0pt}}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isacharasterisk}{\kern0pt}{\isacharcolon}{\kern0pt}{\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharminus}{\kern0pt}l{\isacharparenright}{\kern0pt}\ {\isacharasterisk}{\kern0pt}\ dist\ {\isacharparenleft}{\kern0pt}{\isasymnu}\isactrlsub b\ {\isacharquery}{\kern0pt}p{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ {\isacharquery}{\kern0pt}v{\isacharparenright}{\kern0pt}\ {\isasymle}\ l\ {\isacharasterisk}{\kern0pt}\ dist\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ {\isacharquery}{\kern0pt}v{\isacharparenright}{\kern0pt}\ {\isacharquery}{\kern0pt}v{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ left{\isacharunderscore}{\kern0pt}diff{\isacharunderscore}{\kern0pt}distrib{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}cases\ {\isachardoublequoteopen}l\ {\isacharequal}{\kern0pt}\ {\isadigit{0}}{\isachardoublequoteclose}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{case}\isamarkupfalse%
-\ True\isanewline
-\ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assms\ {\isacharasterisk}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{case}\isamarkupfalse%
-\ False\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isacharasterisk}{\kern0pt}{\isacharasterisk}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}dist\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ {\isacharquery}{\kern0pt}v{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}mpi{\isacharunderscore}{\kern0pt}val\ n{\isacharparenright}{\kern0pt}\ {\isacharless}{\kern0pt}\ eps\ {\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}\ {\isacharminus}{\kern0pt}\ l{\isacharparenright}{\kern0pt}\ {\isacharslash}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isadigit{2}}\ {\isacharasterisk}{\kern0pt}\ l{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ False\ le{\isacharunderscore}{\kern0pt}neq{\isacharunderscore}{\kern0pt}trans{\isacharbrackleft}{\kern0pt}OF\ zero{\isacharunderscore}{\kern0pt}le{\isacharunderscore}{\kern0pt}disc\ False{\isacharbrackleft}{\kern0pt}symmetric{\isacharbrackright}{\kern0pt}{\isacharbrackright}{\kern0pt}\ assms\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ dist{\isacharunderscore}{\kern0pt}commute\ pos{\isacharunderscore}{\kern0pt}less{\isacharunderscore}{\kern0pt}divide{\isacharunderscore}{\kern0pt}eq\ Groups{\isachardot}{\kern0pt}mult{\isacharunderscore}{\kern0pt}ac{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}dist\ {\isacharparenleft}{\kern0pt}{\isasymnu}\isactrlsub b\ {\isacharquery}{\kern0pt}p{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ {\isacharquery}{\kern0pt}v{\isacharparenright}{\kern0pt}\ {\isasymle}\ {\isacharparenleft}{\kern0pt}l{\isacharslash}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharminus}{\kern0pt}l{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharasterisk}{\kern0pt}\ dist\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ {\isacharquery}{\kern0pt}v{\isacharparenright}{\kern0pt}\ {\isacharquery}{\kern0pt}v{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isacharasterisk}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ mult{\isachardot}{\kern0pt}commute\ pos{\isacharunderscore}{\kern0pt}le{\isacharunderscore}{\kern0pt}divide{\isacharunderscore}{\kern0pt}eq{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isasymle}\ {\isacharparenleft}{\kern0pt}l{\isacharslash}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharminus}{\kern0pt}l{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}eps\ {\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}\ {\isacharminus}{\kern0pt}\ l{\isacharparenright}{\kern0pt}\ {\isacharslash}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isadigit{2}}\ {\isacharasterisk}{\kern0pt}\ l{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isacharasterisk}{\kern0pt}{\isacharasterisk}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}fastforce\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ mult{\isacharunderscore}{\kern0pt}left{\isacharunderscore}{\kern0pt}mono\ simp{\isacharcolon}{\kern0pt}\ divide{\isacharunderscore}{\kern0pt}nonneg{\isacharunderscore}{\kern0pt}pos{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isacharequal}{\kern0pt}\ eps\ {\isacharslash}{\kern0pt}\ {\isadigit{2}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ False\ disc{\isacharunderscore}{\kern0pt}lt{\isacharunderscore}{\kern0pt}one\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ order{\isachardot}{\kern0pt}strict{\isacharunderscore}{\kern0pt}iff{\isacharunderscore}{\kern0pt}order{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{finally}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}dist\ {\isacharparenleft}{\kern0pt}{\isasymnu}\isactrlsub b\ {\isacharquery}{\kern0pt}p{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ {\isacharquery}{\kern0pt}v{\isacharparenright}{\kern0pt}\ {\isasymle}\ eps\ {\isacharslash}{\kern0pt}\ {\isadigit{2}}{\isachardoublequoteclose}\isacommand{{\isachardot}{\kern0pt}}\isamarkupfalse%
-\ \ \ \ \isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ mpi{\isacharunderscore}{\kern0pt}pol{\isacharunderscore}{\kern0pt}opt{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}{\isadigit{2}}\ {\isacharasterisk}{\kern0pt}\ l\ {\isacharasterisk}{\kern0pt}\ dist\ {\isacharparenleft}{\kern0pt}mpi{\isacharunderscore}{\kern0pt}val\ n{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ {\isacharparenleft}{\kern0pt}mpi{\isacharunderscore}{\kern0pt}val\ n{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharless}{\kern0pt}\ eps\ {\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}\ {\isacharminus}{\kern0pt}\ l{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ {\isachardoublequoteopen}eps\ {\isachargreater}{\kern0pt}\ {\isadigit{0}}{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}dist\ {\isacharparenleft}{\kern0pt}{\isasymnu}\isactrlsub b\ {\isacharparenleft}{\kern0pt}mk{\isacharunderscore}{\kern0pt}stationary{\isacharunderscore}{\kern0pt}det\ {\isacharparenleft}{\kern0pt}mpi{\isacharunderscore}{\kern0pt}pol\ n{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt{\isacharparenright}{\kern0pt}\ {\isacharless}{\kern0pt}\ eps{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}dist\ {\isacharparenleft}{\kern0pt}{\isasymnu}\isactrlsub b\ {\isacharparenleft}{\kern0pt}mk{\isacharunderscore}{\kern0pt}stationary{\isacharunderscore}{\kern0pt}det\ {\isacharparenleft}{\kern0pt}mpi{\isacharunderscore}{\kern0pt}pol\ n{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt{\isacharparenright}{\kern0pt}\ {\isasymle}\ eps{\isacharslash}{\kern0pt}{\isadigit{2}}\ {\isacharplus}{\kern0pt}\ dist\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ {\isacharparenleft}{\kern0pt}mpi{\isacharunderscore}{\kern0pt}val\ n{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ mpi{\isacharunderscore}{\kern0pt}pol{\isacharunderscore}{\kern0pt}eps{\isacharunderscore}{\kern0pt}opt{\isacharbrackleft}{\kern0pt}OF\ assms{\isacharbrackright}{\kern0pt}\ dist{\isacharunderscore}{\kern0pt}commute\ dist{\isacharunderscore}{\kern0pt}triangle{\isacharunderscore}{\kern0pt}le\ add{\isacharunderscore}{\kern0pt}right{\isacharunderscore}{\kern0pt}mono{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ dist{\isacharunderscore}{\kern0pt}{\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}opt{\isacharunderscore}{\kern0pt}eps\ assms\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ fastforce\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ mpi{\isacharunderscore}{\kern0pt}val{\isacharunderscore}{\kern0pt}term{\isacharunderscore}{\kern0pt}ex{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}v{\isadigit{0}}\ {\isasymle}\ {\isasymL}\isactrlsub b\ v{\isadigit{0}}{\isachardoublequoteclose}\ {\isachardoublequoteopen}eps\ {\isachargreater}{\kern0pt}\ {\isadigit{0}}{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}{\isasymexists}n{\isachardot}{\kern0pt}\ {\isadigit{2}}\ {\isacharasterisk}{\kern0pt}\ l\ {\isacharasterisk}{\kern0pt}\ dist\ {\isacharparenleft}{\kern0pt}mpi{\isacharunderscore}{\kern0pt}val\ n{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ {\isacharparenleft}{\kern0pt}mpi{\isacharunderscore}{\kern0pt}val\ n{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharless}{\kern0pt}\ eps\ {\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}\ {\isacharminus}{\kern0pt}\ l{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{note}\isamarkupfalse%
-\ dist{\isacharunderscore}{\kern0pt}{\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}lt{\isacharunderscore}{\kern0pt}dist{\isacharunderscore}{\kern0pt}opt\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isasymlambda}n{\isachardot}{\kern0pt}\ dist\ {\isacharparenleft}{\kern0pt}mpi{\isacharunderscore}{\kern0pt}val\ n{\isacharparenright}{\kern0pt}\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt{\isacharparenright}{\kern0pt}\ {\isasymlonglonglongrightarrow}\ {\isadigit{0}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ mpi{\isacharunderscore}{\kern0pt}conv{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}{\isacharbrackleft}{\kern0pt}OF\ assms{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}{\isacharbrackright}{\kern0pt}\ tendsto{\isacharunderscore}{\kern0pt}dist{\isacharunderscore}{\kern0pt}iff\ \isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isasymlambda}n{\isachardot}{\kern0pt}\ dist\ {\isacharparenleft}{\kern0pt}mpi{\isacharunderscore}{\kern0pt}val\ n{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ {\isacharparenleft}{\kern0pt}mpi{\isacharunderscore}{\kern0pt}val\ n{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymlonglonglongrightarrow}\ {\isadigit{0}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ dist{\isacharunderscore}{\kern0pt}{\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}lt{\isacharunderscore}{\kern0pt}dist{\isacharunderscore}{\kern0pt}opt\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ metric{\isacharunderscore}{\kern0pt}LIMSEQ{\isacharunderscore}{\kern0pt}I\ intro{\isacharcolon}{\kern0pt}\ tendsto{\isacharunderscore}{\kern0pt}sandwich{\isacharbrackleft}{\kern0pt}of\ {\isachardoublequoteopen}{\isasymlambda}{\isacharunderscore}{\kern0pt}{\isachardot}{\kern0pt}\ {\isadigit{0}}{\isachardoublequoteclose}\ {\isacharunderscore}{\kern0pt}\ {\isacharunderscore}{\kern0pt}\ {\isachardoublequoteopen}{\isasymlambda}n{\isachardot}{\kern0pt}\ {\isadigit{2}}\ {\isacharasterisk}{\kern0pt}\ dist\ {\isacharparenleft}{\kern0pt}mpi{\isacharunderscore}{\kern0pt}val\ n{\isacharparenright}{\kern0pt}\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt{\isachardoublequoteclose}{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymforall}e\ {\isachargreater}{\kern0pt}{\isadigit{0}}{\isachardot}{\kern0pt}\ {\isasymexists}n{\isachardot}{\kern0pt}\ dist\ {\isacharparenleft}{\kern0pt}mpi{\isacharunderscore}{\kern0pt}val\ n{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ {\isacharparenleft}{\kern0pt}mpi{\isacharunderscore}{\kern0pt}val\ n{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharless}{\kern0pt}\ e{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}fastforce\ dest{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ metric{\isacharunderscore}{\kern0pt}LIMSEQ{\isacharunderscore}{\kern0pt}D{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}l\ {\isasymnoteq}\ {\isadigit{0}}\ {\isasymLongrightarrow}\ {\isasymexists}n{\isachardot}{\kern0pt}\ dist\ {\isacharparenleft}{\kern0pt}mpi{\isacharunderscore}{\kern0pt}val\ n{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ {\isacharparenleft}{\kern0pt}mpi{\isacharunderscore}{\kern0pt}val\ n{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharless}{\kern0pt}\ eps\ {\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}\ {\isacharminus}{\kern0pt}\ l{\isacharparenright}{\kern0pt}\ {\isacharslash}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isadigit{2}}\ {\isacharasterisk}{\kern0pt}\ l{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ assms\ order{\isachardot}{\kern0pt}not{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}order{\isacharunderscore}{\kern0pt}implies{\isacharunderscore}{\kern0pt}strict{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymexists}n{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isadigit{2}}\ {\isacharasterisk}{\kern0pt}\ l{\isacharparenright}{\kern0pt}\ {\isacharasterisk}{\kern0pt}\ dist\ {\isacharparenleft}{\kern0pt}mpi{\isacharunderscore}{\kern0pt}val\ n{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ {\isacharparenleft}{\kern0pt}mpi{\isacharunderscore}{\kern0pt}val\ n{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharless}{\kern0pt}\ eps\ {\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}\ {\isacharminus}{\kern0pt}\ l{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assms\ le{\isacharunderscore}{\kern0pt}neq{\isacharunderscore}{\kern0pt}trans{\isacharbrackleft}{\kern0pt}OF\ zero{\isacharunderscore}{\kern0pt}le{\isacharunderscore}{\kern0pt}disc{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}cases\ {\isachardoublequoteopen}l\ {\isacharequal}{\kern0pt}\ {\isadigit{0}}{\isachardoublequoteclose}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ mult{\isachardot}{\kern0pt}commute\ pos{\isacharunderscore}{\kern0pt}less{\isacharunderscore}{\kern0pt}divide{\isacharunderscore}{\kern0pt}eq{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isacommand{end}\isamarkupfalse%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Unbounded MPI%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{context}\isamarkupfalse%
-\isanewline
-\ \ \isakeyword{fixes}\ eps\ {\isasymdelta}\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ real\ \isakeyword{and}\ M\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ nat\isanewline
-\isakeyword{begin}\isanewline
-\isanewline
-\isacommand{function}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}domintros{\isacharparenright}{\kern0pt}\ mpi{\isacharunderscore}{\kern0pt}algo\ \isakeyword{where}\ {\isachardoublequoteopen}mpi{\isacharunderscore}{\kern0pt}algo\ d\ v\ m\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}\isanewline
-\ \ if\ {\isadigit{2}}\ {\isacharasterisk}{\kern0pt}\ l\ {\isacharasterisk}{\kern0pt}\ dist\ v\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ v{\isacharparenright}{\kern0pt}\ {\isacharless}{\kern0pt}\ \ eps\ {\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}\ {\isacharminus}{\kern0pt}\ l{\isacharparenright}{\kern0pt}\isanewline
-\ \ then\ {\isacharparenleft}{\kern0pt}policy{\isacharunderscore}{\kern0pt}improvement\ d\ v{\isacharcomma}{\kern0pt}\ v{\isacharparenright}{\kern0pt}\isanewline
-\ \ else\ mpi{\isacharunderscore}{\kern0pt}algo\ {\isacharparenleft}{\kern0pt}policy{\isacharunderscore}{\kern0pt}improvement\ d\ v{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}L{\isacharunderscore}{\kern0pt}pow\ v\ {\isacharparenleft}{\kern0pt}policy{\isacharunderscore}{\kern0pt}improvement\ d\ v{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}m\ {\isadigit{0}}\ v{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymlambda}n{\isachardot}{\kern0pt}\ m\ {\isacharparenleft}{\kern0pt}Suc\ n{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\begin{isamarkuptext}%
-We define a tailrecursive version of \isa{mpi} which more closely resembles \isa{mpi{\isacharunderscore}{\kern0pt}algo}.%
-\end{isamarkuptext}\isamarkuptrue%
-\isacommand{fun}\isamarkupfalse%
-\ mpi{\isacharprime}{\kern0pt}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}mpi{\isacharprime}{\kern0pt}\ d\ v\ {\isadigit{0}}\ m\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}policy{\isacharunderscore}{\kern0pt}improvement\ d\ v{\isacharcomma}{\kern0pt}\ v{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ {\isacharbar}{\kern0pt}\isanewline
-\ \ {\isachardoublequoteopen}mpi{\isacharprime}{\kern0pt}\ d\ v\ {\isacharparenleft}{\kern0pt}Suc\ n{\isacharparenright}{\kern0pt}\ m\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}\isanewline
-\ \ let\ d{\isacharprime}{\kern0pt}\ {\isacharequal}{\kern0pt}\ policy{\isacharunderscore}{\kern0pt}improvement\ d\ v{\isacharsemicolon}{\kern0pt}\ v{\isacharprime}{\kern0pt}\ {\isacharequal}{\kern0pt}\ L{\isacharunderscore}{\kern0pt}pow\ v\ d{\isacharprime}{\kern0pt}\ {\isacharparenleft}{\kern0pt}m\ {\isadigit{0}}\ v{\isacharparenright}{\kern0pt}\ in\ mpi{\isacharprime}{\kern0pt}\ d{\isacharprime}{\kern0pt}\ v{\isacharprime}{\kern0pt}\ n\ {\isacharparenleft}{\kern0pt}{\isasymlambda}n{\isachardot}{\kern0pt}\ m\ {\isacharparenleft}{\kern0pt}Suc\ n{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ mpi{\isacharunderscore}{\kern0pt}Suc{\isacharprime}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}d\ {\isasymin}\ D\isactrlsub D{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}mpi\ d\ v\ m\ {\isacharparenleft}{\kern0pt}Suc\ n{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ mpi\ {\isacharparenleft}{\kern0pt}policy{\isacharunderscore}{\kern0pt}improvement\ d\ v{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}L{\isacharunderscore}{\kern0pt}pow\ v\ {\isacharparenleft}{\kern0pt}policy{\isacharunderscore}{\kern0pt}improvement\ d\ v{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}m\ {\isadigit{0}}\ v{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymlambda}a{\isachardot}{\kern0pt}\ m\ {\isacharparenleft}{\kern0pt}Suc\ a{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ n{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ assms\ policy{\isacharunderscore}{\kern0pt}improvement{\isacharunderscore}{\kern0pt}is{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}induction\ n\ rule{\isacharcolon}{\kern0pt}\ nat{\isachardot}{\kern0pt}induct{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ Let{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ \isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}d\ {\isasymin}\ D\isactrlsub D{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}mpi\ d\ v\ m\ n\ {\isacharequal}{\kern0pt}\ mpi{\isacharprime}{\kern0pt}\ d\ v\ n\ m{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ assms\isanewline
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}induction\ n\ arbitrary{\isacharcolon}{\kern0pt}\ d\ v\ m\ rule{\isacharcolon}{\kern0pt}\ nat{\isachardot}{\kern0pt}induct{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{case}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}Suc\ nat{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}case\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ policy{\isacharunderscore}{\kern0pt}improvement{\isacharunderscore}{\kern0pt}is{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ Let{\isacharunderscore}{\kern0pt}def\ mpi{\isacharunderscore}{\kern0pt}Suc{\isacharprime}{\kern0pt}{\isacharbrackleft}{\kern0pt}OF\ Suc{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isacharbrackright}{\kern0pt}\ Suc{\isachardot}{\kern0pt}IH{\isacharbrackleft}{\kern0pt}symmetric{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ termination{\isacharunderscore}{\kern0pt}mpi{\isacharunderscore}{\kern0pt}algo{\isacharcolon}{\kern0pt}\ \isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}eps\ {\isachargreater}{\kern0pt}\ {\isadigit{0}}{\isachardoublequoteclose}\ {\isachardoublequoteopen}d\ {\isasymin}\ D\isactrlsub D{\isachardoublequoteclose}\ {\isachardoublequoteopen}v\ {\isasymle}\ {\isasymL}\isactrlsub b\ v{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}mpi{\isacharunderscore}{\kern0pt}algo{\isacharunderscore}{\kern0pt}dom\ {\isacharparenleft}{\kern0pt}d{\isacharcomma}{\kern0pt}\ v{\isacharcomma}{\kern0pt}\ m{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{define}\isamarkupfalse%
-\ n\ \isakeyword{where}\ {\isachardoublequoteopen}n\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}LEAST\ n{\isachardot}{\kern0pt}\ {\isadigit{2}}\ {\isacharasterisk}{\kern0pt}\ l\ {\isacharasterisk}{\kern0pt}\ dist\ {\isacharparenleft}{\kern0pt}mpi{\isacharunderscore}{\kern0pt}val\ d\ v\ m\ n{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ {\isacharparenleft}{\kern0pt}mpi{\isacharunderscore}{\kern0pt}val\ d\ v\ m\ n{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharless}{\kern0pt}\ eps\ {\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}\ {\isacharminus}{\kern0pt}\ l{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ {\isacharparenleft}{\kern0pt}\isakeyword{is}\ {\isachardoublequoteopen}n\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}LEAST\ n{\isachardot}{\kern0pt}\ {\isacharquery}{\kern0pt}P\ d\ v\ m\ n{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ least{\isadigit{0}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymexists}n{\isachardot}{\kern0pt}\ P\ n\ {\isasymLongrightarrow}\ {\isacharparenleft}{\kern0pt}LEAST\ n{\isachardot}{\kern0pt}\ P\ n{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isadigit{0}}\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ nat{\isacharparenright}{\kern0pt}\ {\isasymLongrightarrow}\ P\ {\isadigit{0}}{\isachardoublequoteclose}\ \ \isakeyword{for}\ P\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ LeastI{\isacharunderscore}{\kern0pt}ex{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ n{\isacharunderscore}{\kern0pt}def\ assms\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}induction\ n\ arbitrary{\isacharcolon}{\kern0pt}\ v\ d\ m{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{case}\isamarkupfalse%
-\ {\isadigit{0}}\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isadigit{2}}\ {\isacharasterisk}{\kern0pt}\ l\ {\isacharasterisk}{\kern0pt}\ dist\ {\isacharparenleft}{\kern0pt}mpi{\isacharunderscore}{\kern0pt}val\ d\ v\ m\ {\isadigit{0}}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ {\isacharparenleft}{\kern0pt}mpi{\isacharunderscore}{\kern0pt}val\ d\ v\ m\ {\isadigit{0}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharless}{\kern0pt}\ eps\ {\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}\ {\isacharminus}{\kern0pt}\ l{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ least{\isadigit{0}}\ mpi{\isacharunderscore}{\kern0pt}val{\isacharunderscore}{\kern0pt}term{\isacharunderscore}{\kern0pt}ex\ {\isadigit{0}}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharcomma}{\kern0pt}\ lifting{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}case\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isadigit{0}}\ mpi{\isacharunderscore}{\kern0pt}algo{\isachardot}{\kern0pt}domintros\ mpi{\isacharunderscore}{\kern0pt}val{\isacharunderscore}{\kern0pt}zero\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharcomma}{\kern0pt}\ opaque{\isacharunderscore}{\kern0pt}lifting{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{case}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}Suc\ n\ v\ d\ m{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}d\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}policy{\isacharunderscore}{\kern0pt}improvement\ d\ v{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}Suc\ n\ {\isacharequal}{\kern0pt}\ Suc\ {\isacharparenleft}{\kern0pt}LEAST\ n{\isachardot}{\kern0pt}\ {\isadigit{2}}\ {\isacharasterisk}{\kern0pt}\ l\ {\isacharasterisk}{\kern0pt}\ dist\ {\isacharparenleft}{\kern0pt}mpi{\isacharunderscore}{\kern0pt}val\ d\ v\ m\ {\isacharparenleft}{\kern0pt}Suc\ n{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ {\isacharparenleft}{\kern0pt}mpi{\isacharunderscore}{\kern0pt}val\ d\ v\ m\ {\isacharparenleft}{\kern0pt}Suc\ n{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharless}{\kern0pt}\ eps\ {\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}\ {\isacharminus}{\kern0pt}\ l{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ mpi{\isacharunderscore}{\kern0pt}val{\isacharunderscore}{\kern0pt}term{\isacharunderscore}{\kern0pt}ex{\isacharbrackleft}{\kern0pt}OF\ Suc{\isachardot}{\kern0pt}prems{\isacharparenleft}{\kern0pt}{\isadigit{3}}{\isacharparenright}{\kern0pt}\ {\isacartoucheopen}v\ {\isasymle}\ {\isasymL}\isactrlsub b\ v{\isacartoucheclose}\ {\isacartoucheopen}{\isadigit{0}}\ {\isacharless}{\kern0pt}\ eps{\isacartoucheclose}{\isacharcomma}{\kern0pt}\ of\ m{\isacharbrackright}{\kern0pt}\ Suc{\isachardot}{\kern0pt}prems\ \isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}subst\ Nat{\isachardot}{\kern0pt}Least{\isacharunderscore}{\kern0pt}Suc{\isacharbrackleft}{\kern0pt}symmetric{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}auto\ intro{\isacharcolon}{\kern0pt}\ LeastI{\isacharunderscore}{\kern0pt}ex{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}n\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}LEAST\ n{\isachardot}{\kern0pt}\ {\isadigit{2}}\ {\isacharasterisk}{\kern0pt}\ l\ {\isacharasterisk}{\kern0pt}\ dist\ {\isacharparenleft}{\kern0pt}mpi{\isacharunderscore}{\kern0pt}val\ d\ v\ m\ {\isacharparenleft}{\kern0pt}Suc\ n{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ {\isacharparenleft}{\kern0pt}mpi{\isacharunderscore}{\kern0pt}val\ d\ v\ m\ {\isacharparenleft}{\kern0pt}Suc\ n{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharless}{\kern0pt}\ eps\ {\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}\ {\isacharminus}{\kern0pt}\ l{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ n{\isacharunderscore}{\kern0pt}eq{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}n\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ {\isacharparenleft}{\kern0pt}LEAST\ n{\isachardot}{\kern0pt}\ {\isadigit{2}}\ {\isacharasterisk}{\kern0pt}\ l\ {\isacharasterisk}{\kern0pt}\ dist\ {\isacharparenleft}{\kern0pt}mpi{\isacharunderscore}{\kern0pt}val\ {\isacharquery}{\kern0pt}d\ {\isacharparenleft}{\kern0pt}L{\isacharunderscore}{\kern0pt}pow\ v\ {\isacharquery}{\kern0pt}d\ {\isacharparenleft}{\kern0pt}m\ {\isadigit{0}}\ v{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymlambda}a{\isachardot}{\kern0pt}\ m\ {\isacharparenleft}{\kern0pt}Suc\ a{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ n{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ {\isacharparenleft}{\kern0pt}mpi{\isacharunderscore}{\kern0pt}val\ {\isacharquery}{\kern0pt}d\ {\isacharparenleft}{\kern0pt}L{\isacharunderscore}{\kern0pt}pow\ v\ {\isacharquery}{\kern0pt}d\ {\isacharparenleft}{\kern0pt}m\ {\isadigit{0}}\ v{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymlambda}a{\isachardot}{\kern0pt}\ m\ {\isacharparenleft}{\kern0pt}Suc\ a{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ n{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isacharless}{\kern0pt}\ eps\ {\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}\ {\isacharminus}{\kern0pt}\ l{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ Suc{\isachardot}{\kern0pt}prems\ mpi{\isacharunderscore}{\kern0pt}Suc{\isacharprime}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ is{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det{\isacharunderscore}{\kern0pt}pi\ mpi{\isacharunderscore}{\kern0pt}val{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymnot}\ {\isadigit{2}}\ {\isacharasterisk}{\kern0pt}\ l\ {\isacharasterisk}{\kern0pt}\ dist\ v\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ v{\isacharparenright}{\kern0pt}\ {\isacharless}{\kern0pt}\ eps\ {\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}\ {\isacharminus}{\kern0pt}\ l{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ Suc\ mpi{\isacharunderscore}{\kern0pt}val{\isacharunderscore}{\kern0pt}zero\ \isacommand{by}\isamarkupfalse%
-\ force\isanewline
-\ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}mpi{\isacharunderscore}{\kern0pt}algo{\isacharunderscore}{\kern0pt}dom\ {\isacharparenleft}{\kern0pt}{\isacharquery}{\kern0pt}d{\isacharcomma}{\kern0pt}\ L{\isacharunderscore}{\kern0pt}pow\ v\ {\isacharquery}{\kern0pt}d\ {\isacharparenleft}{\kern0pt}m\ {\isadigit{0}}\ v{\isacharparenright}{\kern0pt}{\isacharcomma}{\kern0pt}\ {\isasymlambda}a{\isachardot}{\kern0pt}\ m\ {\isacharparenleft}{\kern0pt}Suc\ a{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ Suc{\isachardot}{\kern0pt}IH{\isacharbrackleft}{\kern0pt}OF\ n{\isacharunderscore}{\kern0pt}eq\ {\isacartoucheopen}{\isadigit{0}}\ {\isacharless}{\kern0pt}\ eps{\isacartoucheclose}{\isacharbrackright}{\kern0pt}\ Suc{\isachardot}{\kern0pt}prems\ is{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det{\isacharunderscore}{\kern0pt}pi\ L{\isacharunderscore}{\kern0pt}pow{\isacharunderscore}{\kern0pt}{\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}mono{\isacharunderscore}{\kern0pt}inv\ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}case\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ mpi{\isacharunderscore}{\kern0pt}algo{\isachardot}{\kern0pt}domintros\ \isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{abbreviation}\isamarkupfalse%
-\ {\isachardoublequoteopen}mpi{\isacharunderscore}{\kern0pt}alg{\isacharunderscore}{\kern0pt}rec\ d\ v\ m\ {\isasymequiv}\ \isanewline
-\ \ \ \ {\isacharparenleft}{\kern0pt}if\ {\isadigit{2}}\ {\isacharasterisk}{\kern0pt}\ l\ {\isacharasterisk}{\kern0pt}\ dist\ v\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ v{\isacharparenright}{\kern0pt}\ {\isacharless}{\kern0pt}\ eps\ {\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}\ {\isacharminus}{\kern0pt}\ l{\isacharparenright}{\kern0pt}\ then\ {\isacharparenleft}{\kern0pt}policy{\isacharunderscore}{\kern0pt}improvement\ d\ v{\isacharcomma}{\kern0pt}\ v{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ else\ mpi{\isacharunderscore}{\kern0pt}algo\ {\isacharparenleft}{\kern0pt}policy{\isacharunderscore}{\kern0pt}improvement\ d\ v{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}L{\isacharunderscore}{\kern0pt}pow\ v\ {\isacharparenleft}{\kern0pt}policy{\isacharunderscore}{\kern0pt}improvement\ d\ v{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}m\ {\isadigit{0}}\ v{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymlambda}n{\isachardot}{\kern0pt}\ m\ {\isacharparenleft}{\kern0pt}Suc\ n{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ mpi{\isacharunderscore}{\kern0pt}algo{\isacharunderscore}{\kern0pt}def{\isacharprime}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}d\ {\isasymin}\ D\isactrlsub D{\isachardoublequoteclose}\ {\isachardoublequoteopen}v\ {\isasymle}\ {\isasymL}\isactrlsub b\ v{\isachardoublequoteclose}\ {\isachardoublequoteopen}eps\ {\isachargreater}{\kern0pt}\ {\isadigit{0}}{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}mpi{\isacharunderscore}{\kern0pt}algo\ d\ v\ m\ {\isacharequal}{\kern0pt}\ mpi{\isacharunderscore}{\kern0pt}alg{\isacharunderscore}{\kern0pt}rec\ d\ v\ m{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ mpi{\isacharunderscore}{\kern0pt}algo{\isachardot}{\kern0pt}psimps\ termination{\isacharunderscore}{\kern0pt}mpi{\isacharunderscore}{\kern0pt}algo\ assms\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ mpi{\isacharunderscore}{\kern0pt}algo{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}mpi{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}d\ {\isasymin}\ D\isactrlsub D{\isachardoublequoteclose}\ {\isachardoublequoteopen}v\ {\isasymle}\ {\isasymL}\isactrlsub b\ v{\isachardoublequoteclose}\ {\isachardoublequoteopen}eps\ {\isachargreater}{\kern0pt}\ {\isadigit{0}}{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}mpi{\isacharunderscore}{\kern0pt}algo\ d\ v\ m\ {\isacharequal}{\kern0pt}\ mpi\ d\ v\ m\ {\isacharparenleft}{\kern0pt}LEAST\ n{\isachardot}{\kern0pt}\ {\isadigit{2}}\ {\isacharasterisk}{\kern0pt}\ l\ {\isacharasterisk}{\kern0pt}\ dist\ {\isacharparenleft}{\kern0pt}mpi{\isacharunderscore}{\kern0pt}val\ d\ v\ m\ n{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ {\isacharparenleft}{\kern0pt}mpi{\isacharunderscore}{\kern0pt}val\ d\ v\ m\ n{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharless}{\kern0pt}\ eps\ {\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}\ {\isacharminus}{\kern0pt}\ l{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{define}\isamarkupfalse%
-\ n\ \isakeyword{where}\ {\isachardoublequoteopen}n\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}LEAST\ n{\isachardot}{\kern0pt}\ {\isadigit{2}}\ {\isacharasterisk}{\kern0pt}\ l\ {\isacharasterisk}{\kern0pt}\ dist\ {\isacharparenleft}{\kern0pt}mpi{\isacharunderscore}{\kern0pt}val\ d\ v\ m\ n{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ {\isacharparenleft}{\kern0pt}mpi{\isacharunderscore}{\kern0pt}val\ d\ v\ m\ n{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharless}{\kern0pt}\ eps\ {\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}\ {\isacharminus}{\kern0pt}\ l{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ {\isacharparenleft}{\kern0pt}\isakeyword{is}\ {\isachardoublequoteopen}n\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}LEAST\ n{\isachardot}{\kern0pt}\ {\isacharquery}{\kern0pt}P\ d\ v\ m\ n{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ n{\isacharunderscore}{\kern0pt}def\ assms\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}induction\ n\ arbitrary{\isacharcolon}{\kern0pt}\ d\ v\ m{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{case}\isamarkupfalse%
-\ {\isadigit{0}}\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharquery}{\kern0pt}P\ d\ v\ m\ {\isadigit{0}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharcomma}{\kern0pt}\ lifting{\isacharparenright}{\kern0pt}\ assms{\isacharparenleft}{\kern0pt}{\isadigit{3}}{\isacharparenright}{\kern0pt}\ LeastI{\isacharunderscore}{\kern0pt}ex\ {\isadigit{0}}\ mpi{\isacharunderscore}{\kern0pt}val{\isacharunderscore}{\kern0pt}term{\isacharunderscore}{\kern0pt}ex{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}case\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assms\ {\isadigit{0}}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ mpi{\isacharunderscore}{\kern0pt}val{\isacharunderscore}{\kern0pt}def\ mpi{\isacharunderscore}{\kern0pt}algo{\isacharunderscore}{\kern0pt}def{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{case}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}Suc\ n{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ not{\isadigit{0}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymnot}\ {\isacharparenleft}{\kern0pt}{\isadigit{2}}\ {\isacharasterisk}{\kern0pt}\ l\ {\isacharasterisk}{\kern0pt}\ dist\ v\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ v{\isacharparenright}{\kern0pt}\ {\isacharless}{\kern0pt}\ eps\ {\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}\ {\isacharminus}{\kern0pt}\ l{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ Suc{\isacharparenleft}{\kern0pt}{\isadigit{3}}{\isacharparenright}{\kern0pt}\ mpi{\isacharunderscore}{\kern0pt}val{\isacharunderscore}{\kern0pt}zero\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \isacommand{obtain}\isamarkupfalse%
-\ n{\isacharprime}{\kern0pt}\ \isakeyword{where}\ {\isachardoublequoteopen}{\isadigit{2}}\ {\isacharasterisk}{\kern0pt}\ l\ {\isacharasterisk}{\kern0pt}\ dist\ {\isacharparenleft}{\kern0pt}mpi{\isacharunderscore}{\kern0pt}val\ d\ v\ m\ n{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ {\isacharparenleft}{\kern0pt}mpi{\isacharunderscore}{\kern0pt}val\ d\ v\ m\ n{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharless}{\kern0pt}\ eps\ {\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}\ {\isacharminus}{\kern0pt}\ l{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ mpi{\isacharunderscore}{\kern0pt}val{\isacharunderscore}{\kern0pt}term{\isacharunderscore}{\kern0pt}ex{\isacharbrackleft}{\kern0pt}OF\ Suc{\isacharparenleft}{\kern0pt}{\isadigit{3}}{\isacharparenright}{\kern0pt}\ Suc{\isacharparenleft}{\kern0pt}{\isadigit{4}}{\isacharparenright}{\kern0pt}{\isacharcomma}{\kern0pt}\ of\ {\isacharunderscore}{\kern0pt}\ m{\isacharbrackright}{\kern0pt}\ assms\ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}n\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}LEAST\ n{\isachardot}{\kern0pt}\ {\isacharquery}{\kern0pt}P\ d\ v\ m\ {\isacharparenleft}{\kern0pt}Suc\ n{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ Suc{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}\ Suc\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}subst\ {\isacharparenleft}{\kern0pt}asm{\isacharparenright}{\kern0pt}\ Least{\isacharunderscore}{\kern0pt}Suc{\isacharparenright}{\kern0pt}\ auto\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}n\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}LEAST\ n{\isachardot}{\kern0pt}\ {\isacharquery}{\kern0pt}P\ {\isacharparenleft}{\kern0pt}policy{\isacharunderscore}{\kern0pt}improvement\ d\ v{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}L{\isacharunderscore}{\kern0pt}pow\ v\ {\isacharparenleft}{\kern0pt}policy{\isacharunderscore}{\kern0pt}improvement\ d\ v{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}m\ {\isadigit{0}}\ v{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymlambda}n{\isachardot}{\kern0pt}\ m\ {\isacharparenleft}{\kern0pt}Suc\ n{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ n{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ Suc{\isacharparenleft}{\kern0pt}{\isadigit{3}}{\isacharparenright}{\kern0pt}\ policy{\isacharunderscore}{\kern0pt}improvement{\isacharunderscore}{\kern0pt}is{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\ mpi{\isacharunderscore}{\kern0pt}Suc{\isacharprime}{\kern0pt}\ \isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ mpi{\isacharunderscore}{\kern0pt}val{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}mpi{\isacharunderscore}{\kern0pt}algo\ d\ v\ m\ {\isacharequal}{\kern0pt}\ mpi\ d\ v\ m\ {\isacharparenleft}{\kern0pt}Suc\ n{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ mpi{\isacharunderscore}{\kern0pt}algo{\isacharunderscore}{\kern0pt}def{\isacharprime}{\kern0pt}{\isacharbrackleft}{\kern0pt}OF\ Suc{\isachardot}{\kern0pt}prems{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharminus}{\kern0pt}{\isadigit{4}}{\isacharparenright}{\kern0pt}{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ Suc{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}\ Suc{\isachardot}{\kern0pt}prems{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharminus}{\kern0pt}{\isadigit{4}}{\isacharparenright}{\kern0pt}\ is{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det{\isacharunderscore}{\kern0pt}pi\ mpi{\isacharunderscore}{\kern0pt}Suc{\isacharprime}{\kern0pt}\ not{\isadigit{0}}\ L{\isacharunderscore}{\kern0pt}pow{\isacharunderscore}{\kern0pt}{\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}mono{\isacharunderscore}{\kern0pt}inv\ \isacommand{by}\isamarkupfalse%
-\ force\isanewline
-\ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}case\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ Suc{\isachardot}{\kern0pt}prems{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}\ \isacommand{by}\isamarkupfalse%
-\ presburger\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ mpi{\isacharunderscore}{\kern0pt}algo{\isacharunderscore}{\kern0pt}opt{\isacharcolon}{\kern0pt}\ \isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}v{\isadigit{0}}\ {\isasymle}\ {\isasymL}\isactrlsub b\ v{\isadigit{0}}{\isachardoublequoteclose}\ {\isachardoublequoteopen}eps\ {\isachargreater}{\kern0pt}\ {\isadigit{0}}{\isachardoublequoteclose}\ {\isachardoublequoteopen}d\ {\isasymin}\ D\isactrlsub D{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}dist\ {\isacharparenleft}{\kern0pt}{\isasymnu}\isactrlsub b\ {\isacharparenleft}{\kern0pt}mk{\isacharunderscore}{\kern0pt}stationary{\isacharunderscore}{\kern0pt}det\ {\isacharparenleft}{\kern0pt}fst\ {\isacharparenleft}{\kern0pt}mpi{\isacharunderscore}{\kern0pt}algo\ d\ v{\isadigit{0}}\ m{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt\ {\isacharless}{\kern0pt}\ eps{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}P\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}{\isasymlambda}n{\isachardot}{\kern0pt}\ {\isadigit{2}}\ {\isacharasterisk}{\kern0pt}\ l\ {\isacharasterisk}{\kern0pt}\ dist\ {\isacharparenleft}{\kern0pt}mpi{\isacharunderscore}{\kern0pt}val\ d\ v{\isadigit{0}}\ m\ n{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ {\isacharparenleft}{\kern0pt}mpi{\isacharunderscore}{\kern0pt}val\ d\ v{\isadigit{0}}\ m\ n{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharless}{\kern0pt}\ \ eps\ {\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}\ {\isacharminus}{\kern0pt}\ l{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}n\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}Least\ {\isacharquery}{\kern0pt}P{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}mpi{\isacharunderscore}{\kern0pt}algo\ d\ v{\isadigit{0}}\ m\ {\isacharequal}{\kern0pt}\ mpi\ d\ v{\isadigit{0}}\ m\ {\isacharquery}{\kern0pt}n{\isachardoublequoteclose}\ \isakeyword{and}\ {\isachardoublequoteopen}{\isacharquery}{\kern0pt}P\ {\isacharquery}{\kern0pt}n{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ mpi{\isacharunderscore}{\kern0pt}algo{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}mpi\ LeastI{\isacharunderscore}{\kern0pt}ex{\isacharbrackleft}{\kern0pt}OF\ mpi{\isacharunderscore}{\kern0pt}val{\isacharunderscore}{\kern0pt}term{\isacharunderscore}{\kern0pt}ex{\isacharbrackright}{\kern0pt}\ assms\ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assms\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ mpi{\isacharunderscore}{\kern0pt}pol{\isacharunderscore}{\kern0pt}opt\ mpi{\isacharunderscore}{\kern0pt}pol{\isacharunderscore}{\kern0pt}def{\isacharbrackleft}{\kern0pt}symmetric{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{end}\isamarkupfalse%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Initial Value Estimate \isa{v{\isadigit{0}}{\isacharunderscore}{\kern0pt}mpi}%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\begin{isamarkuptext}%
-We define an initial estimate of the value function for which Modified Policy Iteration 
-  always terminates.%
-\end{isamarkuptext}\isamarkuptrue%
-\isacommand{abbreviation}\isamarkupfalse%
-\ {\isachardoublequoteopen}r{\isacharunderscore}{\kern0pt}min\ {\isasymequiv}\ {\isacharparenleft}{\kern0pt}{\isasymSqinter}s{\isacharprime}{\kern0pt}\ a{\isachardot}{\kern0pt}\ r\ {\isacharparenleft}{\kern0pt}s{\isacharprime}{\kern0pt}{\isacharcomma}{\kern0pt}\ a{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ {\isachardoublequoteopen}v{\isadigit{0}}{\isacharunderscore}{\kern0pt}mpi\ s\ {\isacharequal}{\kern0pt}\ r{\isacharunderscore}{\kern0pt}min\ {\isacharslash}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}\ {\isacharminus}{\kern0pt}\ l{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{lift{\isacharunderscore}{\kern0pt}definition}\isamarkupfalse%
-\ v{\isadigit{0}}{\isacharunderscore}{\kern0pt}mpi\isactrlsub b\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}s\ {\isasymRightarrow}\isactrlsub b\ real{\isachardoublequoteclose}\ \isakeyword{is}\ {\isachardoublequoteopen}v{\isadigit{0}}{\isacharunderscore}{\kern0pt}mpi{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ fastforce%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ v{\isadigit{0}}{\isacharunderscore}{\kern0pt}mpi\isactrlsub b{\isacharunderscore}{\kern0pt}le{\isacharunderscore}{\kern0pt}{\isasymL}\isactrlsub b{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}v{\isadigit{0}}{\isacharunderscore}{\kern0pt}mpi\isactrlsub b\ {\isasymle}\ {\isasymL}\isactrlsub b\ v{\isadigit{0}}{\isacharunderscore}{\kern0pt}mpi\isactrlsub b{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}rule\ less{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}bfunI{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\ x\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}r{\isacharunderscore}{\kern0pt}min\ {\isasymle}\ r\ {\isacharparenleft}{\kern0pt}s{\isacharcomma}{\kern0pt}\ a{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{for}\ s\ a\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}fastforce\ intro{\isacharcolon}{\kern0pt}\ cInf{\isacharunderscore}{\kern0pt}lower{\isadigit{2}}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}r{\isacharunderscore}{\kern0pt}min\ {\isasymle}\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharminus}{\kern0pt}l{\isacharparenright}{\kern0pt}\ {\isacharasterisk}{\kern0pt}\ r\ {\isacharparenleft}{\kern0pt}s{\isacharcomma}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isacharplus}{\kern0pt}\ l\ {\isacharasterisk}{\kern0pt}\ r{\isacharunderscore}{\kern0pt}min{\isachardoublequoteclose}\ \isakeyword{for}\ s\ a\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ disc{\isacharunderscore}{\kern0pt}lt{\isacharunderscore}{\kern0pt}one\ zero{\isacharunderscore}{\kern0pt}le{\isacharunderscore}{\kern0pt}disc\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}meson\ order{\isacharunderscore}{\kern0pt}less{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}le\ order{\isacharunderscore}{\kern0pt}refl\ segment{\isacharunderscore}{\kern0pt}bound{\isacharunderscore}{\kern0pt}lemma{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}r{\isacharunderscore}{\kern0pt}min\ {\isacharslash}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}\ {\isacharminus}{\kern0pt}\ l{\isacharparenright}{\kern0pt}\ {\isasymle}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharminus}{\kern0pt}l{\isacharparenright}{\kern0pt}\ {\isacharasterisk}{\kern0pt}\ r\ {\isacharparenleft}{\kern0pt}s{\isacharcomma}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isacharplus}{\kern0pt}\ l\ {\isacharasterisk}{\kern0pt}\ r{\isacharunderscore}{\kern0pt}min{\isacharparenright}{\kern0pt}\ {\isacharslash}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}\ {\isacharminus}{\kern0pt}\ l{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{for}\ s\ a\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ order{\isacharunderscore}{\kern0pt}less{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}le{\isacharbrackleft}{\kern0pt}OF\ disc{\isacharunderscore}{\kern0pt}lt{\isacharunderscore}{\kern0pt}one{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ divide{\isacharunderscore}{\kern0pt}right{\isacharunderscore}{\kern0pt}mono{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}r{\isacharunderscore}{\kern0pt}min\ {\isacharslash}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}\ {\isacharminus}{\kern0pt}\ l{\isacharparenright}{\kern0pt}\ {\isasymle}\ r\ {\isacharparenleft}{\kern0pt}s{\isacharcomma}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isacharplus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}l\ {\isacharasterisk}{\kern0pt}\ r{\isacharunderscore}{\kern0pt}min{\isacharparenright}{\kern0pt}\ {\isacharslash}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}\ {\isacharminus}{\kern0pt}\ l{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{for}\ s\ a\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ disc{\isacharunderscore}{\kern0pt}lt{\isacharunderscore}{\kern0pt}one\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ add{\isacharunderscore}{\kern0pt}divide{\isacharunderscore}{\kern0pt}distrib{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}v{\isadigit{0}}{\isacharunderscore}{\kern0pt}mpi\isactrlsub b\ x\ {\isasymle}\ {\isasymL}\isactrlsub b\ v{\isadigit{0}}{\isacharunderscore}{\kern0pt}mpi\isactrlsub b\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ {\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}SUP{\isacharunderscore}{\kern0pt}L\isactrlsub a\ v{\isadigit{0}}{\isacharunderscore}{\kern0pt}mpi\isactrlsub b{\isachardot}{\kern0pt}rep{\isacharunderscore}{\kern0pt}eq\ v{\isadigit{0}}{\isacharunderscore}{\kern0pt}mpi{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ A{\isacharunderscore}{\kern0pt}ne\ intro{\isacharcolon}{\kern0pt}\ cSUP{\isacharunderscore}{\kern0pt}upper{\isadigit{2}}{\isacharbrackleft}{\kern0pt}\isakeyword{where}\ x\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}arb{\isacharunderscore}{\kern0pt}act\ {\isacharparenleft}{\kern0pt}A\ x{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{An Instance of Modified Policy Iteration with a Valid Conservative Initial Value Estimate%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{definition}\isamarkupfalse%
-\ {\isachardoublequoteopen}mpi{\isacharunderscore}{\kern0pt}user\ eps\ m\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}\isanewline
-\ \ if\ eps\ {\isasymle}\ {\isadigit{0}}\ then\ undefined\ else\ mpi{\isacharunderscore}{\kern0pt}algo\ eps\ {\isacharparenleft}{\kern0pt}{\isasymlambda}x{\isachardot}{\kern0pt}\ arb{\isacharunderscore}{\kern0pt}act\ {\isacharparenleft}{\kern0pt}A\ x{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ v{\isadigit{0}}{\isacharunderscore}{\kern0pt}mpi\isactrlsub b\ m{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ mpi{\isacharunderscore}{\kern0pt}user{\isacharunderscore}{\kern0pt}eq{\isacharcolon}{\kern0pt}\ \isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}eps\ {\isachargreater}{\kern0pt}\ {\isadigit{0}}{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}mpi{\isacharunderscore}{\kern0pt}user\ eps\ {\isacharequal}{\kern0pt}\ mpi{\isacharunderscore}{\kern0pt}alg{\isacharunderscore}{\kern0pt}rec\ eps\ {\isacharparenleft}{\kern0pt}{\isasymlambda}x{\isachardot}{\kern0pt}\ arb{\isacharunderscore}{\kern0pt}act\ {\isacharparenleft}{\kern0pt}A\ x{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ v{\isadigit{0}}{\isacharunderscore}{\kern0pt}mpi\isactrlsub b{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ v{\isadigit{0}}{\isacharunderscore}{\kern0pt}mpi\isactrlsub b{\isacharunderscore}{\kern0pt}le{\isacharunderscore}{\kern0pt}{\isasymL}\isactrlsub b\ assms\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ mpi{\isacharunderscore}{\kern0pt}user{\isacharunderscore}{\kern0pt}def\ mpi{\isacharunderscore}{\kern0pt}algo{\isacharunderscore}{\kern0pt}def{\isacharprime}{\kern0pt}\ A{\isacharunderscore}{\kern0pt}ne\ is{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ mpi{\isacharunderscore}{\kern0pt}user{\isacharunderscore}{\kern0pt}opt{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}eps\ {\isachargreater}{\kern0pt}\ {\isadigit{0}}{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}dist\ {\isacharparenleft}{\kern0pt}{\isasymnu}\isactrlsub b\ {\isacharparenleft}{\kern0pt}mk{\isacharunderscore}{\kern0pt}stationary{\isacharunderscore}{\kern0pt}det\ {\isacharparenleft}{\kern0pt}fst\ {\isacharparenleft}{\kern0pt}mpi{\isacharunderscore}{\kern0pt}user\ eps\ n{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt\ {\isacharless}{\kern0pt}\ eps{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ mpi{\isacharunderscore}{\kern0pt}user{\isacharunderscore}{\kern0pt}def\ \isacommand{using}\isamarkupfalse%
-\ assms\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ intro{\isacharcolon}{\kern0pt}\ mpi{\isacharunderscore}{\kern0pt}algo{\isacharunderscore}{\kern0pt}opt\ simp{\isacharcolon}{\kern0pt}\ is{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det{\isacharunderscore}{\kern0pt}def\ A{\isacharunderscore}{\kern0pt}ne\ v{\isadigit{0}}{\isacharunderscore}{\kern0pt}mpi\isactrlsub b{\isacharunderscore}{\kern0pt}le{\isacharunderscore}{\kern0pt}{\isasymL}\isactrlsub b{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{end}\isamarkupfalse%
-\isanewline
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-\isadelimtheory
-\isanewline
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-\endisadelimtheory
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-\isatagtheory
-\isacommand{end}\isamarkupfalse%
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-\endisatagtheory
-{\isafoldtheory}%
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-\isadelimtheory
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-\endisadelimtheory
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-\end{isabellebody}%
-\endinput
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diff --git a/thys/MDP-Algorithms/output/document/Policy_Iteration.tex b/thys/MDP-Algorithms/output/document/Policy_Iteration.tex
deleted file mode 100644
--- a/thys/MDP-Algorithms/output/document/Policy_Iteration.tex
+++ /dev/null
@@ -1,1335 +0,0 @@
-%
-\begin{isabellebody}%
-\setisabellecontext{Policy{\isacharunderscore}{\kern0pt}Iteration}%
-%
-\isadelimtheory
-\isanewline
-\isanewline
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{theory}\isamarkupfalse%
-\ Policy{\isacharunderscore}{\kern0pt}Iteration\isanewline
-\ \ \isakeyword{imports}\ {\isachardoublequoteopen}MDP{\isacharminus}{\kern0pt}Rewards{\isachardot}{\kern0pt}MDP{\isacharunderscore}{\kern0pt}reward{\isachardoublequoteclose}\isanewline
-\isanewline
-\isakeyword{begin}%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsection{Policy Iteration%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\begin{isamarkuptext}%
-The Policy Iteration algorithms provides another way to find optimal policies under the expected 
-total reward criterion.
-It differs from Value Iteration in that it continuously improves an initial guess for an optimal 
-decision rule. Its execution can be subdivided into two alternating steps: policy evaluation and 
-policy improvement.
-
-Policy evaluation means the calculation of the value of the current decision rule.
-
-During the improvement phase, we choose the decision rule with the maximum value for L, 
-while we prefer to keep the old action selection in case of ties.%
-\end{isamarkuptext}\isamarkuptrue%
-\isacommand{context}\isamarkupfalse%
-\ MDP{\isacharunderscore}{\kern0pt}att{\isacharunderscore}{\kern0pt}{\isasymL}\ \isakeyword{begin}\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ {\isachardoublequoteopen}policy{\isacharunderscore}{\kern0pt}eval\ d\ {\isacharequal}{\kern0pt}\ {\isasymnu}\isactrlsub b\ {\isacharparenleft}{\kern0pt}mk{\isacharunderscore}{\kern0pt}stationary{\isacharunderscore}{\kern0pt}det\ d{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isacommand{end}\isamarkupfalse%
-\isanewline
-\isanewline
-\isacommand{context}\isamarkupfalse%
-\ MDP{\isacharunderscore}{\kern0pt}act\isanewline
-\isakeyword{begin}\isanewline
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ {\isachardoublequoteopen}policy{\isacharunderscore}{\kern0pt}improvement\ d\ v\ s\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}\isanewline
-\ \ if\ is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max\ {\isacharparenleft}{\kern0pt}{\isasymlambda}a{\isachardot}{\kern0pt}\ L\isactrlsub a\ a\ {\isacharparenleft}{\kern0pt}apply{\isacharunderscore}{\kern0pt}bfun\ v{\isacharparenright}{\kern0pt}\ s{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymlambda}a{\isachardot}{\kern0pt}\ a\ {\isasymin}\ A\ s{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}d\ s{\isacharparenright}{\kern0pt}\ \isanewline
-\ \ then\ d\ s\isanewline
-\ \ else\ arb{\isacharunderscore}{\kern0pt}act\ {\isacharparenleft}{\kern0pt}opt{\isacharunderscore}{\kern0pt}acts\ v\ s{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ {\isachardoublequoteopen}policy{\isacharunderscore}{\kern0pt}step\ d\ {\isacharequal}{\kern0pt}\ policy{\isacharunderscore}{\kern0pt}improvement\ d\ {\isacharparenleft}{\kern0pt}policy{\isacharunderscore}{\kern0pt}eval\ d{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isanewline
-\isacommand{function}\isamarkupfalse%
-\ policy{\isacharunderscore}{\kern0pt}iteration\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}s\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a{\isacharparenright}{\kern0pt}\ {\isasymRightarrow}\ {\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}s\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}policy{\isacharunderscore}{\kern0pt}iteration\ d\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}\isanewline
-\ \ let\ d{\isacharprime}{\kern0pt}\ {\isacharequal}{\kern0pt}\ policy{\isacharunderscore}{\kern0pt}step\ d\ in\isanewline
-\ \ if\ d\ {\isacharequal}{\kern0pt}\ d{\isacharprime}{\kern0pt}\ {\isasymor}\ {\isasymnot}is{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\ d\ then\ d\ else\ policy{\isacharunderscore}{\kern0pt}iteration\ d{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\begin{isamarkuptext}%
-The policy iteration algorithm as stated above does require that the supremum in \isa{{\isasymL}\isactrlsub b} is
-always attained.%
-\end{isamarkuptext}\isamarkuptrue%
-%
-\begin{isamarkuptext}%
-Each policy improvement returns a valid decision rule.%
-\end{isamarkuptext}\isamarkuptrue%
-\isacommand{lemma}\isamarkupfalse%
-\ is{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det{\isacharunderscore}{\kern0pt}pi{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}is{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\ {\isacharparenleft}{\kern0pt}policy{\isacharunderscore}{\kern0pt}improvement\ d\ v{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ policy{\isacharunderscore}{\kern0pt}improvement{\isacharunderscore}{\kern0pt}def\ is{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det{\isacharunderscore}{\kern0pt}def\ is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ some{\isacharunderscore}{\kern0pt}opt{\isacharunderscore}{\kern0pt}acts{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ policy{\isacharunderscore}{\kern0pt}improvement{\isacharunderscore}{\kern0pt}is{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}d\ {\isasymin}\ D\isactrlsub D\ {\isasymLongrightarrow}\ policy{\isacharunderscore}{\kern0pt}improvement\ d\ v\ {\isasymin}\ D\isactrlsub D{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ policy{\isacharunderscore}{\kern0pt}improvement{\isacharunderscore}{\kern0pt}def\ is{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \isacommand{using}\isamarkupfalse%
-\ some{\isacharunderscore}{\kern0pt}opt{\isacharunderscore}{\kern0pt}acts{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ policy{\isacharunderscore}{\kern0pt}improvement{\isacharunderscore}{\kern0pt}improving{\isacharcolon}{\kern0pt}\ \isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}d\ {\isasymin}\ D\isactrlsub D{\isachardoublequoteclose}\ \isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}{\isasymnu}{\isacharunderscore}{\kern0pt}improving\ v\ {\isacharparenleft}{\kern0pt}mk{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\ {\isacharparenleft}{\kern0pt}policy{\isacharunderscore}{\kern0pt}improvement\ d\ v{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymL}\isactrlsub b\ v\ x\ {\isacharequal}{\kern0pt}\ L\ {\isacharparenleft}{\kern0pt}mk{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\ {\isacharparenleft}{\kern0pt}policy{\isacharunderscore}{\kern0pt}improvement\ d\ v{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ v\ x{\isachardoublequoteclose}\ \isakeyword{for}\ x\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ is{\isacharunderscore}{\kern0pt}opt{\isacharunderscore}{\kern0pt}act{\isacharunderscore}{\kern0pt}some\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}fastforce\ simp{\isacharcolon}{\kern0pt}\ thm{\isacharunderscore}{\kern0pt}{\isadigit{6}}{\isacharunderscore}{\kern0pt}{\isadigit{2}}{\isacharunderscore}{\kern0pt}{\isadigit{1}}{\isadigit{0}}{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}aux{\isacharprime}{\kern0pt}\ L{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}L\isactrlsub a{\isacharunderscore}{\kern0pt}det\ is{\isacharunderscore}{\kern0pt}opt{\isacharunderscore}{\kern0pt}act{\isacharunderscore}{\kern0pt}def\ policy{\isacharunderscore}{\kern0pt}improvement{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}SUP{\isacharbrackleft}{\kern0pt}symmetric{\isacharcomma}{\kern0pt}\ of\ {\isacharunderscore}{\kern0pt}\ {\isacharunderscore}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}policy{\isacharunderscore}{\kern0pt}improvement\ d\ v\ x{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}{\isacharbrackright}{\kern0pt}\ {\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ policy{\isacharunderscore}{\kern0pt}improvement{\isacharunderscore}{\kern0pt}is{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\ assms\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ {\isasymnu}{\isacharunderscore}{\kern0pt}improving{\isacharunderscore}{\kern0pt}alt{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ eval{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}step{\isacharunderscore}{\kern0pt}L{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}is{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\ d{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}L\ {\isacharparenleft}{\kern0pt}mk{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\ {\isacharparenleft}{\kern0pt}policy{\isacharunderscore}{\kern0pt}step\ d{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}policy{\isacharunderscore}{\kern0pt}eval\ d{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isasymL}\isactrlsub b\ {\isacharparenleft}{\kern0pt}policy{\isacharunderscore}{\kern0pt}eval\ d{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ policy{\isacharunderscore}{\kern0pt}step{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \isacommand{using}\isamarkupfalse%
-\ assms\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ {\isasymnu}{\isacharunderscore}{\kern0pt}improving{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}{\isasymL}\isactrlsub b{\isacharbrackleft}{\kern0pt}OF\ policy{\isacharunderscore}{\kern0pt}improvement{\isacharunderscore}{\kern0pt}improving{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\begin{isamarkuptext}%
-The sequence of policies generated by policy iteration has monotonically increasing 
-discounted reward.%
-\end{isamarkuptext}\isamarkuptrue%
-\isacommand{lemma}\isamarkupfalse%
-\ policy{\isacharunderscore}{\kern0pt}eval{\isacharunderscore}{\kern0pt}mon{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}is{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\ d{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}policy{\isacharunderscore}{\kern0pt}eval\ d\ {\isasymle}\ policy{\isacharunderscore}{\kern0pt}eval\ {\isacharparenleft}{\kern0pt}policy{\isacharunderscore}{\kern0pt}step\ d{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}d{\isacharprime}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}mk{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\ {\isacharparenleft}{\kern0pt}policy{\isacharunderscore}{\kern0pt}step\ d{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}dp\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}mk{\isacharunderscore}{\kern0pt}stationary{\isacharunderscore}{\kern0pt}det\ d{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}P\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}{\isasymSum}t{\isachardot}{\kern0pt}\ l\ {\isacharcircum}{\kern0pt}\ t\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isasymP}\isactrlsub {\isadigit{1}}\ {\isacharquery}{\kern0pt}d{\isacharprime}{\kern0pt}\ {\isacharcircum}{\kern0pt}{\isacharcircum}{\kern0pt}\ t{\isachardoublequoteclose}\isanewline
-\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}L\ {\isacharparenleft}{\kern0pt}mk{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\ d{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}policy{\isacharunderscore}{\kern0pt}eval\ d{\isacharparenright}{\kern0pt}\ {\isasymle}\ L\ {\isacharquery}{\kern0pt}d{\isacharprime}{\kern0pt}\ {\isacharparenleft}{\kern0pt}policy{\isacharunderscore}{\kern0pt}eval\ d{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assms\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ L{\isacharunderscore}{\kern0pt}le{\isacharunderscore}{\kern0pt}{\isasymL}\isactrlsub b\ eval{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}step{\isacharunderscore}{\kern0pt}L{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}policy{\isacharunderscore}{\kern0pt}eval\ d\ {\isasymle}\ L\ {\isacharquery}{\kern0pt}d{\isacharprime}{\kern0pt}\ {\isacharparenleft}{\kern0pt}policy{\isacharunderscore}{\kern0pt}eval\ d{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ L{\isacharunderscore}{\kern0pt}{\isasymnu}{\isacharunderscore}{\kern0pt}fix\ policy{\isacharunderscore}{\kern0pt}eval{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymnu}\isactrlsub b\ {\isacharquery}{\kern0pt}dp\ {\isasymle}\ r{\isacharunderscore}{\kern0pt}dec\isactrlsub b\ {\isacharquery}{\kern0pt}d{\isacharprime}{\kern0pt}\ {\isacharplus}{\kern0pt}\ l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isasymP}\isactrlsub {\isadigit{1}}\ {\isacharquery}{\kern0pt}d{\isacharprime}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymnu}\isactrlsub b\ {\isacharquery}{\kern0pt}dp{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ policy{\isacharunderscore}{\kern0pt}eval{\isacharunderscore}{\kern0pt}def\ L{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}id{\isacharunderscore}{\kern0pt}blinfun\ {\isacharminus}{\kern0pt}\ l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isasymP}\isactrlsub {\isadigit{1}}\ {\isacharquery}{\kern0pt}d{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymnu}\isactrlsub b\ {\isacharquery}{\kern0pt}dp{\isacharparenright}{\kern0pt}\ {\isasymle}\ r{\isacharunderscore}{\kern0pt}dec\isactrlsub b\ {\isacharquery}{\kern0pt}d{\isacharprime}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ blinfun{\isachardot}{\kern0pt}diff{\isacharunderscore}{\kern0pt}left\ diff{\isacharunderscore}{\kern0pt}le{\isacharunderscore}{\kern0pt}eq\ scaleR{\isacharunderscore}{\kern0pt}blinfun{\isachardot}{\kern0pt}rep{\isacharunderscore}{\kern0pt}eq{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharquery}{\kern0pt}P\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}id{\isacharunderscore}{\kern0pt}blinfun\ {\isacharminus}{\kern0pt}\ l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isasymP}\isactrlsub {\isadigit{1}}\ {\isacharquery}{\kern0pt}d{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymnu}\isactrlsub b\ {\isacharquery}{\kern0pt}dp{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymle}\ {\isacharquery}{\kern0pt}P\ {\isacharparenleft}{\kern0pt}r{\isacharunderscore}{\kern0pt}dec\isactrlsub b\ {\isacharquery}{\kern0pt}d{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ lemma{\isacharunderscore}{\kern0pt}{\isadigit{6}}{\isacharunderscore}{\kern0pt}{\isadigit{1}}{\isacharunderscore}{\kern0pt}{\isadigit{2}}{\isacharunderscore}{\kern0pt}b\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymnu}\isactrlsub b\ {\isacharquery}{\kern0pt}dp\ {\isasymle}\ {\isacharquery}{\kern0pt}P\ {\isacharparenleft}{\kern0pt}r{\isacharunderscore}{\kern0pt}dec\isactrlsub b\ {\isacharquery}{\kern0pt}d{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ inv{\isacharunderscore}{\kern0pt}norm{\isacharunderscore}{\kern0pt}le{\isacharprime}{\kern0pt}{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isacharbrackleft}{\kern0pt}OF\ norm{\isacharunderscore}{\kern0pt}{\isasymP}\isactrlsub {\isadigit{1}}{\isacharunderscore}{\kern0pt}l{\isacharunderscore}{\kern0pt}less{\isacharbrackright}{\kern0pt}\ blincomp{\isacharunderscore}{\kern0pt}scaleR{\isacharunderscore}{\kern0pt}right\ suminf{\isacharunderscore}{\kern0pt}cong\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}mono{\isacharunderscore}{\kern0pt}tags{\isacharcomma}{\kern0pt}\ lifting{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ policy{\isacharunderscore}{\kern0pt}eval{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ {\isasymnu}{\isacharunderscore}{\kern0pt}stationary{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\begin{isamarkuptext}%
-If policy iteration terminates, i.e. \isa{d\ {\isacharequal}{\kern0pt}\ policy{\isacharunderscore}{\kern0pt}step\ d}, then it does so with optimal value.%
-\end{isamarkuptext}\isamarkuptrue%
-\isacommand{lemma}\isamarkupfalse%
-\ policy{\isacharunderscore}{\kern0pt}step{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}opt{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}is{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\ d{\isachardoublequoteclose}\ {\isachardoublequoteopen}d\ {\isacharequal}{\kern0pt}\ policy{\isacharunderscore}{\kern0pt}step\ d{\isachardoublequoteclose}\ \isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}{\isasymnu}\isactrlsub b\ {\isacharparenleft}{\kern0pt}mk{\isacharunderscore}{\kern0pt}stationary\ {\isacharparenleft}{\kern0pt}mk{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\ d{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}policy{\isacharunderscore}{\kern0pt}eval\ d\ {\isacharequal}{\kern0pt}\ {\isasymL}\isactrlsub b\ {\isacharparenleft}{\kern0pt}policy{\isacharunderscore}{\kern0pt}eval\ d{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ policy{\isacharunderscore}{\kern0pt}eval{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ L{\isacharunderscore}{\kern0pt}{\isasymnu}{\isacharunderscore}{\kern0pt}fix\ assms\ eval{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}step{\isacharunderscore}{\kern0pt}L{\isacharbrackleft}{\kern0pt}unfolded\ policy{\isacharunderscore}{\kern0pt}eval{\isacharunderscore}{\kern0pt}def{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ fastforce\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ policy{\isacharunderscore}{\kern0pt}eval{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isasymL}{\isacharunderscore}{\kern0pt}fix{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}opt\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{end}\isamarkupfalse%
-%
-\begin{isamarkuptext}%
-We prove termination of policy iteration only if both the state and action sets are finite.%
-\end{isamarkuptext}\isamarkuptrue%
-\isacommand{locale}\isamarkupfalse%
-\ MDP{\isacharunderscore}{\kern0pt}PI{\isacharunderscore}{\kern0pt}finite\ {\isacharequal}{\kern0pt}\ MDP{\isacharunderscore}{\kern0pt}act\ A\ K\ r\ l\ arb{\isacharunderscore}{\kern0pt}act\isanewline
-\ \ \isakeyword{for}\isanewline
-\ \ \ \ A\ \isakeyword{and}\isanewline
-\ \ \ \ K\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}s\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}countable\ {\isasymtimes}\ {\isacharprime}{\kern0pt}a\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}countable\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}s\ pmf{\isachardoublequoteclose}\ \isakeyword{and}\ r\ l\ arb{\isacharunderscore}{\kern0pt}act\ {\isacharplus}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ fin{\isacharunderscore}{\kern0pt}states{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite\ {\isacharparenleft}{\kern0pt}UNIV\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isacharprime}{\kern0pt}s\ set{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{and}\ fin{\isacharunderscore}{\kern0pt}actions{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymAnd}s{\isachardot}{\kern0pt}\ finite\ {\isacharparenleft}{\kern0pt}A\ s{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isakeyword{begin}%
-\begin{isamarkuptext}%
-If the state and action sets are both finite, 
-  then so is the set of deterministic decision rules \isa{D\isactrlsub D}%
-\end{isamarkuptext}\isamarkuptrue%
-\isacommand{lemma}\isamarkupfalse%
-\ finite{\isacharunderscore}{\kern0pt}D\isactrlsub D{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite\ D\isactrlsub D{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}set\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}d{\isachardot}{\kern0pt}\ {\isasymforall}x\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isacharprime}{\kern0pt}s{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}x\ {\isasymin}\ UNIV\ {\isasymlongrightarrow}\ d\ x\ {\isasymin}\ {\isacharparenleft}{\kern0pt}{\isasymUnion}s{\isachardot}{\kern0pt}\ A\ s{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymand}\ {\isacharparenleft}{\kern0pt}x\ {\isasymnotin}\ UNIV\ {\isasymlongrightarrow}\ d\ x\ {\isacharequal}{\kern0pt}\ undefined{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}finite\ {\isacharparenleft}{\kern0pt}{\isasymUnion}s{\isachardot}{\kern0pt}\ A\ s{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ fin{\isacharunderscore}{\kern0pt}actions\ fin{\isacharunderscore}{\kern0pt}states\ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}finite\ {\isacharquery}{\kern0pt}set{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ fin{\isacharunderscore}{\kern0pt}states\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}fastforce\ intro{\isacharcolon}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}set{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}finite{\isacharunderscore}{\kern0pt}funs{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}D\isactrlsub D\ {\isasymsubseteq}\ {\isacharquery}{\kern0pt}set{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ is{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det{\isacharunderscore}{\kern0pt}def\ \isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ finite{\isacharunderscore}{\kern0pt}subset\ \isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ finite{\isacharunderscore}{\kern0pt}rel{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite\ {\isacharbraceleft}{\kern0pt}{\isacharparenleft}{\kern0pt}u{\isacharcomma}{\kern0pt}\ v{\isacharparenright}{\kern0pt}{\isachardot}{\kern0pt}\ is{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\ u\ {\isasymand}\ is{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\ v\ {\isasymand}\ {\isasymnu}\isactrlsub b\ {\isacharparenleft}{\kern0pt}mk{\isacharunderscore}{\kern0pt}stationary{\isacharunderscore}{\kern0pt}det\ u{\isacharparenright}{\kern0pt}\ {\isachargreater}{\kern0pt}\ \isanewline
-\ \ {\isasymnu}\isactrlsub b\ {\isacharparenleft}{\kern0pt}mk{\isacharunderscore}{\kern0pt}stationary{\isacharunderscore}{\kern0pt}det\ v{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-{\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ aux{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite\ {\isacharbraceleft}{\kern0pt}{\isacharparenleft}{\kern0pt}u{\isacharcomma}{\kern0pt}\ v{\isacharparenright}{\kern0pt}{\isachardot}{\kern0pt}\ is{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\ u\ {\isasymand}\ is{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\ v{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ intro{\isacharcolon}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}subset{\isacharbrackleft}{\kern0pt}OF\ {\isacharunderscore}{\kern0pt}\ aux{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\begin{isamarkuptext}%
-This auxiliary lemma shows that policy iteration terminates if no improvement to the value of 
-the policy could be made, as then the policy remains unchanged.%
-\end{isamarkuptext}\isamarkuptrue%
-\isacommand{lemma}\isamarkupfalse%
-\ eval{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}eq{\isacharcolon}{\kern0pt}\ \isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}policy{\isacharunderscore}{\kern0pt}eval\ d\ {\isacharequal}{\kern0pt}\ policy{\isacharunderscore}{\kern0pt}eval\ {\isacharparenleft}{\kern0pt}policy{\isacharunderscore}{\kern0pt}step\ d{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ {\isachardoublequoteopen}is{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\ d{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}d\ {\isacharequal}{\kern0pt}\ policy{\isacharunderscore}{\kern0pt}step\ d{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}policy{\isacharunderscore}{\kern0pt}eval\ d\ s\ {\isacharequal}{\kern0pt}\ policy{\isacharunderscore}{\kern0pt}eval\ {\isacharparenleft}{\kern0pt}policy{\isacharunderscore}{\kern0pt}step\ d{\isacharparenright}{\kern0pt}\ s{\isachardoublequoteclose}\ \isakeyword{for}\ s\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assms\ \isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}policy{\isacharunderscore}{\kern0pt}eval\ d\ {\isacharequal}{\kern0pt}\ L\ {\isacharparenleft}{\kern0pt}mk{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\ d{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}policy{\isacharunderscore}{\kern0pt}eval\ {\isacharparenleft}{\kern0pt}policy{\isacharunderscore}{\kern0pt}step\ d{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ policy{\isacharunderscore}{\kern0pt}eval{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ L{\isacharunderscore}{\kern0pt}{\isasymnu}{\isacharunderscore}{\kern0pt}fix\ \isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ assms{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}{\isacharbrackleft}{\kern0pt}symmetric{\isacharcomma}{\kern0pt}\ unfolded\ policy{\isacharunderscore}{\kern0pt}eval{\isacharunderscore}{\kern0pt}def{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}policy{\isacharunderscore}{\kern0pt}eval\ d\ {\isacharequal}{\kern0pt}\ {\isasymL}\isactrlsub b\ {\isacharparenleft}{\kern0pt}policy{\isacharunderscore}{\kern0pt}eval\ d{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ L{\isacharunderscore}{\kern0pt}{\isasymnu}{\isacharunderscore}{\kern0pt}fix\ policy{\isacharunderscore}{\kern0pt}eval{\isacharunderscore}{\kern0pt}def\ assms\ eval{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}step{\isacharunderscore}{\kern0pt}L{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}L\ {\isacharparenleft}{\kern0pt}mk{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\ d{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}policy{\isacharunderscore}{\kern0pt}eval\ d{\isacharparenright}{\kern0pt}\ s\ {\isacharequal}{\kern0pt}\ {\isasymL}\isactrlsub b\ {\isacharparenleft}{\kern0pt}policy{\isacharunderscore}{\kern0pt}eval\ d{\isacharparenright}{\kern0pt}\ s{\isachardoublequoteclose}\ \isakeyword{for}\ s\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isacartoucheopen}policy{\isacharunderscore}{\kern0pt}eval\ d\ {\isacharequal}{\kern0pt}\ L\ {\isacharparenleft}{\kern0pt}mk{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\ d{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}policy{\isacharunderscore}{\kern0pt}eval\ {\isacharparenleft}{\kern0pt}policy{\isacharunderscore}{\kern0pt}step\ d{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacartoucheclose}\ assms{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}\ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max\ {\isacharparenleft}{\kern0pt}{\isasymlambda}a{\isachardot}{\kern0pt}\ L\isactrlsub a\ a\ {\isacharparenleft}{\kern0pt}{\isasymnu}\isactrlsub b\ {\isacharparenleft}{\kern0pt}mk{\isacharunderscore}{\kern0pt}stationary\ {\isacharparenleft}{\kern0pt}mk{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\ d{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ s{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymlambda}a{\isachardot}{\kern0pt}\ a\ {\isasymin}\ A\ s{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}d\ s{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{for}\ s\isanewline
-\ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ L{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}L\isactrlsub a{\isacharunderscore}{\kern0pt}det\isanewline
-\ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ policy{\isacharunderscore}{\kern0pt}eval{\isacharunderscore}{\kern0pt}def\ {\isasymL}\isactrlsub b{\isachardot}{\kern0pt}rep{\isacharunderscore}{\kern0pt}eq\ {\isasymL}{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}SUP{\isacharunderscore}{\kern0pt}det\ SUP{\isacharunderscore}{\kern0pt}step{\isacharunderscore}{\kern0pt}det{\isacharunderscore}{\kern0pt}eq\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assms{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}\ is{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det{\isacharunderscore}{\kern0pt}def\ L\isactrlsub a{\isacharunderscore}{\kern0pt}le\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp\ del{\isacharcolon}{\kern0pt}\ {\isasymnu}\isactrlsub b{\isachardot}{\kern0pt}rep{\isacharunderscore}{\kern0pt}eq\ simp{\isacharcolon}{\kern0pt}\ {\isasymnu}\isactrlsub b{\isachardot}{\kern0pt}rep{\isacharunderscore}{\kern0pt}eq{\isacharbrackleft}{\kern0pt}symmetric{\isacharbrackright}{\kern0pt}\ \isanewline
-\ \ \ \ \ \ \ \ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ SUP{\isacharunderscore}{\kern0pt}is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max\ boundedI{\isacharbrackleft}{\kern0pt}of\ {\isacharunderscore}{\kern0pt}\ {\isachardoublequoteopen}r\isactrlsub M\ {\isacharplus}{\kern0pt}\ l\ {\isacharasterisk}{\kern0pt}\ norm\ {\isacharunderscore}{\kern0pt}{\isachardoublequoteclose}{\isacharbrackright}{\kern0pt}\ bounded{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}bdd{\isacharunderscore}{\kern0pt}above{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ policy{\isacharunderscore}{\kern0pt}eval{\isacharunderscore}{\kern0pt}def\ policy{\isacharunderscore}{\kern0pt}step{\isacharunderscore}{\kern0pt}def\ policy{\isacharunderscore}{\kern0pt}improvement{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\begin{isamarkuptext}%
-We are now ready to prove termination in the context of finite state-action spaces.
-Intuitively, the algorithm terminates as there are only finitely many decision rules,
-and in each recursive call the value of the decision rule increases.%
-\end{isamarkuptext}\isamarkuptrue%
-\isacommand{termination}\isamarkupfalse%
-\ policy{\isacharunderscore}{\kern0pt}iteration\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}relation\ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}{\isacharparenleft}{\kern0pt}u{\isacharcomma}{\kern0pt}\ v{\isacharparenright}{\kern0pt}{\isachardot}{\kern0pt}\ u\ {\isasymin}\ D\isactrlsub D\ {\isasymand}\ v\ {\isasymin}\ D\isactrlsub D\ {\isasymand}\ {\isasymnu}\isactrlsub b\ {\isacharparenleft}{\kern0pt}mk{\isacharunderscore}{\kern0pt}stationary{\isacharunderscore}{\kern0pt}det\ u{\isacharparenright}{\kern0pt}\ {\isachargreater}{\kern0pt}\ {\isasymnu}\isactrlsub b\ {\isacharparenleft}{\kern0pt}mk{\isacharunderscore}{\kern0pt}stationary{\isacharunderscore}{\kern0pt}det\ v{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}wf\ {\isacharbraceleft}{\kern0pt}{\isacharparenleft}{\kern0pt}u{\isacharcomma}{\kern0pt}\ v{\isacharparenright}{\kern0pt}{\isachardot}{\kern0pt}\ u\ {\isasymin}\ D\isactrlsub D\ {\isasymand}\ v\ {\isasymin}\ D\isactrlsub D\ {\isasymand}\ {\isasymnu}\isactrlsub b\ {\isacharparenleft}{\kern0pt}mk{\isacharunderscore}{\kern0pt}stationary{\isacharunderscore}{\kern0pt}det\ v{\isacharparenright}{\kern0pt}\ {\isacharless}{\kern0pt}\ {\isasymnu}\isactrlsub b\ {\isacharparenleft}{\kern0pt}mk{\isacharunderscore}{\kern0pt}stationary{\isacharunderscore}{\kern0pt}det\ u{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ finite{\isacharunderscore}{\kern0pt}rel\ \isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}acyclic{\isacharunderscore}{\kern0pt}wf\ acyclicI{\isacharunderscore}{\kern0pt}order{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\ d\ x\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\ h{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isacharequal}{\kern0pt}\ policy{\isacharunderscore}{\kern0pt}step\ d{\isachardoublequoteclose}\ {\isachardoublequoteopen}{\isasymnot}\ {\isacharparenleft}{\kern0pt}d\ {\isacharequal}{\kern0pt}\ x\ {\isasymor}\ {\isasymnot}\ is{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\ d{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}is{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\ d\ {\isasymLongrightarrow}\ {\isasymnu}\isactrlsub b\ {\isacharparenleft}{\kern0pt}mk{\isacharunderscore}{\kern0pt}stationary{\isacharunderscore}{\kern0pt}det\ d{\isacharparenright}{\kern0pt}\ {\isasymle}\ {\isasymnu}\isactrlsub b\ {\isacharparenleft}{\kern0pt}mk{\isacharunderscore}{\kern0pt}stationary{\isacharunderscore}{\kern0pt}det\ {\isacharparenleft}{\kern0pt}policy{\isacharunderscore}{\kern0pt}step\ d{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ policy{\isacharunderscore}{\kern0pt}eval{\isacharunderscore}{\kern0pt}mon\ \ \isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ policy{\isacharunderscore}{\kern0pt}eval{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}is{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\ d\ {\isasymLongrightarrow}\ d\ {\isasymnoteq}\ policy{\isacharunderscore}{\kern0pt}step\ d\ {\isasymLongrightarrow}\isanewline
-\ \ \ \ {\isasymnu}\isactrlsub b\ {\isacharparenleft}{\kern0pt}mk{\isacharunderscore}{\kern0pt}stationary{\isacharunderscore}{\kern0pt}det\ d{\isacharparenright}{\kern0pt}\ {\isacharless}{\kern0pt}\ {\isasymnu}\isactrlsub b\ {\isacharparenleft}{\kern0pt}mk{\isacharunderscore}{\kern0pt}stationary{\isacharunderscore}{\kern0pt}det\ {\isacharparenleft}{\kern0pt}policy{\isacharunderscore}{\kern0pt}step\ d{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ eval{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}eq\ policy{\isacharunderscore}{\kern0pt}eval{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}force\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ order{\isachardot}{\kern0pt}not{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}order{\isacharunderscore}{\kern0pt}implies{\isacharunderscore}{\kern0pt}strict{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ d{\isacharparenright}{\kern0pt}\ {\isasymin}\ {\isacharbraceleft}{\kern0pt}{\isacharparenleft}{\kern0pt}u{\isacharcomma}{\kern0pt}\ v{\isacharparenright}{\kern0pt}{\isachardot}{\kern0pt}\ u\ {\isasymin}\ D\isactrlsub D\ {\isasymand}\ v\ {\isasymin}\ D\isactrlsub D\ {\isasymand}\ {\isasymnu}\isactrlsub b\ {\isacharparenleft}{\kern0pt}mk{\isacharunderscore}{\kern0pt}stationary{\isacharunderscore}{\kern0pt}det\ v{\isacharparenright}{\kern0pt}\ {\isacharless}{\kern0pt}\ {\isasymnu}\isactrlsub b\ {\isacharparenleft}{\kern0pt}mk{\isacharunderscore}{\kern0pt}stationary{\isacharunderscore}{\kern0pt}det\ u{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ is{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det{\isacharunderscore}{\kern0pt}pi\ policy{\isacharunderscore}{\kern0pt}step{\isacharunderscore}{\kern0pt}def\ h\ \isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\begin{isamarkuptext}%
-The termination proof gives us access to the induction rule/simplification lemmas associated 
-with the \isa{policy{\isacharunderscore}{\kern0pt}iteration} definition.
-Thus we can prove that the algorithm finds an optimal policy.%
-\end{isamarkuptext}\isamarkuptrue%
-\isacommand{lemma}\isamarkupfalse%
-\ is{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det{\isacharunderscore}{\kern0pt}pi{\isacharprime}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}d\ {\isasymin}\ D\isactrlsub D\ {\isasymLongrightarrow}\ is{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\ {\isacharparenleft}{\kern0pt}policy{\isacharunderscore}{\kern0pt}iteration\ d{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ is{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det{\isacharunderscore}{\kern0pt}pi\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}induction\ d\ rule{\isacharcolon}{\kern0pt}\ policy{\isacharunderscore}{\kern0pt}iteration{\isachardot}{\kern0pt}induct{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ Let{\isacharunderscore}{\kern0pt}def\ policy{\isacharunderscore}{\kern0pt}step{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ pi{\isacharunderscore}{\kern0pt}pi{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}d\ {\isasymin}\ D\isactrlsub D\ {\isasymLongrightarrow}\ policy{\isacharunderscore}{\kern0pt}step\ {\isacharparenleft}{\kern0pt}policy{\isacharunderscore}{\kern0pt}iteration\ d{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ policy{\isacharunderscore}{\kern0pt}iteration\ d{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ is{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det{\isacharunderscore}{\kern0pt}pi\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}induction\ d\ rule{\isacharcolon}{\kern0pt}\ policy{\isacharunderscore}{\kern0pt}iteration{\isachardot}{\kern0pt}induct{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ policy{\isacharunderscore}{\kern0pt}step{\isacharunderscore}{\kern0pt}def\ Let{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ policy{\isacharunderscore}{\kern0pt}iteration{\isacharunderscore}{\kern0pt}correct{\isacharcolon}{\kern0pt}\ \isanewline
-\ \ {\isachardoublequoteopen}d\ {\isasymin}\ D\isactrlsub D\ {\isasymLongrightarrow}\ {\isasymnu}\isactrlsub b\ {\isacharparenleft}{\kern0pt}mk{\isacharunderscore}{\kern0pt}stationary{\isacharunderscore}{\kern0pt}det\ {\isacharparenleft}{\kern0pt}policy{\isacharunderscore}{\kern0pt}iteration\ d{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt{\isachardoublequoteclose}\ \isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}induction\ d\ rule{\isacharcolon}{\kern0pt}\ policy{\isacharunderscore}{\kern0pt}iteration{\isachardot}{\kern0pt}induct{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ {\isacharparenleft}{\kern0pt}fastforce\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ policy{\isacharunderscore}{\kern0pt}step{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}opt\ is{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det{\isacharunderscore}{\kern0pt}pi{\isacharprime}{\kern0pt}\ simp\ del{\isacharcolon}{\kern0pt}\ policy{\isacharunderscore}{\kern0pt}iteration{\isachardot}{\kern0pt}simps{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isacommand{end}\isamarkupfalse%
-\isanewline
-\isanewline
-\isacommand{context}\isamarkupfalse%
-\ MDP{\isacharunderscore}{\kern0pt}finite{\isacharunderscore}{\kern0pt}type\ \isakeyword{begin}%
-\begin{isamarkuptext}%
-The following proofs concern code generation, i.e. how to represent \isa{{\isasymP}\isactrlsub {\isadigit{1}}} as a matrix.%
-\end{isamarkuptext}\isamarkuptrue%
-\isacommand{sublocale}\isamarkupfalse%
-\ MDP{\isacharunderscore}{\kern0pt}att{\isacharunderscore}{\kern0pt}{\isasymL}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ A{\isacharunderscore}{\kern0pt}ne\ finite{\isacharunderscore}{\kern0pt}is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max\ MDP{\isacharunderscore}{\kern0pt}att{\isacharunderscore}{\kern0pt}{\isasymL}{\isacharunderscore}{\kern0pt}def\ MDP{\isacharunderscore}{\kern0pt}att{\isacharunderscore}{\kern0pt}{\isasymL}{\isacharunderscore}{\kern0pt}axioms{\isacharunderscore}{\kern0pt}def\ max{\isacharunderscore}{\kern0pt}L{\isacharunderscore}{\kern0pt}ex{\isacharunderscore}{\kern0pt}def\ \isanewline
-\ \ \ \ \ \ has{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}def\ MDP{\isacharunderscore}{\kern0pt}reward{\isacharunderscore}{\kern0pt}axioms{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\ \isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ {\isachardoublequoteopen}fun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ f\ {\isacharequal}{\kern0pt}\ matrix\ {\isacharparenleft}{\kern0pt}{\isasymlambda}v{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymchi}\ j{\isachardot}{\kern0pt}\ f\ {\isacharparenleft}{\kern0pt}vec{\isacharunderscore}{\kern0pt}nth\ v{\isacharparenright}{\kern0pt}\ j{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ {\isachardoublequoteopen}Ek{\isacharunderscore}{\kern0pt}mat\ d\ {\isacharequal}{\kern0pt}\ fun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ {\isacharparenleft}{\kern0pt}{\isasymlambda}v{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}{\isasymP}\isactrlsub {\isadigit{1}}\ d{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}Bfun\ v{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ {\isachardoublequoteopen}nu{\isacharunderscore}{\kern0pt}inv{\isacharunderscore}{\kern0pt}mat\ d\ {\isacharequal}{\kern0pt}\ fun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}{\isasymlambda}v{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}id{\isacharunderscore}{\kern0pt}blinfun\ {\isacharminus}{\kern0pt}\ l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isasymP}\isactrlsub {\isadigit{1}}\ d{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}Bfun\ v{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ {\isachardoublequoteopen}nu{\isacharunderscore}{\kern0pt}mat\ d\ {\isacharequal}{\kern0pt}\ fun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ {\isacharparenleft}{\kern0pt}{\isasymlambda}v{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}{\isasymSum}i{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isasymP}\isactrlsub {\isadigit{1}}\ d{\isacharparenright}{\kern0pt}\ {\isacharcircum}{\kern0pt}{\isacharcircum}{\kern0pt}\ i{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}Bfun\ v{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ apply{\isacharunderscore}{\kern0pt}nu{\isacharunderscore}{\kern0pt}inv{\isacharunderscore}{\kern0pt}mat{\isacharcolon}{\kern0pt}\ \isanewline
-\ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}id{\isacharunderscore}{\kern0pt}blinfun\ {\isacharminus}{\kern0pt}\ l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isasymP}\isactrlsub {\isadigit{1}}\ d{\isacharparenright}{\kern0pt}\ v\ {\isacharequal}{\kern0pt}\ Bfun\ {\isacharparenleft}{\kern0pt}{\isasymlambda}i{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}nu{\isacharunderscore}{\kern0pt}inv{\isacharunderscore}{\kern0pt}mat\ d{\isacharparenright}{\kern0pt}\ {\isacharasterisk}{\kern0pt}v\ {\isacharparenleft}{\kern0pt}vec{\isacharunderscore}{\kern0pt}lambda\ v{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isachardollar}{\kern0pt}\ i{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ eq{\isacharunderscore}{\kern0pt}onpI{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}P\ x\ {\isasymLongrightarrow}\ eq{\isacharunderscore}{\kern0pt}onp\ P\ x\ x{\isachardoublequoteclose}\ \isakeyword{for}\ P\ x\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ eq{\isacharunderscore}{\kern0pt}onp{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}Real{\isacharunderscore}{\kern0pt}Vector{\isacharunderscore}{\kern0pt}Spaces{\isachardot}{\kern0pt}linear\ {\isacharparenleft}{\kern0pt}{\isasymlambda}v{\isachardot}{\kern0pt}\ vec{\isacharunderscore}{\kern0pt}lambda\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}id{\isacharunderscore}{\kern0pt}blinfun\ {\isacharminus}{\kern0pt}\ l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isasymP}\isactrlsub {\isadigit{1}}\ d{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}bfun{\isachardot}{\kern0pt}Bfun\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}{\isachardollar}{\kern0pt}{\isacharparenright}{\kern0pt}\ v{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp\ del{\isacharcolon}{\kern0pt}\ real{\isacharunderscore}{\kern0pt}scaleR{\isacharunderscore}{\kern0pt}def\ intro{\isacharcolon}{\kern0pt}\ linearI\isanewline
-\ \ \ \ \ \ \ \ simp{\isacharcolon}{\kern0pt}\ scaleR{\isacharunderscore}{\kern0pt}vec{\isacharunderscore}{\kern0pt}def\ eq{\isacharunderscore}{\kern0pt}onpI\ plus{\isacharunderscore}{\kern0pt}vec{\isacharunderscore}{\kern0pt}def\ vec{\isacharunderscore}{\kern0pt}lambda{\isacharunderscore}{\kern0pt}inverse\ plus{\isacharunderscore}{\kern0pt}bfun{\isachardot}{\kern0pt}abs{\isacharunderscore}{\kern0pt}eq{\isacharbrackleft}{\kern0pt}symmetric{\isacharbrackright}{\kern0pt}\ \isanewline
-\ \ \ \ \ \ \ \ scaleR{\isacharunderscore}{\kern0pt}bfun{\isachardot}{\kern0pt}abs{\isacharunderscore}{\kern0pt}eq{\isacharbrackleft}{\kern0pt}symmetric{\isacharbrackright}{\kern0pt}\ blinfun{\isachardot}{\kern0pt}scaleR{\isacharunderscore}{\kern0pt}right\ blinfun{\isachardot}{\kern0pt}add{\isacharunderscore}{\kern0pt}right{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ Ek{\isacharunderscore}{\kern0pt}mat{\isacharunderscore}{\kern0pt}def\ fun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}def\ nu{\isacharunderscore}{\kern0pt}inv{\isacharunderscore}{\kern0pt}mat{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ apply{\isacharunderscore}{\kern0pt}bfun{\isacharunderscore}{\kern0pt}inverse\ vec{\isacharunderscore}{\kern0pt}lambda{\isacharunderscore}{\kern0pt}inverse{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ bounded{\isacharunderscore}{\kern0pt}linear{\isacharunderscore}{\kern0pt}vec{\isacharunderscore}{\kern0pt}lambda{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}bounded{\isacharunderscore}{\kern0pt}linear\ {\isacharparenleft}{\kern0pt}{\isasymlambda}x{\isachardot}{\kern0pt}\ vec{\isacharunderscore}{\kern0pt}lambda\ {\isacharparenleft}{\kern0pt}x\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isacharprime}{\kern0pt}s\ {\isasymRightarrow}\isactrlsub b\ real{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}intro\ bounded{\isacharunderscore}{\kern0pt}linear{\isacharunderscore}{\kern0pt}intro{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\ x\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}s\ {\isasymRightarrow}\isactrlsub b\ real{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}sqrt\ {\isacharparenleft}{\kern0pt}{\isasymSum}\ i\ {\isasymin}\ UNIV\ {\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}apply{\isacharunderscore}{\kern0pt}bfun\ x\ i{\isacharparenright}{\kern0pt}\isactrlsup {\isadigit{2}}{\isacharparenright}{\kern0pt}\ {\isasymle}\ {\isacharparenleft}{\kern0pt}{\isasymSum}\ i\ {\isasymin}\ UNIV\ {\isachardot}{\kern0pt}\ {\isasymbar}{\isacharparenleft}{\kern0pt}apply{\isacharunderscore}{\kern0pt}bfun\ x\ i{\isacharparenright}{\kern0pt}{\isasymbar}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ L{\isadigit{2}}{\isacharunderscore}{\kern0pt}set{\isacharunderscore}{\kern0pt}le{\isacharunderscore}{\kern0pt}sum{\isacharunderscore}{\kern0pt}abs\ \isanewline
-\ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ L{\isadigit{2}}{\isacharunderscore}{\kern0pt}set{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isasymSum}\ i\ {\isasymin}\ UNIV\ {\isachardot}{\kern0pt}\ {\isasymbar}{\isacharparenleft}{\kern0pt}apply{\isacharunderscore}{\kern0pt}bfun\ x\ i{\isacharparenright}{\kern0pt}{\isasymbar}{\isacharparenright}{\kern0pt}\ {\isasymle}\ {\isacharparenleft}{\kern0pt}card\ {\isacharparenleft}{\kern0pt}UNIV\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isacharprime}{\kern0pt}s\ set{\isacharparenright}{\kern0pt}\ {\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSqunion}xa{\isachardot}{\kern0pt}\ {\isasymbar}apply{\isacharunderscore}{\kern0pt}bfun\ x\ xa{\isasymbar}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ cSup{\isacharunderscore}{\kern0pt}upper\ sum{\isacharunderscore}{\kern0pt}bounded{\isacharunderscore}{\kern0pt}above{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{finally}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}norm\ {\isacharparenleft}{\kern0pt}vec{\isacharunderscore}{\kern0pt}lambda\ {\isacharparenleft}{\kern0pt}apply{\isacharunderscore}{\kern0pt}bfun\ x{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymle}\ norm\ x\ {\isacharasterisk}{\kern0pt}\ CARD{\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}s{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ norm{\isacharunderscore}{\kern0pt}vec{\isacharunderscore}{\kern0pt}def\ norm{\isacharunderscore}{\kern0pt}bfun{\isacharunderscore}{\kern0pt}def\ dist{\isacharunderscore}{\kern0pt}bfun{\isacharunderscore}{\kern0pt}def\ L{\isadigit{2}}{\isacharunderscore}{\kern0pt}set{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp\ add{\isacharcolon}{\kern0pt}\ mult{\isachardot}{\kern0pt}commute{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ plus{\isacharunderscore}{\kern0pt}vec{\isacharunderscore}{\kern0pt}def\ scaleR{\isacharunderscore}{\kern0pt}vec{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ bounded{\isacharunderscore}{\kern0pt}linear{\isacharunderscore}{\kern0pt}vec{\isacharunderscore}{\kern0pt}lambda{\isacharunderscore}{\kern0pt}blinfun{\isacharcolon}{\kern0pt}\ \isanewline
-\ \ \isakeyword{fixes}\ f\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}s\ {\isasymRightarrow}\isactrlsub b\ real{\isacharparenright}{\kern0pt}\ {\isasymRightarrow}\isactrlsub L\ {\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}s\ {\isasymRightarrow}\isactrlsub b\ real{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}bounded{\isacharunderscore}{\kern0pt}linear\ {\isacharparenleft}{\kern0pt}{\isasymlambda}v{\isachardot}{\kern0pt}\ vec{\isacharunderscore}{\kern0pt}lambda\ {\isacharparenleft}{\kern0pt}apply{\isacharunderscore}{\kern0pt}bfun\ {\isacharparenleft}{\kern0pt}blinfun{\isacharunderscore}{\kern0pt}apply\ f\ {\isacharparenleft}{\kern0pt}bfun{\isachardot}{\kern0pt}Bfun\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}{\isachardollar}{\kern0pt}{\isacharparenright}{\kern0pt}\ v{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ blinfun{\isachardot}{\kern0pt}bounded{\isacharunderscore}{\kern0pt}linear{\isacharunderscore}{\kern0pt}right\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}fastforce\ intro{\isacharcolon}{\kern0pt}\ bounded{\isacharunderscore}{\kern0pt}linear{\isacharunderscore}{\kern0pt}compose{\isacharbrackleft}{\kern0pt}OF\ bounded{\isacharunderscore}{\kern0pt}linear{\isacharunderscore}{\kern0pt}vec{\isacharunderscore}{\kern0pt}lambda{\isacharbrackright}{\kern0pt}\ \isanewline
-\ \ \ \ \ \ bounded{\isacharunderscore}{\kern0pt}linear{\isacharunderscore}{\kern0pt}bfun{\isacharunderscore}{\kern0pt}nth\ bounded{\isacharunderscore}{\kern0pt}linear{\isacharunderscore}{\kern0pt}compose{\isacharbrackleft}{\kern0pt}of\ f{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ invertible{\isacharunderscore}{\kern0pt}nu{\isacharunderscore}{\kern0pt}inv{\isacharunderscore}{\kern0pt}max{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}invertible\ {\isacharparenleft}{\kern0pt}nu{\isacharunderscore}{\kern0pt}inv{\isacharunderscore}{\kern0pt}mat\ d{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ nu{\isacharunderscore}{\kern0pt}inv{\isacharunderscore}{\kern0pt}mat{\isacharunderscore}{\kern0pt}def\ fun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ matrix{\isacharunderscore}{\kern0pt}invertible\ inv{\isacharunderscore}{\kern0pt}norm{\isacharunderscore}{\kern0pt}le{\isacharprime}{\kern0pt}\ vec{\isacharunderscore}{\kern0pt}lambda{\isacharunderscore}{\kern0pt}inverse\ apply{\isacharunderscore}{\kern0pt}bfun{\isacharunderscore}{\kern0pt}inverse\ \isanewline
-\ \ \ \ \ \ bounded{\isacharunderscore}{\kern0pt}linear{\isachardot}{\kern0pt}linear{\isacharbrackleft}{\kern0pt}OF\ bounded{\isacharunderscore}{\kern0pt}linear{\isacharunderscore}{\kern0pt}vec{\isacharunderscore}{\kern0pt}lambda{\isacharunderscore}{\kern0pt}blinfun{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \ \ \ \ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ exI{\isacharbrackleft}{\kern0pt}of\ {\isacharunderscore}{\kern0pt}\ {\isachardoublequoteopen}{\isasymlambda}v{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymchi}\ j{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymlambda}v{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSum}i{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isasymP}\isactrlsub {\isadigit{1}}\ d{\isacharparenright}{\kern0pt}\ {\isacharcircum}{\kern0pt}{\isacharcircum}{\kern0pt}\ i{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}Bfun\ v{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}vec{\isacharunderscore}{\kern0pt}nth\ v{\isacharparenright}{\kern0pt}\ j{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{end}\isamarkupfalse%
-\isanewline
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ {\isachardoublequoteopen}least{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max\ f\ P\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}LEAST\ x{\isachardot}{\kern0pt}\ is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max\ f\ P\ x{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{locale}\isamarkupfalse%
-\ MDP{\isacharunderscore}{\kern0pt}ord\ {\isacharequal}{\kern0pt}\ MDP{\isacharunderscore}{\kern0pt}finite{\isacharunderscore}{\kern0pt}type\ A\ K\ r\ l\isanewline
-\ \ \isakeyword{for}\ A\ \isakeyword{and}\isanewline
-\ \ \ \ K\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}s\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}finite{\isacharcomma}{\kern0pt}\ wellorder{\isacharbraceright}{\kern0pt}\ {\isasymtimes}\ {\isacharprime}{\kern0pt}a\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}finite{\isacharcomma}{\kern0pt}\ wellorder{\isacharbraceright}{\kern0pt}\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}s\ pmf{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isakeyword{and}\ r\ l\isanewline
-\isakeyword{begin}\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ {\isasymL}{\isacharunderscore}{\kern0pt}fin{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}det{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymL}\ v\ s\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSqunion}a\ {\isasymin}\ A\ s{\isachardot}{\kern0pt}\ L\isactrlsub a\ a\ v\ s{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ SUP{\isacharunderscore}{\kern0pt}step{\isacharunderscore}{\kern0pt}det{\isacharunderscore}{\kern0pt}eq\ {\isasymL}{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}SUP{\isacharunderscore}{\kern0pt}det{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ {\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}fin{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}det{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymL}\isactrlsub b\ v\ s\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSqunion}a\ {\isasymin}\ A\ s{\isachardot}{\kern0pt}\ L\isactrlsub a\ a\ v\ s{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ SUP{\isacharunderscore}{\kern0pt}step{\isacharunderscore}{\kern0pt}det{\isacharunderscore}{\kern0pt}eq\ {\isasymL}\isactrlsub b{\isachardot}{\kern0pt}rep{\isacharunderscore}{\kern0pt}eq\ {\isasymL}{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}SUP{\isacharunderscore}{\kern0pt}det{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{sublocale}\isamarkupfalse%
-\ MDP{\isacharunderscore}{\kern0pt}PI{\isacharunderscore}{\kern0pt}finite\ A\ K\ r\ l\ {\isachardoublequoteopen}{\isasymlambda}X{\isachardot}{\kern0pt}\ Least\ {\isacharparenleft}{\kern0pt}{\isasymlambda}x{\isachardot}{\kern0pt}\ x\ {\isasymin}\ X{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ unfold{\isacharunderscore}{\kern0pt}locales\ {\isacharparenleft}{\kern0pt}auto\ intro{\isacharcolon}{\kern0pt}\ LeastI{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{end}\isamarkupfalse%
-\isanewline
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{end}\isamarkupfalse%
-%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\end{isabellebody}%
-\endinput
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diff --git a/thys/MDP-Algorithms/output/document/Splitting_Methods.tex b/thys/MDP-Algorithms/output/document/Splitting_Methods.tex
deleted file mode 100644
--- a/thys/MDP-Algorithms/output/document/Splitting_Methods.tex
+++ /dev/null
@@ -1,9722 +0,0 @@
-%
-\begin{isabellebody}%
-\setisabellecontext{Splitting{\isacharunderscore}{\kern0pt}Methods}%
-%
-\isadelimtheory
-\isanewline
-\isanewline
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{theory}\isamarkupfalse%
-\ Splitting{\isacharunderscore}{\kern0pt}Methods\isanewline
-\ \ \isakeyword{imports}\ \isanewline
-\ \ \ \ Blinfun{\isacharunderscore}{\kern0pt}Matrix\isanewline
-\ \ \ \ Value{\isacharunderscore}{\kern0pt}Iteration\isanewline
-\ \ \ \ Policy{\isacharunderscore}{\kern0pt}Iteration\isanewline
-\isakeyword{begin}%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsection{Value Iteration using Splitting Methods%
-}
-\isamarkuptrue%
-%
-\isamarkupsubsection{Regular Splittings for Matrices and Bounded Linear Functions%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{definition}\isamarkupfalse%
-\ {\isachardoublequoteopen}is{\isacharunderscore}{\kern0pt}splitting{\isacharunderscore}{\kern0pt}mat\ X\ Q\ R\ {\isasymlongleftrightarrow}\isanewline
-\ \ X\ {\isacharequal}{\kern0pt}\ Q\ {\isacharminus}{\kern0pt}\ R\ {\isasymand}\ invertible\ Q\ {\isasymand}\ {\isadigit{0}}\ {\isasymle}\ matrix{\isacharunderscore}{\kern0pt}inv\ Q\ {\isasymand}\ {\isadigit{0}}\ {\isasymle}\ R{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ {\isachardoublequoteopen}is{\isacharunderscore}{\kern0pt}splitting{\isacharunderscore}{\kern0pt}blin\ X\ Q\ R\ {\isasymlongleftrightarrow}\ is{\isacharunderscore}{\kern0pt}splitting{\isacharunderscore}{\kern0pt}mat\ {\isacharparenleft}{\kern0pt}blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ X{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ Q{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ R{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ is{\isacharunderscore}{\kern0pt}splitting{\isacharunderscore}{\kern0pt}blin{\isacharunderscore}{\kern0pt}def{\isacharprime}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}is{\isacharunderscore}{\kern0pt}splitting{\isacharunderscore}{\kern0pt}blin\ X\ Q\ R\ {\isasymlongleftrightarrow}\isanewline
-\ \ X\ {\isacharequal}{\kern0pt}\ Q\ {\isacharminus}{\kern0pt}\ R\ {\isasymand}\ invertible\isactrlsub L\ Q\ {\isasymand}\ nonneg{\isacharunderscore}{\kern0pt}blinfun\ {\isacharparenleft}{\kern0pt}inv\isactrlsub L\ Q{\isacharparenright}{\kern0pt}\ {\isasymand}\ nonneg{\isacharunderscore}{\kern0pt}blinfun\ R{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ X\ {\isacharequal}{\kern0pt}\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ Q\ {\isacharminus}{\kern0pt}\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ R\ {\isasymlongleftrightarrow}\ X\ {\isacharequal}{\kern0pt}\ Q\ {\isacharminus}{\kern0pt}\ R{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ \ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}diff\ matrix{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}blinfun{\isacharunderscore}{\kern0pt}inv\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ is{\isacharunderscore}{\kern0pt}splitting{\isacharunderscore}{\kern0pt}blin{\isacharunderscore}{\kern0pt}def\ is{\isacharunderscore}{\kern0pt}splitting{\isacharunderscore}{\kern0pt}mat{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}inverse{\isacharbrackleft}{\kern0pt}of\ Q{\isacharbrackright}{\kern0pt}\ matrix{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}blinfun{\isacharunderscore}{\kern0pt}inv\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}fastforce\ simp{\isacharcolon}{\kern0pt}\ invertible{\isacharunderscore}{\kern0pt}invertible\isactrlsub L{\isacharunderscore}{\kern0pt}I{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ is{\isacharunderscore}{\kern0pt}splitting{\isacharunderscore}{\kern0pt}blinD{\isacharbrackleft}{\kern0pt}dest{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\ \isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}is{\isacharunderscore}{\kern0pt}splitting{\isacharunderscore}{\kern0pt}blin\ X\ Q\ R{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}X\ {\isacharequal}{\kern0pt}\ Q\ {\isacharminus}{\kern0pt}\ R{\isachardoublequoteclose}\ {\isachardoublequoteopen}invertible\isactrlsub L\ Q{\isachardoublequoteclose}\ {\isachardoublequoteopen}nonneg{\isacharunderscore}{\kern0pt}blinfun\ {\isacharparenleft}{\kern0pt}inv\isactrlsub L\ Q{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ {\isachardoublequoteopen}nonneg{\isacharunderscore}{\kern0pt}blinfun\ R{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ is{\isacharunderscore}{\kern0pt}splitting{\isacharunderscore}{\kern0pt}blin{\isacharunderscore}{\kern0pt}def{\isacharprime}{\kern0pt}\ assms\ \isacommand{by}\isamarkupfalse%
-\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Splitting Methods for MDPs%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{locale}\isamarkupfalse%
-\ MDP{\isacharunderscore}{\kern0pt}QR\ {\isacharequal}{\kern0pt}\ MDP{\isacharunderscore}{\kern0pt}finite{\isacharunderscore}{\kern0pt}type\ A\ K\ r\ l\ \isanewline
-\ \ \isakeyword{for}\ A\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}s\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ finite\ {\isasymRightarrow}\ {\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}a\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ finite{\isacharparenright}{\kern0pt}\ set{\isachardoublequoteclose}\ \isanewline
-\ \ \ \ \isakeyword{and}\ K\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}s\ {\isasymtimes}\ {\isacharprime}{\kern0pt}a{\isacharparenright}{\kern0pt}\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}s\ pmf{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isakeyword{and}\ \ r\ l\ {\isacharplus}{\kern0pt}\isanewline
-\ \ \isakeyword{fixes}\ Q\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}s\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a{\isacharparenright}{\kern0pt}\ {\isasymRightarrow}\ {\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}s\ {\isasymRightarrow}\isactrlsub b\ real{\isacharparenright}{\kern0pt}\ {\isasymRightarrow}\isactrlsub L\ {\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}s\ {\isasymRightarrow}\isactrlsub b\ real{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{fixes}\ R\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}s\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a{\isacharparenright}{\kern0pt}\ {\isasymRightarrow}\ {\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}s\ {\isasymRightarrow}\isactrlsub b\ real{\isacharparenright}{\kern0pt}\ {\isasymRightarrow}\isactrlsub L\ {\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}s\ {\isasymRightarrow}\isactrlsub b\ real{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{assumes}\ is{\isacharunderscore}{\kern0pt}splitting{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymAnd}d{\isachardot}{\kern0pt}\ d\ {\isasymin}\ D\isactrlsub D\ {\isasymLongrightarrow}\ is{\isacharunderscore}{\kern0pt}splitting{\isacharunderscore}{\kern0pt}blin\ {\isacharparenleft}{\kern0pt}id{\isacharunderscore}{\kern0pt}blinfun\ {\isacharminus}{\kern0pt}\ l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isasymP}\isactrlsub {\isadigit{1}}\ {\isacharparenleft}{\kern0pt}mk{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\ d{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}Q\ d{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}R\ d{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{assumes}\ QR{\isacharunderscore}{\kern0pt}contraction{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isasymSqunion}d{\isasymin}D\isactrlsub D{\isachardot}{\kern0pt}\ norm\ {\isacharparenleft}{\kern0pt}inv\isactrlsub L\ {\isacharparenleft}{\kern0pt}Q\ d{\isacharparenright}{\kern0pt}\ o\isactrlsub L\ R\ d{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharless}{\kern0pt}\ {\isadigit{1}}{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{assumes}\ arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}ex{\isacharunderscore}{\kern0pt}split{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymexists}d{\isachardot}{\kern0pt}\ {\isasymforall}s{\isachardot}{\kern0pt}\ is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max\ {\isacharparenleft}{\kern0pt}{\isasymlambda}d{\isachardot}{\kern0pt}\ inv\isactrlsub L\ {\isacharparenleft}{\kern0pt}Q\ d{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}r{\isacharunderscore}{\kern0pt}dec\isactrlsub b\ {\isacharparenleft}{\kern0pt}mk{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\ d{\isacharparenright}{\kern0pt}\ {\isacharplus}{\kern0pt}\ R\ d\ v{\isacharparenright}{\kern0pt}\ s{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymlambda}d{\isachardot}{\kern0pt}\ d\ {\isasymin}\ D\isactrlsub D{\isacharparenright}{\kern0pt}\ d{\isachardoublequoteclose}\isanewline
-\isakeyword{begin}\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ inv{\isacharunderscore}{\kern0pt}Q{\isacharunderscore}{\kern0pt}mono{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}d\ {\isasymin}\ D\isactrlsub D\ {\isasymLongrightarrow}\ u\ {\isasymle}\ v\ {\isasymLongrightarrow}\ {\isacharparenleft}{\kern0pt}inv\isactrlsub L\ {\isacharparenleft}{\kern0pt}Q\ d{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ u\ {\isasymle}\ {\isacharparenleft}{\kern0pt}inv\isactrlsub L\ {\isacharparenleft}{\kern0pt}Q\ d{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ v{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ is{\isacharunderscore}{\kern0pt}splitting\ \isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ nonneg{\isacharunderscore}{\kern0pt}blinfun{\isacharunderscore}{\kern0pt}mono{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ splitting{\isacharunderscore}{\kern0pt}eq{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}d\ {\isasymin}\ D\isactrlsub D\ {\isasymLongrightarrow}\ Q\ d\ {\isacharminus}{\kern0pt}\ R\ d\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}id{\isacharunderscore}{\kern0pt}blinfun\ {\isacharminus}{\kern0pt}\ l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isasymP}\isactrlsub {\isadigit{1}}\ {\isacharparenleft}{\kern0pt}mk{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\ d{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ is{\isacharunderscore}{\kern0pt}splitting\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ fastforce%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ Q{\isacharunderscore}{\kern0pt}nonneg{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}d\ {\isasymin}\ D\isactrlsub D\ {\isasymLongrightarrow}\ {\isadigit{0}}\ {\isasymle}\ v\ {\isasymLongrightarrow}\ {\isadigit{0}}\ {\isasymle}\ inv\isactrlsub L\ {\isacharparenleft}{\kern0pt}Q\ d{\isacharparenright}{\kern0pt}\ v{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ is{\isacharunderscore}{\kern0pt}splitting\ nonneg{\isacharunderscore}{\kern0pt}blinfun{\isacharunderscore}{\kern0pt}nonneg\ \isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ Q{\isacharunderscore}{\kern0pt}invertible{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}d\ {\isasymin}\ D\isactrlsub D\ {\isasymLongrightarrow}\ invertible\isactrlsub L\ {\isacharparenleft}{\kern0pt}Q\ d{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ is{\isacharunderscore}{\kern0pt}splitting\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ R{\isacharunderscore}{\kern0pt}nonneg{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}d\ {\isasymin}\ D\isactrlsub D\ {\isasymLongrightarrow}\ {\isadigit{0}}\ {\isasymle}\ v\ {\isasymLongrightarrow}\ {\isadigit{0}}\ {\isasymle}\ R\ d\ v{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ is{\isacharunderscore}{\kern0pt}splitting{\isacharunderscore}{\kern0pt}blinD{\isacharbrackleft}{\kern0pt}OF\ is{\isacharunderscore}{\kern0pt}splitting{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}fastforce\ simp{\isacharcolon}{\kern0pt}\ nonneg{\isacharunderscore}{\kern0pt}blinfun{\isacharunderscore}{\kern0pt}nonneg\ intro{\isacharcolon}{\kern0pt}\ nonneg{\isacharunderscore}{\kern0pt}blinfun{\isacharunderscore}{\kern0pt}mono{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ R{\isacharunderscore}{\kern0pt}mono{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}d\ {\isasymin}\ D\isactrlsub D\ {\isasymLongrightarrow}\ u\ {\isasymle}\ v\ {\isasymLongrightarrow}\ {\isacharparenleft}{\kern0pt}R\ d{\isacharparenright}{\kern0pt}\ u\ {\isasymle}\ {\isacharparenleft}{\kern0pt}R\ d{\isacharparenright}{\kern0pt}\ v{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ R{\isacharunderscore}{\kern0pt}nonneg{\isacharbrackleft}{\kern0pt}of\ d\ {\isachardoublequoteopen}v\ {\isacharminus}{\kern0pt}\ u{\isachardoublequoteclose}{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ blinfun{\isachardot}{\kern0pt}bilinear{\isacharunderscore}{\kern0pt}simps{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ QR{\isacharunderscore}{\kern0pt}nonneg{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}d\ {\isasymin}\ D\isactrlsub D\ {\isasymLongrightarrow}\ {\isadigit{0}}\ {\isasymle}\ v\ {\isasymLongrightarrow}\ {\isadigit{0}}\ {\isasymle}\ {\isacharparenleft}{\kern0pt}inv\isactrlsub L\ {\isacharparenleft}{\kern0pt}Q\ d{\isacharparenright}{\kern0pt}\ o\isactrlsub L\ R\ d{\isacharparenright}{\kern0pt}\ v{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ Q{\isacharunderscore}{\kern0pt}nonneg\ R{\isacharunderscore}{\kern0pt}nonneg{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ QR{\isacharunderscore}{\kern0pt}mono{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}d\ {\isasymin}\ D\isactrlsub D\ {\isasymLongrightarrow}\ u\ {\isasymle}\ v\ {\isasymLongrightarrow}\ {\isacharparenleft}{\kern0pt}inv\isactrlsub L\ {\isacharparenleft}{\kern0pt}Q\ d{\isacharparenright}{\kern0pt}\ o\isactrlsub L\ R\ d{\isacharparenright}{\kern0pt}\ u\ {\isasymle}\ {\isacharparenleft}{\kern0pt}inv\isactrlsub L\ {\isacharparenleft}{\kern0pt}Q\ d{\isacharparenright}{\kern0pt}\ o\isactrlsub L\ R\ d{\isacharparenright}{\kern0pt}\ v{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ QR{\isacharunderscore}{\kern0pt}nonneg{\isacharbrackleft}{\kern0pt}of\ d\ {\isachardoublequoteopen}v\ {\isacharminus}{\kern0pt}\ u{\isachardoublequoteclose}{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ blinfun{\isachardot}{\kern0pt}bilinear{\isacharunderscore}{\kern0pt}simps{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ norm{\isacharunderscore}{\kern0pt}QR{\isacharunderscore}{\kern0pt}less{\isacharunderscore}{\kern0pt}one{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}d\ {\isasymin}\ D\isactrlsub D\ {\isasymLongrightarrow}\ norm\ {\isacharparenleft}{\kern0pt}inv\isactrlsub L\ {\isacharparenleft}{\kern0pt}Q\ d{\isacharparenright}{\kern0pt}\ o\isactrlsub L\ R\ d{\isacharparenright}{\kern0pt}\ {\isacharless}{\kern0pt}\ {\isadigit{1}}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ QR{\isacharunderscore}{\kern0pt}contraction\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ cSUP{\isacharunderscore}{\kern0pt}lessD{\isacharbrackleft}{\kern0pt}of\ {\isachardoublequoteopen}{\isasymlambda}d{\isachardot}{\kern0pt}\ norm\ {\isacharparenleft}{\kern0pt}inv\isactrlsub L\ {\isacharparenleft}{\kern0pt}Q\ d{\isacharparenright}{\kern0pt}\ o\isactrlsub L\ R\ d{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ splitting{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}d\ {\isasymin}\ D\isactrlsub D\ {\isasymLongrightarrow}\ id{\isacharunderscore}{\kern0pt}blinfun\ {\isacharminus}{\kern0pt}\ l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isasymP}\isactrlsub {\isadigit{1}}\ {\isacharparenleft}{\kern0pt}mk{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\ d{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ Q\ d\ {\isacharminus}{\kern0pt}\ R\ d{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ is{\isacharunderscore}{\kern0pt}splitting\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Discount Factor \isa{QR{\isacharunderscore}{\kern0pt}disc}%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{abbreviation}\isamarkupfalse%
-\ {\isachardoublequoteopen}QR{\isacharunderscore}{\kern0pt}disc\ {\isasymequiv}\ {\isacharparenleft}{\kern0pt}{\isasymSqunion}d\ {\isasymin}\ D\isactrlsub D{\isachardot}{\kern0pt}\ norm\ {\isacharparenleft}{\kern0pt}inv\isactrlsub L\ {\isacharparenleft}{\kern0pt}Q\ d{\isacharparenright}{\kern0pt}\ o\isactrlsub L\ R\ d{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ QR{\isacharunderscore}{\kern0pt}le{\isacharunderscore}{\kern0pt}QR{\isacharunderscore}{\kern0pt}disc{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}d\ {\isasymin}\ D\isactrlsub D\ {\isasymLongrightarrow}\ norm\ {\isacharparenleft}{\kern0pt}inv\isactrlsub L\ {\isacharparenleft}{\kern0pt}Q\ d{\isacharparenright}{\kern0pt}\ o\isactrlsub L\ {\isacharparenleft}{\kern0pt}R\ d{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymle}\ QR{\isacharunderscore}{\kern0pt}disc{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ intro{\isacharcolon}{\kern0pt}\ cSUP{\isacharunderscore}{\kern0pt}upper{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ a{\isacharunderscore}{\kern0pt}nonneg{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isadigit{0}}\ {\isasymle}\ QR{\isacharunderscore}{\kern0pt}disc{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ QR{\isacharunderscore}{\kern0pt}contraction\ norm{\isacharunderscore}{\kern0pt}ge{\isacharunderscore}{\kern0pt}zero\ ex{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}fastforce\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ cSUP{\isacharunderscore}{\kern0pt}upper{\isadigit{2}}{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Bellman-Operator%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{abbreviation}\isamarkupfalse%
-\ {\isachardoublequoteopen}L{\isacharunderscore}{\kern0pt}split\ d\ v\ {\isasymequiv}\ inv\isactrlsub L\ {\isacharparenleft}{\kern0pt}Q\ d{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}r{\isacharunderscore}{\kern0pt}dec\isactrlsub b\ {\isacharparenleft}{\kern0pt}mk{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\ d{\isacharparenright}{\kern0pt}\ {\isacharplus}{\kern0pt}\ R\ d\ v{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymL}{\isacharunderscore}{\kern0pt}split\ v\ s\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSqunion}d\ {\isasymin}\ D\isactrlsub D{\isachardot}{\kern0pt}\ L{\isacharunderscore}{\kern0pt}split\ d\ v\ s{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ {\isasymL}{\isacharunderscore}{\kern0pt}split{\isacharunderscore}{\kern0pt}bfun{\isacharunderscore}{\kern0pt}aux{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}d\ {\isasymin}\ D\isactrlsub D{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}norm\ {\isacharparenleft}{\kern0pt}L{\isacharunderscore}{\kern0pt}split\ d\ v{\isacharparenright}{\kern0pt}\ {\isasymle}\ {\isacharparenleft}{\kern0pt}{\isasymSqunion}d\ {\isasymin}\ D\isactrlsub D{\isachardot}{\kern0pt}\ norm\ {\isacharparenleft}{\kern0pt}inv\isactrlsub L\ {\isacharparenleft}{\kern0pt}Q\ d{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharasterisk}{\kern0pt}\ r\isactrlsub M\ {\isacharplus}{\kern0pt}\ norm\ v{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}norm\ {\isacharparenleft}{\kern0pt}L{\isacharunderscore}{\kern0pt}split\ d\ v{\isacharparenright}{\kern0pt}\ {\isasymle}\ norm\ {\isacharparenleft}{\kern0pt}inv\isactrlsub L\ {\isacharparenleft}{\kern0pt}Q\ d{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}r{\isacharunderscore}{\kern0pt}dec\isactrlsub b\ {\isacharparenleft}{\kern0pt}mk{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\ d{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharplus}{\kern0pt}\ norm\ {\isacharparenleft}{\kern0pt}inv\isactrlsub L\ {\isacharparenleft}{\kern0pt}Q\ d{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}R\ d\ v{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ blinfun{\isachardot}{\kern0pt}add{\isacharunderscore}{\kern0pt}right\ norm{\isacharunderscore}{\kern0pt}triangle{\isacharunderscore}{\kern0pt}ineq{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isasymle}\ norm\ {\isacharparenleft}{\kern0pt}inv\isactrlsub L\ {\isacharparenleft}{\kern0pt}Q\ d{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}r{\isacharunderscore}{\kern0pt}dec\isactrlsub b\ {\isacharparenleft}{\kern0pt}mk{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\ d{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharplus}{\kern0pt}\ norm\ {\isacharparenleft}{\kern0pt}inv\isactrlsub L\ {\isacharparenleft}{\kern0pt}Q\ d{\isacharparenright}{\kern0pt}\ o\isactrlsub L\ R\ d{\isacharparenright}{\kern0pt}\ {\isacharasterisk}{\kern0pt}\ norm\ v{\isachardoublequoteclose}\ \ \ \ \isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ blinfun{\isacharunderscore}{\kern0pt}apply{\isacharunderscore}{\kern0pt}blinfun{\isacharunderscore}{\kern0pt}compose{\isacharbrackleft}{\kern0pt}symmetric{\isacharbrackright}{\kern0pt}\ norm{\isacharunderscore}{\kern0pt}blinfun\ simp\ del{\isacharcolon}{\kern0pt}\ blinfun{\isacharunderscore}{\kern0pt}apply{\isacharunderscore}{\kern0pt}blinfun{\isacharunderscore}{\kern0pt}compose{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isasymle}\ norm\ {\isacharparenleft}{\kern0pt}inv\isactrlsub L\ {\isacharparenleft}{\kern0pt}Q\ d{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}r{\isacharunderscore}{\kern0pt}dec\isactrlsub b\ {\isacharparenleft}{\kern0pt}mk{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\ d{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharplus}{\kern0pt}\ norm\ v{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ norm{\isacharunderscore}{\kern0pt}QR{\isacharunderscore}{\kern0pt}less{\isacharunderscore}{\kern0pt}one\ assms\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}fastforce\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ mult{\isacharunderscore}{\kern0pt}left{\isacharunderscore}{\kern0pt}le{\isacharunderscore}{\kern0pt}one{\isacharunderscore}{\kern0pt}le{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isasymle}\ norm\ {\isacharparenleft}{\kern0pt}inv\isactrlsub L\ {\isacharparenleft}{\kern0pt}Q\ d{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharasterisk}{\kern0pt}\ r\isactrlsub M\ {\isacharplus}{\kern0pt}\ norm\ v{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ order{\isachardot}{\kern0pt}trans{\isacharbrackleft}{\kern0pt}OF\ norm{\isacharunderscore}{\kern0pt}blinfun{\isacharbrackright}{\kern0pt}\ mult{\isacharunderscore}{\kern0pt}left{\isacharunderscore}{\kern0pt}mono\ simp{\isacharcolon}{\kern0pt}\ \ norm{\isacharunderscore}{\kern0pt}r{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}le{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isasymle}\ {\isacharparenleft}{\kern0pt}{\isasymSqunion}d\ {\isasymin}\ D\isactrlsub D{\isachardot}{\kern0pt}\ norm\ {\isacharparenleft}{\kern0pt}inv\isactrlsub L\ {\isacharparenleft}{\kern0pt}Q\ d{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharasterisk}{\kern0pt}\ r\isactrlsub M\ {\isacharplus}{\kern0pt}\ norm\ v{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ mult{\isacharunderscore}{\kern0pt}right{\isacharunderscore}{\kern0pt}mono\ cSUP{\isacharunderscore}{\kern0pt}upper\ assms\ simp{\isacharcolon}{\kern0pt}\ r\isactrlsub M{\isacharunderscore}{\kern0pt}nonneg{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{finally}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isacommand{{\isachardot}{\kern0pt}}\isamarkupfalse%
-\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lift{\isacharunderscore}{\kern0pt}definition}\isamarkupfalse%
-\ {\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}split\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}s\ {\isasymRightarrow}\isactrlsub b\ real{\isacharparenright}{\kern0pt}\ {\isasymRightarrow}\ {\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}s\ {\isasymRightarrow}\isactrlsub b\ real{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{is}\ {\isasymL}{\isacharunderscore}{\kern0pt}split\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ fastforce%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ {\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}split{\isacharunderscore}{\kern0pt}def{\isacharprime}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}split\ v\ s\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSqunion}d{\isasymin}D\isactrlsub D{\isachardot}{\kern0pt}\ L{\isacharunderscore}{\kern0pt}split\ d\ v\ s{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ {\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}split{\isachardot}{\kern0pt}rep{\isacharunderscore}{\kern0pt}eq\ {\isasymL}{\isacharunderscore}{\kern0pt}split{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ {\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}split{\isacharunderscore}{\kern0pt}contraction{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}dist\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}split\ v{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}split\ u{\isacharparenright}{\kern0pt}\ {\isasymle}\ QR{\isacharunderscore}{\kern0pt}disc\ {\isacharasterisk}{\kern0pt}\ dist\ v\ u{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ aux{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}split\ v\ s\ {\isacharminus}{\kern0pt}\ {\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}split\ u\ s\ {\isasymle}\ QR{\isacharunderscore}{\kern0pt}disc\ {\isacharasterisk}{\kern0pt}\ norm\ {\isacharparenleft}{\kern0pt}v\ {\isacharminus}{\kern0pt}\ u{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{if}\ h{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}split\ u\ s\ {\isasymle}\ {\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}split\ v\ s{\isachardoublequoteclose}\ \isakeyword{for}\ u\ v\ s\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{obtain}\isamarkupfalse%
-\ d\ \isakeyword{where}\ d{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max\ {\isacharparenleft}{\kern0pt}{\isasymlambda}d{\isachardot}{\kern0pt}\ inv\isactrlsub L\ {\isacharparenleft}{\kern0pt}Q\ d{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}r{\isacharunderscore}{\kern0pt}dec\isactrlsub b\ {\isacharparenleft}{\kern0pt}mk{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\ d{\isacharparenright}{\kern0pt}\ {\isacharplus}{\kern0pt}\ R\ d\ v{\isacharparenright}{\kern0pt}\ s{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymlambda}d{\isachardot}{\kern0pt}\ d\ {\isasymin}\ D\isactrlsub D{\isacharparenright}{\kern0pt}\ d{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ finite{\isacharunderscore}{\kern0pt}is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharbrackleft}{\kern0pt}of\ {\isachardoublequoteopen}D\isactrlsub D{\isachardoublequoteclose}{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isacharasterisk}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}inv\isactrlsub L\ {\isacharparenleft}{\kern0pt}Q\ d{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}r{\isacharunderscore}{\kern0pt}dec\isactrlsub b\ {\isacharparenleft}{\kern0pt}mk{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\ d{\isacharparenright}{\kern0pt}\ {\isacharplus}{\kern0pt}\ R\ d\ u{\isacharparenright}{\kern0pt}\ s\ {\isasymle}\ {\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}split\ u\ s{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ d\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ {\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}split{\isacharunderscore}{\kern0pt}def{\isacharprime}{\kern0pt}\ is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}linorder\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ cSUP{\isacharunderscore}{\kern0pt}upper{\isadigit{2}}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}inv\isactrlsub L\ {\isacharparenleft}{\kern0pt}Q\ d{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}r{\isacharunderscore}{\kern0pt}dec\isactrlsub b\ {\isacharparenleft}{\kern0pt}mk{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\ d{\isacharparenright}{\kern0pt}\ {\isacharplus}{\kern0pt}\ R\ d\ v{\isacharparenright}{\kern0pt}\ s\ {\isacharequal}{\kern0pt}\ {\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}split\ v\ s{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ {\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}split{\isacharunderscore}{\kern0pt}def{\isacharprime}{\kern0pt}\ arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}SUP{\isacharbrackleft}{\kern0pt}OF\ d{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}split\ v\ s\ {\isacharminus}{\kern0pt}\ {\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}split\ u\ s\ {\isacharequal}{\kern0pt}\ inv\isactrlsub L\ {\isacharparenleft}{\kern0pt}Q\ d{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}r{\isacharunderscore}{\kern0pt}dec\isactrlsub b\ {\isacharparenleft}{\kern0pt}mk{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\ d{\isacharparenright}{\kern0pt}\ {\isacharplus}{\kern0pt}\ R\ d\ v{\isacharparenright}{\kern0pt}\ s\ {\isacharminus}{\kern0pt}\ {\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}split\ u\ s{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isasymle}\ {\isacharparenleft}{\kern0pt}inv\isactrlsub L\ {\isacharparenleft}{\kern0pt}Q\ d{\isacharparenright}{\kern0pt}\ o\isactrlsub L\ R\ d{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}v\ {\isacharminus}{\kern0pt}\ u{\isacharparenright}{\kern0pt}\ s{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isacharasterisk}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ blinfun{\isachardot}{\kern0pt}bilinear{\isacharunderscore}{\kern0pt}simps{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isasymle}\ norm\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}inv\isactrlsub L\ {\isacharparenleft}{\kern0pt}Q\ d{\isacharparenright}{\kern0pt}\ o\isactrlsub L\ R\ d{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharasterisk}{\kern0pt}\ norm\ {\isacharparenleft}{\kern0pt}v\ {\isacharminus}{\kern0pt}\ u{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}fastforce\ intro{\isacharcolon}{\kern0pt}\ order{\isachardot}{\kern0pt}trans{\isacharbrackleft}{\kern0pt}OF\ le{\isacharunderscore}{\kern0pt}norm{\isacharunderscore}{\kern0pt}bfun\ norm{\isacharunderscore}{\kern0pt}blinfun{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isasymle}\ QR{\isacharunderscore}{\kern0pt}disc\ {\isacharasterisk}{\kern0pt}\ norm\ {\isacharparenleft}{\kern0pt}v\ {\isacharminus}{\kern0pt}\ u{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ QR{\isacharunderscore}{\kern0pt}contraction\ d\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}linorder\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ mult{\isacharunderscore}{\kern0pt}right{\isacharunderscore}{\kern0pt}mono\ cSUP{\isacharunderscore}{\kern0pt}upper{\isadigit{2}}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{finally}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isacommand{{\isachardot}{\kern0pt}}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymbar}{\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}split\ v\ {\isacharminus}{\kern0pt}\ {\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}split\ u{\isacharparenright}{\kern0pt}\ s{\isasymbar}\ {\isasymle}\ QR{\isacharunderscore}{\kern0pt}disc\ {\isacharasterisk}{\kern0pt}\ dist\ v\ u{\isachardoublequoteclose}\ \isakeyword{for}\ s\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ aux\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}cases\ {\isachardoublequoteopen}{\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}split\ v\ s\ {\isasymle}\ {\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}split\ u\ s{\isachardoublequoteclose}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}fastforce\ simp{\isacharcolon}{\kern0pt}\ dist{\isacharunderscore}{\kern0pt}norm\ norm{\isacharunderscore}{\kern0pt}minus{\isacharunderscore}{\kern0pt}commute{\isacharparenright}{\kern0pt}{\isacharplus}{\kern0pt}\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ cSUP{\isacharunderscore}{\kern0pt}least\ simp{\isacharcolon}{\kern0pt}\ dist{\isacharunderscore}{\kern0pt}bfun{\isachardot}{\kern0pt}rep{\isacharunderscore}{\kern0pt}eq\ dist{\isacharunderscore}{\kern0pt}real{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ {\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}lim{\isacharcolon}{\kern0pt}\isanewline
-\ \ {\isachardoublequoteopen}{\isasymexists}{\isacharbang}{\kern0pt}v{\isachardot}{\kern0pt}\ {\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}split\ v\ {\isacharequal}{\kern0pt}\ v{\isachardoublequoteclose}\isanewline
-\ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isasymlambda}n{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}split\ {\isacharcircum}{\kern0pt}{\isacharcircum}{\kern0pt}\ n{\isacharparenright}{\kern0pt}\ v{\isacharparenright}{\kern0pt}\ {\isasymlonglonglongrightarrow}\ {\isacharparenleft}{\kern0pt}THE\ v{\isachardot}{\kern0pt}\ {\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}split\ v\ {\isacharequal}{\kern0pt}\ v{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ banach{\isacharprime}{\kern0pt}{\isacharbrackleft}{\kern0pt}of\ {\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}split{\isacharbrackright}{\kern0pt}\ a{\isacharunderscore}{\kern0pt}nonneg\ QR{\isacharunderscore}{\kern0pt}contraction\ {\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}split{\isacharunderscore}{\kern0pt}contraction\isanewline
-\ \ \isacommand{unfolding}\isamarkupfalse%
-\ is{\isacharunderscore}{\kern0pt}contraction{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ {\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}split{\isacharunderscore}{\kern0pt}tendsto{\isacharunderscore}{\kern0pt}opt{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isasymlambda}n{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}split\ {\isacharcircum}{\kern0pt}{\isacharcircum}{\kern0pt}\ n{\isacharparenright}{\kern0pt}\ v{\isacharparenright}{\kern0pt}\ {\isasymlonglonglongrightarrow}\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{obtain}\isamarkupfalse%
-\ L\ \isakeyword{where}\ l{\isacharunderscore}{\kern0pt}fix{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}split\ L\ {\isacharequal}{\kern0pt}\ L{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}lim{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymnu}\isactrlsub b\ {\isacharparenleft}{\kern0pt}mk{\isacharunderscore}{\kern0pt}stationary{\isacharunderscore}{\kern0pt}det\ d{\isacharparenright}{\kern0pt}\ {\isasymle}\ L{\isachardoublequoteclose}\ \isakeyword{if}\ d{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}d\ {\isasymin}\ D\isactrlsub D{\isachardoublequoteclose}\ \isakeyword{for}\ d\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}QR\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}inv\isactrlsub L\ {\isacharparenleft}{\kern0pt}Q\ d{\isacharparenright}{\kern0pt}\ o\isactrlsub L\ R\ d{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}inv\isactrlsub L\ {\isacharparenleft}{\kern0pt}Q\ d{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}r{\isacharunderscore}{\kern0pt}dec\isactrlsub b\ {\isacharparenleft}{\kern0pt}mk{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\ d{\isacharparenright}{\kern0pt}\ {\isacharplus}{\kern0pt}\ R\ d\ L{\isacharparenright}{\kern0pt}\ {\isasymle}\ {\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}split\ L{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ d\ l{\isacharunderscore}{\kern0pt}fix\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}fastforce\ simp{\isacharcolon}{\kern0pt}\ {\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}split{\isacharunderscore}{\kern0pt}def{\isacharprime}{\kern0pt}\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ cSUP{\isacharunderscore}{\kern0pt}upper{\isadigit{2}}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}inv\isactrlsub L\ {\isacharparenleft}{\kern0pt}Q\ d{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}r{\isacharunderscore}{\kern0pt}dec\isactrlsub b\ {\isacharparenleft}{\kern0pt}mk{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\ d{\isacharparenright}{\kern0pt}\ {\isacharplus}{\kern0pt}\ R\ d\ L{\isacharparenright}{\kern0pt}\ {\isasymle}\ L{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ l{\isacharunderscore}{\kern0pt}fix\ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ aux{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}inv\isactrlsub L\ {\isacharparenleft}{\kern0pt}Q\ d{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}r{\isacharunderscore}{\kern0pt}dec\isactrlsub b\ {\isacharparenleft}{\kern0pt}mk{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\ d{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymle}\ {\isacharparenleft}{\kern0pt}id{\isacharunderscore}{\kern0pt}blinfun\ {\isacharminus}{\kern0pt}\ {\isacharquery}{\kern0pt}QR{\isacharparenright}{\kern0pt}\ L{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ that\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ blinfun{\isachardot}{\kern0pt}bilinear{\isacharunderscore}{\kern0pt}simps\ le{\isacharunderscore}{\kern0pt}diff{\isacharunderscore}{\kern0pt}eq{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ inv{\isacharunderscore}{\kern0pt}eq{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}inv\isactrlsub L\ {\isacharparenleft}{\kern0pt}id{\isacharunderscore}{\kern0pt}blinfun\ {\isacharminus}{\kern0pt}\ {\isacharquery}{\kern0pt}QR{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSum}i{\isachardot}{\kern0pt}\ {\isacharquery}{\kern0pt}QR\ {\isacharcircum}{\kern0pt}{\isacharcircum}{\kern0pt}\ i{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ QR{\isacharunderscore}{\kern0pt}contraction\ d\ norm{\isacharunderscore}{\kern0pt}QR{\isacharunderscore}{\kern0pt}less{\isacharunderscore}{\kern0pt}one\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ inv\isactrlsub L{\isacharunderscore}{\kern0pt}inf{\isacharunderscore}{\kern0pt}sum{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ summable{\isacharunderscore}{\kern0pt}QR{\isacharcolon}{\kern0pt}{\isachardoublequoteopen}summable\ {\isacharparenleft}{\kern0pt}{\isasymlambda}i{\isachardot}{\kern0pt}\ norm\ {\isacharparenleft}{\kern0pt}{\isacharquery}{\kern0pt}QR\ {\isacharcircum}{\kern0pt}{\isacharcircum}{\kern0pt}\ i{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ QR{\isacharunderscore}{\kern0pt}contraction\ d\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}fastforce\ simp{\isacharcolon}{\kern0pt}\ a{\isacharunderscore}{\kern0pt}nonneg\isanewline
-\ \ \ \ \ \ \ \ \ \ intro{\isacharcolon}{\kern0pt}\ summable{\isacharunderscore}{\kern0pt}comparison{\isacharunderscore}{\kern0pt}test{\isacharprime}{\kern0pt}{\isacharbrackleft}{\kern0pt}\isakeyword{where}\ g\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}{\isasymlambda}i{\isachardot}{\kern0pt}\ QR{\isacharunderscore}{\kern0pt}disc{\isacharcircum}{\kern0pt}i{\isachardoublequoteclose}{\isacharbrackright}{\kern0pt}\ \isanewline
-\ \ \ \ \ \ \ \ \ \ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ cSUP{\isacharunderscore}{\kern0pt}upper{\isadigit{2}}\ power{\isacharunderscore}{\kern0pt}mono\ order{\isachardot}{\kern0pt}trans{\isacharbrackleft}{\kern0pt}OF\ norm{\isacharunderscore}{\kern0pt}blinfunpow{\isacharunderscore}{\kern0pt}le{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}summable\ {\isacharparenleft}{\kern0pt}{\isasymlambda}i{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharquery}{\kern0pt}QR\ {\isacharcircum}{\kern0pt}{\isacharcircum}{\kern0pt}\ i{\isacharparenright}{\kern0pt}\ v\ s{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{for}\ v\ s\ \isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}rule\ summable{\isacharunderscore}{\kern0pt}comparison{\isacharunderscore}{\kern0pt}test{\isacharprime}{\kern0pt}{\isacharbrackleft}{\kern0pt}\isakeyword{where}\ g\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}{\isasymlambda}i{\isachardot}{\kern0pt}\ norm\ {\isacharparenleft}{\kern0pt}{\isacharquery}{\kern0pt}QR\ {\isacharcircum}{\kern0pt}{\isacharcircum}{\kern0pt}\ i{\isacharparenright}{\kern0pt}\ {\isacharasterisk}{\kern0pt}\ norm\ v{\isachardoublequoteclose}{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}auto\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ summable{\isacharunderscore}{\kern0pt}QR\ summable{\isacharunderscore}{\kern0pt}norm{\isacharunderscore}{\kern0pt}cancel\ order{\isachardot}{\kern0pt}trans{\isacharbrackleft}{\kern0pt}OF\ abs{\isacharunderscore}{\kern0pt}le{\isacharunderscore}{\kern0pt}norm{\isacharunderscore}{\kern0pt}bfun{\isacharbrackright}{\kern0pt}\ order{\isachardot}{\kern0pt}trans{\isacharbrackleft}{\kern0pt}OF\ norm{\isacharunderscore}{\kern0pt}blinfun{\isacharbrackright}{\kern0pt}\ summable{\isacharunderscore}{\kern0pt}mult{\isadigit{2}}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isadigit{0}}\ {\isasymle}\ v\ {\isasymLongrightarrow}\ {\isadigit{0}}\ {\isasymle}\ {\isacharparenleft}{\kern0pt}{\isasymSum}i{\isacharless}{\kern0pt}n{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharquery}{\kern0pt}QR\ {\isacharcircum}{\kern0pt}{\isacharcircum}{\kern0pt}\ i{\isacharparenright}{\kern0pt}\ v\ s{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{for}\ n\ v\ s\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ blinfunpow{\isacharunderscore}{\kern0pt}nonneg{\isacharbrackleft}{\kern0pt}OF\ QR{\isacharunderscore}{\kern0pt}nonneg{\isacharbrackleft}{\kern0pt}OF\ d{\isacharbrackright}{\kern0pt}{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}induction\ n{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}auto\ simp\ add{\isacharcolon}{\kern0pt}\ less{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}bfun{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isadigit{0}}\ {\isasymle}\ v\ {\isasymLongrightarrow}\ {\isadigit{0}}\ {\isasymle}\ {\isacharparenleft}{\kern0pt}{\isasymSum}i{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}{\isacharquery}{\kern0pt}QR\ {\isacharcircum}{\kern0pt}{\isacharcircum}{\kern0pt}\ i{\isacharparenright}{\kern0pt}\ v\ s{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isachardoublequoteclose}\ \isakeyword{for}\ v\ s\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ summable{\isacharunderscore}{\kern0pt}LIMSEQ\ LIMSEQ{\isacharunderscore}{\kern0pt}le{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isadigit{0}}\ {\isasymle}\ v\ {\isasymLongrightarrow}\ {\isadigit{0}}\ {\isasymle}\ {\isacharparenleft}{\kern0pt}{\isasymSum}i{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}{\isacharquery}{\kern0pt}QR\ {\isacharcircum}{\kern0pt}{\isacharcircum}{\kern0pt}\ i{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ v\ s{\isachardoublequoteclose}\ \isakeyword{for}\ v\ s\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ bounded{\isacharunderscore}{\kern0pt}linear{\isacharunderscore}{\kern0pt}apply{\isacharunderscore}{\kern0pt}bfun\ summable{\isacharunderscore}{\kern0pt}QR\ summable{\isacharunderscore}{\kern0pt}comparison{\isacharunderscore}{\kern0pt}test{\isacharprime}{\kern0pt}\ \isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}subst\ bounded{\isacharunderscore}{\kern0pt}linear{\isachardot}{\kern0pt}suminf{\isacharbrackleft}{\kern0pt}\isakeyword{where}\ f\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isasymlambda}i{\isachardot}{\kern0pt}\ apply{\isacharunderscore}{\kern0pt}bfun\ {\isacharparenleft}{\kern0pt}blinfun{\isacharunderscore}{\kern0pt}apply\ i\ v{\isacharparenright}{\kern0pt}\ s{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}fastforce\ intro{\isacharcolon}{\kern0pt}\ bounded{\isacharunderscore}{\kern0pt}linear{\isacharunderscore}{\kern0pt}compose{\isacharbrackleft}{\kern0pt}of\ {\isachardoublequoteopen}{\isasymlambda}s{\isachardot}{\kern0pt}\ apply{\isacharunderscore}{\kern0pt}bfun\ s\ {\isacharunderscore}{\kern0pt}{\isachardoublequoteclose}{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharplus}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isadigit{0}}\ {\isasymle}\ v\ {\isasymLongrightarrow}\ {\isadigit{0}}\ {\isasymle}\ inv\isactrlsub L\ {\isacharparenleft}{\kern0pt}id{\isacharunderscore}{\kern0pt}blinfun\ {\isacharminus}{\kern0pt}\ {\isacharquery}{\kern0pt}QR{\isacharparenright}{\kern0pt}\ v{\isachardoublequoteclose}\ \isakeyword{for}\ v\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ inv{\isacharunderscore}{\kern0pt}eq\ less{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}bfun{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}inv\isactrlsub L\ {\isacharparenleft}{\kern0pt}id{\isacharunderscore}{\kern0pt}blinfun\ {\isacharminus}{\kern0pt}\ {\isacharquery}{\kern0pt}QR{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}inv\isactrlsub L\ {\isacharparenleft}{\kern0pt}Q\ d{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}r{\isacharunderscore}{\kern0pt}dec\isactrlsub b\ {\isacharparenleft}{\kern0pt}mk{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\ d{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ {\isasymle}\ {\isacharparenleft}{\kern0pt}inv\isactrlsub L\ {\isacharparenleft}{\kern0pt}id{\isacharunderscore}{\kern0pt}blinfun\ {\isacharminus}{\kern0pt}\ {\isacharquery}{\kern0pt}QR{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}id{\isacharunderscore}{\kern0pt}blinfun\ {\isacharminus}{\kern0pt}\ {\isacharquery}{\kern0pt}QR{\isacharparenright}{\kern0pt}\ L{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ aux\ blinfun{\isachardot}{\kern0pt}diff{\isacharunderscore}{\kern0pt}right\ diff{\isacharunderscore}{\kern0pt}ge{\isacharunderscore}{\kern0pt}{\isadigit{0}}{\isacharunderscore}{\kern0pt}iff{\isacharunderscore}{\kern0pt}ge{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}inv\isactrlsub L\ {\isacharparenleft}{\kern0pt}id{\isacharunderscore}{\kern0pt}blinfun\ {\isacharminus}{\kern0pt}\ {\isacharquery}{\kern0pt}QR{\isacharparenright}{\kern0pt}\ o\isactrlsub L\ inv\isactrlsub L\ {\isacharparenleft}{\kern0pt}Q\ d{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}r{\isacharunderscore}{\kern0pt}dec\isactrlsub b\ {\isacharparenleft}{\kern0pt}mk{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\ d{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymle}\ L{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ invertible\isactrlsub L{\isacharunderscore}{\kern0pt}inf{\isacharunderscore}{\kern0pt}sum{\isacharbrackleft}{\kern0pt}OF\ norm{\isacharunderscore}{\kern0pt}QR{\isacharunderscore}{\kern0pt}less{\isacharunderscore}{\kern0pt}one{\isacharbrackleft}{\kern0pt}OF\ that{\isacharbrackright}{\kern0pt}{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}inv\isactrlsub L\ {\isacharparenleft}{\kern0pt}Q\ d\ o\isactrlsub L\ {\isacharparenleft}{\kern0pt}id{\isacharunderscore}{\kern0pt}blinfun\ {\isacharminus}{\kern0pt}\ {\isacharquery}{\kern0pt}QR{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}r{\isacharunderscore}{\kern0pt}dec\isactrlsub b\ {\isacharparenleft}{\kern0pt}mk{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\ d{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymle}\ L{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ d\ norm{\isacharunderscore}{\kern0pt}QR{\isacharunderscore}{\kern0pt}less{\isacharunderscore}{\kern0pt}one\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ inv\isactrlsub L{\isacharunderscore}{\kern0pt}compose{\isacharbrackleft}{\kern0pt}OF\ Q{\isacharunderscore}{\kern0pt}invertible\ invertible\isactrlsub L{\isacharunderscore}{\kern0pt}inf{\isacharunderscore}{\kern0pt}sum{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}inv\isactrlsub L\ {\isacharparenleft}{\kern0pt}Q\ d\ {\isacharminus}{\kern0pt}\ R\ d{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}r{\isacharunderscore}{\kern0pt}dec\isactrlsub b\ {\isacharparenleft}{\kern0pt}mk{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\ d{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymle}\ L{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ Q{\isacharunderscore}{\kern0pt}invertible\ d\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ blinfun{\isacharunderscore}{\kern0pt}compose{\isacharunderscore}{\kern0pt}diff{\isacharunderscore}{\kern0pt}right\ blinfun{\isacharunderscore}{\kern0pt}compose{\isacharunderscore}{\kern0pt}assoc{\isacharbrackleft}{\kern0pt}symmetric{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymnu}\isactrlsub b\ {\isacharparenleft}{\kern0pt}mk{\isacharunderscore}{\kern0pt}stationary{\isacharunderscore}{\kern0pt}det\ d{\isacharparenright}{\kern0pt}\ {\isasymle}\ L{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ {\isasymnu}{\isacharunderscore}{\kern0pt}stationary\ splitting{\isacharbrackleft}{\kern0pt}OF\ that{\isacharcomma}{\kern0pt}\ symmetric{\isacharbrackright}{\kern0pt}\ inv\isactrlsub L{\isacharunderscore}{\kern0pt}inf{\isacharunderscore}{\kern0pt}sum\ blincomp{\isacharunderscore}{\kern0pt}scaleR{\isacharunderscore}{\kern0pt}right{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ opt{\isacharunderscore}{\kern0pt}le{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt\ {\isasymle}\ L{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ thm{\isacharunderscore}{\kern0pt}{\isadigit{6}}{\isacharunderscore}{\kern0pt}{\isadigit{2}}{\isacharunderscore}{\kern0pt}{\isadigit{1}}{\isadigit{0}}\ finite\ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\isanewline
-\ \ \isacommand{obtain}\isamarkupfalse%
-\ d\ \isakeyword{where}\ d{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max\ {\isacharparenleft}{\kern0pt}{\isasymlambda}d{\isachardot}{\kern0pt}\ inv\isactrlsub L\ {\isacharparenleft}{\kern0pt}Q\ d{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}r{\isacharunderscore}{\kern0pt}dec\isactrlsub b\ {\isacharparenleft}{\kern0pt}mk{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\ d{\isacharparenright}{\kern0pt}\ {\isacharplus}{\kern0pt}\ R\ d\ L{\isacharparenright}{\kern0pt}\ s{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymlambda}d{\isachardot}{\kern0pt}\ d\ {\isasymin}\ D\isactrlsub D{\isacharparenright}{\kern0pt}\ d{\isachardoublequoteclose}\ \isakeyword{for}\ s\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}ex{\isacharunderscore}{\kern0pt}split\ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}d\ {\isasymin}\ D\isactrlsub D{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}linorder\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}L\ {\isacharequal}{\kern0pt}\ inv\isactrlsub L\ {\isacharparenleft}{\kern0pt}Q\ d{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}r{\isacharunderscore}{\kern0pt}dec\isactrlsub b\ {\isacharparenleft}{\kern0pt}mk{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\ d{\isacharparenright}{\kern0pt}\ {\isacharplus}{\kern0pt}\ R\ d\ L{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}subst\ l{\isacharunderscore}{\kern0pt}fix{\isacharbrackleft}{\kern0pt}symmetric{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}fastforce\ simp{\isacharcolon}{\kern0pt}\ {\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}split{\isacharunderscore}{\kern0pt}def{\isacharprime}{\kern0pt}\ arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}SUP{\isacharbrackleft}{\kern0pt}OF\ d{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}Q\ d\ L\ {\isacharequal}{\kern0pt}\ r{\isacharunderscore}{\kern0pt}dec\isactrlsub b\ {\isacharparenleft}{\kern0pt}mk{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\ d{\isacharparenright}{\kern0pt}\ {\isacharplus}{\kern0pt}\ R\ d\ L{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ Q{\isacharunderscore}{\kern0pt}invertible\ {\isacartoucheopen}d\ {\isasymin}\ D\isactrlsub D{\isacartoucheclose}\ inv{\isacharunderscore}{\kern0pt}app{\isadigit{2}}{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}id{\isacharunderscore}{\kern0pt}blinfun\ {\isacharminus}{\kern0pt}\ l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isasymP}\isactrlsub {\isadigit{1}}\ {\isacharparenleft}{\kern0pt}mk{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\ d{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ L\ {\isacharequal}{\kern0pt}\ r{\isacharunderscore}{\kern0pt}dec\isactrlsub b\ {\isacharparenleft}{\kern0pt}mk{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\ d{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ splitting{\isacharbrackleft}{\kern0pt}OF\ {\isacartoucheopen}d\ {\isasymin}\ D\isactrlsub D{\isacartoucheclose}{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ blinfun{\isachardot}{\kern0pt}diff{\isacharunderscore}{\kern0pt}left{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}L\ {\isacharequal}{\kern0pt}\ inv\isactrlsub L\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}id{\isacharunderscore}{\kern0pt}blinfun\ {\isacharminus}{\kern0pt}\ l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isasymP}\isactrlsub {\isadigit{1}}\ {\isacharparenleft}{\kern0pt}mk{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\ d{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}r{\isacharunderscore}{\kern0pt}dec\isactrlsub b\ {\isacharparenleft}{\kern0pt}mk{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\ d{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ invertible\isactrlsub L{\isacharunderscore}{\kern0pt}inf{\isacharunderscore}{\kern0pt}sum{\isacharbrackleft}{\kern0pt}OF\ norm{\isacharunderscore}{\kern0pt}{\isasymP}\isactrlsub {\isadigit{1}}{\isacharunderscore}{\kern0pt}l{\isacharunderscore}{\kern0pt}less{\isacharbrackright}{\kern0pt}\ inv{\isacharunderscore}{\kern0pt}app{\isadigit{1}}{\isacharprime}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}L\ {\isacharequal}{\kern0pt}\ {\isasymnu}\isactrlsub b\ {\isacharparenleft}{\kern0pt}mk{\isacharunderscore}{\kern0pt}stationary{\isacharunderscore}{\kern0pt}det\ d{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ inv\isactrlsub L{\isacharunderscore}{\kern0pt}inf{\isacharunderscore}{\kern0pt}sum\ {\isasymnu}{\isacharunderscore}{\kern0pt}stationary\ blincomp{\isacharunderscore}{\kern0pt}scaleR{\isacharunderscore}{\kern0pt}right{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt\ {\isacharequal}{\kern0pt}\ L{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ opt{\isacharunderscore}{\kern0pt}le\ {\isacartoucheopen}d\ {\isasymin}\ D\isactrlsub D{\isacartoucheclose}\ is{\isacharunderscore}{\kern0pt}markovian{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ intro{\isacharcolon}{\kern0pt}\ order{\isachardot}{\kern0pt}antisym{\isacharbrackleft}{\kern0pt}OF\ {\isacharunderscore}{\kern0pt}\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}le{\isacharunderscore}{\kern0pt}opt{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}lim\ l{\isacharunderscore}{\kern0pt}fix\ the{\isadigit{1}}{\isacharunderscore}{\kern0pt}equality{\isacharbrackleft}{\kern0pt}OF\ {\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}lim{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ {\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}split{\isacharunderscore}{\kern0pt}fix{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}split\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt\ {\isacharequal}{\kern0pt}\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ {\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}lim\ {\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}split{\isacharunderscore}{\kern0pt}tendsto{\isacharunderscore}{\kern0pt}opt\ the{\isacharunderscore}{\kern0pt}equality\ limI\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}mono{\isacharunderscore}{\kern0pt}tags{\isacharcomma}{\kern0pt}\ lifting{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ dist{\isacharunderscore}{\kern0pt}{\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}split{\isacharunderscore}{\kern0pt}opt{\isacharunderscore}{\kern0pt}eps{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}eps\ {\isachargreater}{\kern0pt}\ {\isadigit{0}}{\isachardoublequoteclose}\ {\isachardoublequoteopen}{\isadigit{2}}\ {\isacharasterisk}{\kern0pt}\ QR{\isacharunderscore}{\kern0pt}disc\ {\isacharasterisk}{\kern0pt}\ dist\ v\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}split\ v{\isacharparenright}{\kern0pt}\ {\isacharless}{\kern0pt}\ eps\ {\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharminus}{\kern0pt}QR{\isacharunderscore}{\kern0pt}disc{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}dist\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}split\ v{\isacharparenright}{\kern0pt}\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt\ {\isacharless}{\kern0pt}\ eps\ {\isacharslash}{\kern0pt}\ {\isadigit{2}}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isadigit{1}}\ {\isacharminus}{\kern0pt}\ QR{\isacharunderscore}{\kern0pt}disc{\isacharparenright}{\kern0pt}\ {\isacharasterisk}{\kern0pt}\ dist\ v\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt\ {\isasymle}\ dist\ v\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}split\ v{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ dist{\isacharunderscore}{\kern0pt}triangle\ {\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}split{\isacharunderscore}{\kern0pt}contraction{\isacharbrackleft}{\kern0pt}of\ v\ {\isachardoublequoteopen}{\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt{\isachardoublequoteclose}{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}fastforce\ simp{\isacharcolon}{\kern0pt}\ algebra{\isacharunderscore}{\kern0pt}simps\ intro{\isacharcolon}{\kern0pt}\ order{\isachardot}{\kern0pt}trans{\isacharbrackleft}{\kern0pt}OF\ {\isacharunderscore}{\kern0pt}\ add{\isacharunderscore}{\kern0pt}left{\isacharunderscore}{\kern0pt}mono{\isacharbrackleft}{\kern0pt}of\ {\isachardoublequoteopen}dist\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}split\ v{\isacharparenright}{\kern0pt}\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt{\isachardoublequoteclose}{\isacharbrackright}{\kern0pt}{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}dist\ v\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt\ {\isasymle}\ dist\ v\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}split\ v{\isacharparenright}{\kern0pt}\ {\isacharslash}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}\ {\isacharminus}{\kern0pt}\ QR{\isacharunderscore}{\kern0pt}disc{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ QR{\isacharunderscore}{\kern0pt}contraction\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ mult{\isachardot}{\kern0pt}commute\ pos{\isacharunderscore}{\kern0pt}le{\isacharunderscore}{\kern0pt}divide{\isacharunderscore}{\kern0pt}eq{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isadigit{2}}\ {\isacharasterisk}{\kern0pt}\ QR{\isacharunderscore}{\kern0pt}disc\ {\isacharasterisk}{\kern0pt}\ dist\ v\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt\ {\isasymle}\ {\isadigit{2}}\ {\isacharasterisk}{\kern0pt}\ QR{\isacharunderscore}{\kern0pt}disc\ {\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}dist\ v\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}split\ v{\isacharparenright}{\kern0pt}\ {\isacharslash}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}\ {\isacharminus}{\kern0pt}\ QR{\isacharunderscore}{\kern0pt}disc{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}split{\isacharunderscore}{\kern0pt}contraction\ assms\ mult{\isacharunderscore}{\kern0pt}le{\isacharunderscore}{\kern0pt}cancel{\isacharunderscore}{\kern0pt}left{\isacharunderscore}{\kern0pt}pos{\isacharbrackleft}{\kern0pt}of\ {\isachardoublequoteopen}{\isadigit{2}}\ {\isacharasterisk}{\kern0pt}\ QR{\isacharunderscore}{\kern0pt}disc{\isachardoublequoteclose}{\isacharbrackright}{\kern0pt}\ a{\isacharunderscore}{\kern0pt}nonneg\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}fastforce\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ mult{\isacharunderscore}{\kern0pt}left{\isacharunderscore}{\kern0pt}mono{\isacharbrackleft}{\kern0pt}of\ {\isacharunderscore}{\kern0pt}\ {\isacharunderscore}{\kern0pt}\ {\isachardoublequoteopen}{\isadigit{2}}\ {\isacharasterisk}{\kern0pt}\ QR{\isacharunderscore}{\kern0pt}disc{\isachardoublequoteclose}{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isadigit{2}}\ {\isacharasterisk}{\kern0pt}\ QR{\isacharunderscore}{\kern0pt}disc\ {\isacharasterisk}{\kern0pt}\ dist\ v\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt\ {\isacharless}{\kern0pt}\ eps{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ a{\isacharunderscore}{\kern0pt}nonneg\ QR{\isacharunderscore}{\kern0pt}contraction\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ assms{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}\ pos{\isacharunderscore}{\kern0pt}divide{\isacharunderscore}{\kern0pt}less{\isacharunderscore}{\kern0pt}eq\ intro{\isacharcolon}{\kern0pt}\ order{\isachardot}{\kern0pt}strict{\isacharunderscore}{\kern0pt}trans{\isadigit{1}}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}dist\ v\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt\ {\isacharasterisk}{\kern0pt}\ QR{\isacharunderscore}{\kern0pt}disc\ {\isacharless}{\kern0pt}\ eps\ {\isacharslash}{\kern0pt}\ {\isadigit{2}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ argo\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}dist\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}split\ v{\isacharparenright}{\kern0pt}\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt\ {\isacharless}{\kern0pt}\ eps\ {\isacharslash}{\kern0pt}\ {\isadigit{2}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}split{\isacharunderscore}{\kern0pt}contraction{\isacharbrackleft}{\kern0pt}of\ v\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt{\isacharbrackright}{\kern0pt}\ \isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ algebra{\isacharunderscore}{\kern0pt}simps{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ L{\isacharunderscore}{\kern0pt}split{\isacharunderscore}{\kern0pt}fix{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}d\ {\isasymin}\ D\isactrlsub D{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}L{\isacharunderscore}{\kern0pt}split\ d\ {\isacharparenleft}{\kern0pt}{\isasymnu}\isactrlsub b\ {\isacharparenleft}{\kern0pt}mk{\isacharunderscore}{\kern0pt}stationary{\isacharunderscore}{\kern0pt}det\ d{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isasymnu}\isactrlsub b\ {\isacharparenleft}{\kern0pt}mk{\isacharunderscore}{\kern0pt}stationary{\isacharunderscore}{\kern0pt}det\ d{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}d\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}mk{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\ d{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}p\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}mk{\isacharunderscore}{\kern0pt}stationary{\isacharunderscore}{\kern0pt}det\ d{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}Q\ d\ {\isacharminus}{\kern0pt}\ R\ d{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymnu}\isactrlsub b\ {\isacharquery}{\kern0pt}p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ r{\isacharunderscore}{\kern0pt}dec\isactrlsub b\ {\isacharquery}{\kern0pt}d{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ L{\isacharunderscore}{\kern0pt}{\isasymnu}{\isacharunderscore}{\kern0pt}fix{\isacharbrackleft}{\kern0pt}of\ {\isachardoublequoteopen}mk{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\ d{\isachardoublequoteclose}{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ L{\isacharunderscore}{\kern0pt}def\ splitting{\isacharbrackleft}{\kern0pt}OF\ assms{\isacharcomma}{\kern0pt}\ symmetric{\isacharbrackright}{\kern0pt}\ blinfun{\isachardot}{\kern0pt}bilinear{\isacharunderscore}{\kern0pt}simps\ diff{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}eq{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assms\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ blinfun{\isachardot}{\kern0pt}bilinear{\isacharunderscore}{\kern0pt}simps\ diff{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}eq\ inv\isactrlsub L{\isacharunderscore}{\kern0pt}cancel{\isacharunderscore}{\kern0pt}iff{\isacharbrackleft}{\kern0pt}OF\ Q{\isacharunderscore}{\kern0pt}invertible{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ L{\isacharunderscore}{\kern0pt}split{\isacharunderscore}{\kern0pt}contraction{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}d\ {\isasymin}\ D\isactrlsub D{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}dist\ {\isacharparenleft}{\kern0pt}L{\isacharunderscore}{\kern0pt}split\ d\ v{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}L{\isacharunderscore}{\kern0pt}split\ d\ u{\isacharparenright}{\kern0pt}\ {\isasymle}\ QR{\isacharunderscore}{\kern0pt}disc\ {\isacharasterisk}{\kern0pt}\ dist\ v\ u{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ aux{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}L{\isacharunderscore}{\kern0pt}split\ d\ v\ s\ {\isacharminus}{\kern0pt}\ L{\isacharunderscore}{\kern0pt}split\ d\ u\ s\ {\isasymle}\ QR{\isacharunderscore}{\kern0pt}disc\ {\isacharasterisk}{\kern0pt}\ dist\ v\ u{\isachardoublequoteclose}\ \isakeyword{if}\ lea{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}L{\isacharunderscore}{\kern0pt}split\ d\ u\ s{\isacharparenright}{\kern0pt}\ {\isasymle}\ {\isacharparenleft}{\kern0pt}L{\isacharunderscore}{\kern0pt}split\ d\ v\ s{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{for}\ v\ s\ u\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}L{\isacharunderscore}{\kern0pt}split\ d\ v\ s\ {\isacharminus}{\kern0pt}\ L{\isacharunderscore}{\kern0pt}split\ d\ u\ s\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}inv\isactrlsub L\ {\isacharparenleft}{\kern0pt}Q\ d{\isacharparenright}{\kern0pt}\ o\isactrlsub L\ {\isacharparenleft}{\kern0pt}R\ d{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}v\ {\isacharminus}{\kern0pt}\ u{\isacharparenright}{\kern0pt}\ s{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ blinfun{\isachardot}{\kern0pt}bilinear{\isacharunderscore}{\kern0pt}simps{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isasymle}\ norm\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}inv\isactrlsub L\ {\isacharparenleft}{\kern0pt}Q\ d{\isacharparenright}{\kern0pt}\ o\isactrlsub L\ {\isacharparenleft}{\kern0pt}R\ d{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}v\ {\isacharminus}{\kern0pt}\ u{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ le{\isacharunderscore}{\kern0pt}norm{\isacharunderscore}{\kern0pt}bfun{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isasymle}\ norm\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}inv\isactrlsub L\ {\isacharparenleft}{\kern0pt}Q\ d{\isacharparenright}{\kern0pt}\ o\isactrlsub L\ {\isacharparenleft}{\kern0pt}R\ d{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharasterisk}{\kern0pt}\ dist\ v\ u{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp\ only{\isacharcolon}{\kern0pt}\ dist{\isacharunderscore}{\kern0pt}norm\ norm{\isacharunderscore}{\kern0pt}blinfun{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isasymle}\ QR{\isacharunderscore}{\kern0pt}disc\ {\isacharasterisk}{\kern0pt}\ dist\ v\ u{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assms\ QR{\isacharunderscore}{\kern0pt}le{\isacharunderscore}{\kern0pt}QR{\isacharunderscore}{\kern0pt}disc\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ mult{\isacharunderscore}{\kern0pt}right{\isacharunderscore}{\kern0pt}mono{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{finally}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}dist\ {\isacharparenleft}{\kern0pt}L{\isacharunderscore}{\kern0pt}split\ d\ v\ s{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}L{\isacharunderscore}{\kern0pt}split\ d\ u\ s{\isacharparenright}{\kern0pt}\ {\isasymle}\ QR{\isacharunderscore}{\kern0pt}disc\ {\isacharasterisk}{\kern0pt}\ dist\ v\ u{\isachardoublequoteclose}\ \isakeyword{for}\ v\ s\ u\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ aux\ aux{\isacharbrackleft}{\kern0pt}of\ v\ {\isacharunderscore}{\kern0pt}\ u{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}cases\ {\isachardoublequoteopen}L{\isacharunderscore}{\kern0pt}split\ d\ v\ s\ {\isasymge}\ L{\isacharunderscore}{\kern0pt}split\ d\ u\ s{\isachardoublequoteclose}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ dist{\isacharunderscore}{\kern0pt}real{\isacharunderscore}{\kern0pt}def\ dist{\isacharunderscore}{\kern0pt}commute{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}dist\ {\isacharparenleft}{\kern0pt}L{\isacharunderscore}{\kern0pt}split\ d\ v{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}L{\isacharunderscore}{\kern0pt}split\ d\ u{\isacharparenright}{\kern0pt}\ {\isasymle}\ QR{\isacharunderscore}{\kern0pt}disc\ {\isacharasterisk}{\kern0pt}\ dist\ v\ u{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ dist{\isacharunderscore}{\kern0pt}bound{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ find{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}QR{\isacharunderscore}{\kern0pt}error{\isacharunderscore}{\kern0pt}bound{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}eps\ {\isachargreater}{\kern0pt}\ {\isadigit{0}}{\isachardoublequoteclose}\ {\isachardoublequoteopen}{\isadigit{2}}\ {\isacharasterisk}{\kern0pt}\ QR{\isacharunderscore}{\kern0pt}disc\ {\isacharasterisk}{\kern0pt}\ dist\ v\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}split\ v{\isacharparenright}{\kern0pt}\ {\isacharless}{\kern0pt}\ eps\ {\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharminus}{\kern0pt}QR{\isacharunderscore}{\kern0pt}disc{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{assumes}\ am{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymAnd}s{\isachardot}{\kern0pt}\ is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max\ {\isacharparenleft}{\kern0pt}{\isasymlambda}d{\isachardot}{\kern0pt}\ L{\isacharunderscore}{\kern0pt}split\ d\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}split\ v{\isacharparenright}{\kern0pt}\ s{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymlambda}d{\isachardot}{\kern0pt}\ d\ {\isasymin}\ D\isactrlsub D{\isacharparenright}{\kern0pt}\ d{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}dist\ {\isacharparenleft}{\kern0pt}{\isasymnu}\isactrlsub b\ {\isacharparenleft}{\kern0pt}mk{\isacharunderscore}{\kern0pt}stationary{\isacharunderscore}{\kern0pt}det\ d{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt\ {\isacharless}{\kern0pt}\ eps{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}p\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}mk{\isacharunderscore}{\kern0pt}stationary{\isacharunderscore}{\kern0pt}det\ d{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ L{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}{\isasymL}\isactrlsub b{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}L{\isacharunderscore}{\kern0pt}split\ d\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}split\ v{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}split\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}split\ v{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ {\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}split{\isacharunderscore}{\kern0pt}def{\isacharprime}{\kern0pt}\ arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}SUP{\isacharbrackleft}{\kern0pt}OF\ am{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}dist\ {\isacharparenleft}{\kern0pt}{\isasymnu}\isactrlsub b\ {\isacharquery}{\kern0pt}p{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}split\ v{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ dist\ {\isacharparenleft}{\kern0pt}L{\isacharunderscore}{\kern0pt}split\ d\ {\isacharparenleft}{\kern0pt}{\isasymnu}\isactrlsub b\ {\isacharquery}{\kern0pt}p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}split\ v{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ am\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}linorder\ L{\isacharunderscore}{\kern0pt}split{\isacharunderscore}{\kern0pt}fix{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isasymle}\ dist\ {\isacharparenleft}{\kern0pt}L{\isacharunderscore}{\kern0pt}split\ d\ {\isacharparenleft}{\kern0pt}{\isasymnu}\isactrlsub b\ {\isacharquery}{\kern0pt}p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}split\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}split\ v{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharplus}{\kern0pt}\ dist\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}split\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}split\ v{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}split\ v{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ intro{\isacharcolon}{\kern0pt}\ dist{\isacharunderscore}{\kern0pt}triangle{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isacharequal}{\kern0pt}\ dist\ {\isacharparenleft}{\kern0pt}L{\isacharunderscore}{\kern0pt}split\ d\ {\isacharparenleft}{\kern0pt}{\isasymnu}\isactrlsub b\ {\isacharquery}{\kern0pt}p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}L{\isacharunderscore}{\kern0pt}split\ d\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}split\ v{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharplus}{\kern0pt}\ dist\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}split\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}split\ v{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}split\ v{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ L{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}{\isasymL}\isactrlsub b{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isasymle}\ QR{\isacharunderscore}{\kern0pt}disc\ {\isacharasterisk}{\kern0pt}\ dist\ {\isacharparenleft}{\kern0pt}{\isasymnu}\isactrlsub b\ {\isacharquery}{\kern0pt}p{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}split\ v{\isacharparenright}{\kern0pt}\ {\isacharplus}{\kern0pt}\ QR{\isacharunderscore}{\kern0pt}disc\ {\isacharasterisk}{\kern0pt}\ dist\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}split\ v{\isacharparenright}{\kern0pt}\ v{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}split{\isacharunderscore}{\kern0pt}contraction\ L{\isacharunderscore}{\kern0pt}split{\isacharunderscore}{\kern0pt}contraction\ am\ \isacommand{unfolding}\isamarkupfalse%
-\ is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ add{\isacharunderscore}{\kern0pt}mono{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{finally}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ aux{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}dist\ {\isacharparenleft}{\kern0pt}{\isasymnu}\isactrlsub b\ {\isacharquery}{\kern0pt}p{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}split\ v{\isacharparenright}{\kern0pt}\ {\isasymle}\ QR{\isacharunderscore}{\kern0pt}disc\ {\isacharasterisk}{\kern0pt}\ dist\ {\isacharparenleft}{\kern0pt}{\isasymnu}\isactrlsub b\ {\isacharquery}{\kern0pt}p{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}split\ v{\isacharparenright}{\kern0pt}\ {\isacharplus}{\kern0pt}\ QR{\isacharunderscore}{\kern0pt}disc\ {\isacharasterisk}{\kern0pt}\ dist\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}split\ v{\isacharparenright}{\kern0pt}\ v{\isachardoublequoteclose}\ \isacommand{{\isachardot}{\kern0pt}}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}dist\ {\isacharparenleft}{\kern0pt}{\isasymnu}\isactrlsub b\ {\isacharquery}{\kern0pt}p{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}split\ v{\isacharparenright}{\kern0pt}\ {\isacharminus}{\kern0pt}\ QR{\isacharunderscore}{\kern0pt}disc\ {\isacharasterisk}{\kern0pt}\ dist\ {\isacharparenleft}{\kern0pt}{\isasymnu}\isactrlsub b\ {\isacharquery}{\kern0pt}p{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}split\ v{\isacharparenright}{\kern0pt}\ {\isasymle}\ QR{\isacharunderscore}{\kern0pt}disc\ {\isacharasterisk}{\kern0pt}\ dist\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}split\ v{\isacharparenright}{\kern0pt}\ v{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}dist\ {\isacharparenleft}{\kern0pt}{\isasymnu}\isactrlsub b\ {\isacharquery}{\kern0pt}p{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}split\ v{\isacharparenright}{\kern0pt}\ {\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}\ {\isacharminus}{\kern0pt}\ QR{\isacharunderscore}{\kern0pt}disc{\isacharparenright}{\kern0pt}\ {\isasymle}\ QR{\isacharunderscore}{\kern0pt}disc\ {\isacharasterisk}{\kern0pt}\ dist\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}split\ v{\isacharparenright}{\kern0pt}\ v{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ argo\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ \ {\isachardoublequoteopen}{\isadigit{2}}\ {\isacharasterisk}{\kern0pt}\ dist\ {\isacharparenleft}{\kern0pt}{\isasymnu}\isactrlsub b\ {\isacharquery}{\kern0pt}p{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}split\ v{\isacharparenright}{\kern0pt}\ {\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharminus}{\kern0pt}QR{\isacharunderscore}{\kern0pt}disc{\isacharparenright}{\kern0pt}\ {\isasymle}\ {\isadigit{2}}\ {\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}QR{\isacharunderscore}{\kern0pt}disc\ {\isacharasterisk}{\kern0pt}\ dist\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}split\ v{\isacharparenright}{\kern0pt}\ v{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ mult{\isacharunderscore}{\kern0pt}left{\isacharunderscore}{\kern0pt}mono\ \isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isadigit{2}}\ {\isacharasterisk}{\kern0pt}\ dist\ {\isacharparenleft}{\kern0pt}{\isasymnu}\isactrlsub b\ {\isacharquery}{\kern0pt}p{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}split\ v{\isacharparenright}{\kern0pt}\ {\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}\ {\isacharminus}{\kern0pt}\ QR{\isacharunderscore}{\kern0pt}disc{\isacharparenright}{\kern0pt}\ {\isasymle}\ eps\ {\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}\ {\isacharminus}{\kern0pt}\ QR{\isacharunderscore}{\kern0pt}disc{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assms\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ mult{\isacharunderscore}{\kern0pt}left{\isacharunderscore}{\kern0pt}mono\ simp{\isacharcolon}{\kern0pt}\ dist{\isacharunderscore}{\kern0pt}commute\ pos{\isacharunderscore}{\kern0pt}divide{\isacharunderscore}{\kern0pt}le{\isacharunderscore}{\kern0pt}eq{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isadigit{2}}\ {\isacharasterisk}{\kern0pt}\ dist\ {\isacharparenleft}{\kern0pt}{\isasymnu}\isactrlsub b\ {\isacharquery}{\kern0pt}p{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}split\ v{\isacharparenright}{\kern0pt}\ {\isasymle}\ eps{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ QR{\isacharunderscore}{\kern0pt}contraction\ mult{\isacharunderscore}{\kern0pt}right{\isacharunderscore}{\kern0pt}le{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}le\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isadigit{2}}\ {\isacharasterisk}{\kern0pt}\ dist\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}split\ v{\isacharparenright}{\kern0pt}\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt\ {\isacharless}{\kern0pt}\ eps{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ dist{\isacharunderscore}{\kern0pt}{\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}split{\isacharunderscore}{\kern0pt}opt{\isacharunderscore}{\kern0pt}eps\ assms\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ fastforce\isanewline
-\ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\ \isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ dist{\isacharunderscore}{\kern0pt}triangle{\isacharbrackleft}{\kern0pt}of\ {\isachardoublequoteopen}{\isasymnu}\isactrlsub b\ {\isacharquery}{\kern0pt}p{\isachardoublequoteclose}\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt\ {\isachardoublequoteopen}{\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}split\ v{\isachardoublequoteclose}{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isacommand{end}\isamarkupfalse%
-\isanewline
-\isanewline
-\isacommand{context}\isamarkupfalse%
-\ MDP{\isacharunderscore}{\kern0pt}ord\ \isakeyword{begin}\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ inv{\isacharunderscore}{\kern0pt}one{\isacharunderscore}{\kern0pt}sub{\isacharunderscore}{\kern0pt}Q{\isacharprime}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{fixes}\ Q\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}c\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ banach\ {\isasymRightarrow}\isactrlsub L\ {\isacharprime}{\kern0pt}c{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{assumes}\ onorm{\isacharunderscore}{\kern0pt}le{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}norm\ {\isacharparenleft}{\kern0pt}id{\isacharunderscore}{\kern0pt}blinfun\ {\isacharminus}{\kern0pt}\ Q{\isacharparenright}{\kern0pt}\ {\isacharless}{\kern0pt}\ {\isadigit{1}}{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}inv\isactrlsub L\ Q\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSum}i{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}id{\isacharunderscore}{\kern0pt}blinfun\ {\isacharminus}{\kern0pt}\ Q{\isacharparenright}{\kern0pt}{\isacharcircum}{\kern0pt}{\isacharcircum}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ inv\isactrlsub L{\isacharunderscore}{\kern0pt}I\ inv{\isacharunderscore}{\kern0pt}one{\isacharunderscore}{\kern0pt}sub{\isacharunderscore}{\kern0pt}Q\ assms{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\begin{isamarkuptext}%
-An important theorem: allows to compare the rate of convergence for different splittings%
-\end{isamarkuptext}\isamarkuptrue%
-\isacommand{lemma}\isamarkupfalse%
-\ norm{\isacharunderscore}{\kern0pt}splitting{\isacharunderscore}{\kern0pt}le{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}is{\isacharunderscore}{\kern0pt}splitting{\isacharunderscore}{\kern0pt}blin\ {\isacharparenleft}{\kern0pt}id{\isacharunderscore}{\kern0pt}blinfun\ {\isacharminus}{\kern0pt}\ l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isasymP}\isactrlsub {\isadigit{1}}\ d{\isacharparenright}{\kern0pt}\ Q{\isadigit{1}}\ R{\isadigit{1}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isakeyword{and}\ {\isachardoublequoteopen}is{\isacharunderscore}{\kern0pt}splitting{\isacharunderscore}{\kern0pt}blin\ {\isacharparenleft}{\kern0pt}id{\isacharunderscore}{\kern0pt}blinfun\ {\isacharminus}{\kern0pt}\ l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isasymP}\isactrlsub {\isadigit{1}}\ d{\isacharparenright}{\kern0pt}\ Q{\isadigit{2}}\ R{\isadigit{2}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isakeyword{and}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ R{\isadigit{2}}{\isacharparenright}{\kern0pt}\ {\isasymle}\ {\isacharparenleft}{\kern0pt}blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ R{\isadigit{1}}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isakeyword{and}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ R{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isasymle}\ {\isacharparenleft}{\kern0pt}blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ {\isacharparenleft}{\kern0pt}l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isasymP}\isactrlsub {\isadigit{1}}\ d{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}norm\ {\isacharparenleft}{\kern0pt}inv\isactrlsub L\ Q{\isadigit{2}}\ o\isactrlsub L\ R{\isadigit{2}}{\isacharparenright}{\kern0pt}\ {\isasymle}\ norm\ {\isacharparenleft}{\kern0pt}inv\isactrlsub L\ Q{\isadigit{1}}\ o\isactrlsub L\ R{\isadigit{1}}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}R{\isadigit{1}}\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ R{\isadigit{1}}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}R{\isadigit{2}}\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ R{\isadigit{2}}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}Q{\isadigit{1}}\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ Q{\isadigit{1}}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}Q{\isadigit{2}}\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ Q{\isadigit{2}}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ \isanewline
-\ \ \ \ inv{\isacharunderscore}{\kern0pt}Q{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}inv\isactrlsub L\ Q\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSum}i{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}id{\isacharunderscore}{\kern0pt}blinfun\ {\isacharminus}{\kern0pt}\ Q{\isacharparenright}{\kern0pt}{\isacharcircum}{\kern0pt}{\isacharcircum}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ {\isachardoublequoteopen}norm\ {\isacharparenleft}{\kern0pt}id{\isacharunderscore}{\kern0pt}blinfun\ {\isacharminus}{\kern0pt}\ Q{\isacharparenright}{\kern0pt}\ {\isacharless}{\kern0pt}\ {\isadigit{1}}{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ splitting{\isacharunderscore}{\kern0pt}eq{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}id{\isacharunderscore}{\kern0pt}blinfun\ {\isacharminus}{\kern0pt}\ Q\ {\isacharequal}{\kern0pt}\ l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isasymP}\isactrlsub {\isadigit{1}}\ d\ {\isacharminus}{\kern0pt}\ R{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ nonneg{\isacharunderscore}{\kern0pt}Q{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isadigit{0}}\ {\isasymle}\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ {\isacharparenleft}{\kern0pt}id{\isacharunderscore}{\kern0pt}blinfun\ {\isacharminus}{\kern0pt}\ Q{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isakeyword{if}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ R{\isacharparenright}{\kern0pt}\ {\isasymle}\ {\isacharparenleft}{\kern0pt}blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ {\isacharparenleft}{\kern0pt}l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isasymP}\isactrlsub {\isadigit{1}}\ d{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isakeyword{and}\ {\isachardoublequoteopen}is{\isacharunderscore}{\kern0pt}splitting{\isacharunderscore}{\kern0pt}blin\ {\isacharparenleft}{\kern0pt}id{\isacharunderscore}{\kern0pt}blinfun\ {\isacharminus}{\kern0pt}\ l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isasymP}\isactrlsub {\isadigit{1}}\ d{\isacharparenright}{\kern0pt}\ Q\ R{\isachardoublequoteclose}\ \isakeyword{for}\ Q\ R\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}R\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ R{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{show}\isamarkupfalse%
-\ splitting{\isacharunderscore}{\kern0pt}eq{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}id{\isacharunderscore}{\kern0pt}blinfun\ {\isacharminus}{\kern0pt}\ Q\ {\isacharequal}{\kern0pt}\ l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isasymP}\isactrlsub {\isadigit{1}}\ d\ {\isacharminus}{\kern0pt}\ R{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ that\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ eq{\isacharunderscore}{\kern0pt}diff{\isacharunderscore}{\kern0pt}eq\ is{\isacharunderscore}{\kern0pt}splitting{\isacharunderscore}{\kern0pt}blin{\isacharunderscore}{\kern0pt}def{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ R{\isacharunderscore}{\kern0pt}nonneg{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isadigit{0}}\ {\isasymle}\ {\isacharquery}{\kern0pt}R{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ that\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \ \ \isacommand{show}\isamarkupfalse%
-\ nonneg{\isacharunderscore}{\kern0pt}Q{\isacharcolon}{\kern0pt}\ \ {\isachardoublequoteopen}{\isadigit{0}}\ {\isasymle}\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ {\isacharparenleft}{\kern0pt}id{\isacharunderscore}{\kern0pt}blinfun\ {\isacharminus}{\kern0pt}\ Q{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ that\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ splitting{\isacharunderscore}{\kern0pt}eq\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}diff{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ {\isacharparenleft}{\kern0pt}id{\isacharunderscore}{\kern0pt}blinfun\ {\isacharminus}{\kern0pt}\ Q{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymle}\ {\isacharparenleft}{\kern0pt}blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ {\isacharparenleft}{\kern0pt}l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isasymP}\isactrlsub {\isadigit{1}}\ d{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ R{\isacharunderscore}{\kern0pt}nonneg\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ splitting{\isacharunderscore}{\kern0pt}eq\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}diff{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}norm\ {\isacharparenleft}{\kern0pt}id{\isacharunderscore}{\kern0pt}blinfun\ {\isacharminus}{\kern0pt}\ Q{\isacharparenright}{\kern0pt}\ {\isasymle}\ norm\ {\isacharparenleft}{\kern0pt}l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isasymP}\isactrlsub {\isadigit{1}}\ d{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ matrix{\isacharunderscore}{\kern0pt}le{\isacharunderscore}{\kern0pt}norm{\isacharunderscore}{\kern0pt}mono\ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}norm\ {\isacharparenleft}{\kern0pt}id{\isacharunderscore}{\kern0pt}blinfun\ {\isacharminus}{\kern0pt}\ Q{\isacharparenright}{\kern0pt}\ {\isacharless}{\kern0pt}\ {\isadigit{1}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ norm{\isacharunderscore}{\kern0pt}{\isasymP}\isactrlsub {\isadigit{1}}{\isacharunderscore}{\kern0pt}l{\isacharunderscore}{\kern0pt}less\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ order{\isachardot}{\kern0pt}strict{\isacharunderscore}{\kern0pt}trans{\isadigit{1}}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}inv\isactrlsub L\ Q\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSum}i{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}id{\isacharunderscore}{\kern0pt}blinfun\ {\isacharminus}{\kern0pt}\ Q{\isacharparenright}{\kern0pt}\ {\isacharcircum}{\kern0pt}{\isacharcircum}{\kern0pt}\ i{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ inv{\isacharunderscore}{\kern0pt}one{\isacharunderscore}{\kern0pt}sub{\isacharunderscore}{\kern0pt}Q{\isacharprime}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ i{\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}inv\isactrlsub L\ Q{\isadigit{1}}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSum}i{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}id{\isacharunderscore}{\kern0pt}blinfun\ {\isacharminus}{\kern0pt}\ Q{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isacharcircum}{\kern0pt}{\isacharcircum}{\kern0pt}\ i{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ {\isachardoublequoteopen}norm\ {\isacharparenleft}{\kern0pt}id{\isacharunderscore}{\kern0pt}blinfun\ {\isacharminus}{\kern0pt}\ Q{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isacharless}{\kern0pt}\ {\isadigit{1}}{\isachardoublequoteclose}\ \isanewline
-\ \ \ \ \isakeyword{and}\ i{\isadigit{2}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}inv\isactrlsub L\ Q{\isadigit{2}}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSum}i{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}id{\isacharunderscore}{\kern0pt}blinfun\ {\isacharminus}{\kern0pt}\ Q{\isadigit{2}}{\isacharparenright}{\kern0pt}\ {\isacharcircum}{\kern0pt}{\isacharcircum}{\kern0pt}\ i{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ {\isachardoublequoteopen}norm\ {\isacharparenleft}{\kern0pt}id{\isacharunderscore}{\kern0pt}blinfun\ {\isacharminus}{\kern0pt}\ Q{\isadigit{2}}{\isacharparenright}{\kern0pt}\ {\isacharless}{\kern0pt}\ {\isadigit{1}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assms\ \ \isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ intro{\isacharcolon}{\kern0pt}\ inv{\isacharunderscore}{\kern0pt}Q{\isacharbrackleft}{\kern0pt}of\ R{\isadigit{2}}\ Q{\isadigit{2}}{\isacharbrackright}{\kern0pt}\ inv{\isacharunderscore}{\kern0pt}Q{\isacharbrackleft}{\kern0pt}of\ R{\isadigit{1}}\ Q{\isadigit{1}}{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ Q{\isadigit{1}}{\isacharunderscore}{\kern0pt}le{\isacharunderscore}{\kern0pt}Q{\isadigit{2}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ {\isacharparenleft}{\kern0pt}id{\isacharunderscore}{\kern0pt}blinfun\ {\isacharminus}{\kern0pt}\ Q{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isasymle}\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ {\isacharparenleft}{\kern0pt}id{\isacharunderscore}{\kern0pt}blinfun\ {\isacharminus}{\kern0pt}\ Q{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assms\ \isacommand{unfolding}\isamarkupfalse%
-\ is{\isacharunderscore}{\kern0pt}splitting{\isacharunderscore}{\kern0pt}blin{\isacharunderscore}{\kern0pt}def{\isacharprime}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}diff\ eq{\isacharunderscore}{\kern0pt}diff{\isacharunderscore}{\kern0pt}eq\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}add{\isacharparenright}{\kern0pt}\isanewline
-\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ {\isacharparenleft}{\kern0pt}inv\isactrlsub L\ Q{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}{\isasymSum}i{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}id{\isacharunderscore}{\kern0pt}blinfun\ {\isacharminus}{\kern0pt}\ Q{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isacharcircum}{\kern0pt}{\isacharcircum}{\kern0pt}\ i{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ i{\isadigit{1}}\ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isacharequal}{\kern0pt}\ \ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}{\isasymSum}i{\isachardot}{\kern0pt}\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}id{\isacharunderscore}{\kern0pt}blinfun\ {\isacharminus}{\kern0pt}\ Q{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isacharcircum}{\kern0pt}{\isacharcircum}{\kern0pt}\ i{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ bounded{\isacharunderscore}{\kern0pt}linear{\isachardot}{\kern0pt}suminf{\isacharbrackleft}{\kern0pt}OF\ bounded{\isacharunderscore}{\kern0pt}linear{\isacharunderscore}{\kern0pt}blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix{\isacharbrackright}{\kern0pt}\ summable{\isacharunderscore}{\kern0pt}inv{\isacharunderscore}{\kern0pt}Q\ i{\isadigit{1}}{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}\ \isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isasymle}\ {\isacharparenleft}{\kern0pt}{\isasymSum}i{\isachardot}{\kern0pt}\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}id{\isacharunderscore}{\kern0pt}blinfun\ {\isacharminus}{\kern0pt}\ Q{\isadigit{2}}{\isacharparenright}{\kern0pt}\ {\isacharcircum}{\kern0pt}{\isacharcircum}{\kern0pt}\ i{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ le{\isacharunderscore}{\kern0pt}n{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymAnd}n{\isachardot}{\kern0pt}\ {\isadigit{0}}\ {\isasymle}\ n\ {\isasymLongrightarrow}\ {\isacharparenleft}{\kern0pt}{\isasymSum}i{\isacharless}{\kern0pt}n{\isachardot}{\kern0pt}\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}id{\isacharunderscore}{\kern0pt}blinfun\ {\isacharminus}{\kern0pt}\ Q{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isacharcircum}{\kern0pt}{\isacharcircum}{\kern0pt}\ i{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymle}\ {\isacharparenleft}{\kern0pt}{\isasymSum}i{\isacharless}{\kern0pt}n{\isachardot}{\kern0pt}\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}id{\isacharunderscore}{\kern0pt}blinfun\ {\isacharminus}{\kern0pt}\ Q{\isadigit{2}}{\isacharparenright}{\kern0pt}\ {\isacharcircum}{\kern0pt}{\isacharcircum}{\kern0pt}\ i{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assms\ nonneg{\isacharunderscore}{\kern0pt}Q\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ sum{\isacharunderscore}{\kern0pt}mono\ matpow{\isacharunderscore}{\kern0pt}mono\ simp{\isacharcolon}{\kern0pt}\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}matpow\ Q{\isadigit{1}}{\isacharunderscore}{\kern0pt}le{\isacharunderscore}{\kern0pt}Q{\isadigit{2}}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ le{\isacharunderscore}{\kern0pt}n{\isacharunderscore}{\kern0pt}elem{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymAnd}n{\isachardot}{\kern0pt}\ {\isadigit{0}}\ {\isasymle}\ n\ {\isasymLongrightarrow}\ {\isacharparenleft}{\kern0pt}{\isasymSum}i{\isacharless}{\kern0pt}n{\isachardot}{\kern0pt}\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}id{\isacharunderscore}{\kern0pt}blinfun\ {\isacharminus}{\kern0pt}\ Q{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isacharcircum}{\kern0pt}{\isacharcircum}{\kern0pt}\ i{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isachardollar}{\kern0pt}\ i\ {\isachardollar}{\kern0pt}\ j\ {\isasymle}\ {\isacharparenleft}{\kern0pt}{\isasymSum}i{\isacharless}{\kern0pt}n{\isachardot}{\kern0pt}\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}id{\isacharunderscore}{\kern0pt}blinfun\ {\isacharminus}{\kern0pt}\ Q{\isadigit{2}}{\isacharparenright}{\kern0pt}\ {\isacharcircum}{\kern0pt}{\isacharcircum}{\kern0pt}\ i{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isachardollar}{\kern0pt}\ i\ {\isachardollar}{\kern0pt}\ j\ {\isachardoublequoteclose}\ \isakeyword{for}\ i\ j\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ less{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}vec{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isasymlambda}n{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSum}i{\isacharless}{\kern0pt}n{\isachardot}{\kern0pt}\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}id{\isacharunderscore}{\kern0pt}blinfun\ {\isacharminus}{\kern0pt}\ Q{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isacharcircum}{\kern0pt}{\isacharcircum}{\kern0pt}\ i{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymlonglonglongrightarrow}\ {\isacharparenleft}{\kern0pt}{\isasymSum}i{\isachardot}{\kern0pt}\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}id{\isacharunderscore}{\kern0pt}blinfun\ {\isacharminus}{\kern0pt}\ Q{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isacharcircum}{\kern0pt}{\isacharcircum}{\kern0pt}\ i{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ bounded{\isacharunderscore}{\kern0pt}linear{\isachardot}{\kern0pt}summable{\isacharbrackleft}{\kern0pt}of\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix{\isacharbrackright}{\kern0pt}\ summable{\isacharunderscore}{\kern0pt}LIMSEQ\ simp\ add{\isacharcolon}{\kern0pt}\ bounded{\isacharunderscore}{\kern0pt}linear{\isacharunderscore}{\kern0pt}blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ i{\isadigit{1}}{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}\ summable{\isacharunderscore}{\kern0pt}inv{\isacharunderscore}{\kern0pt}Q{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ le{\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isasymlambda}n{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSum}i{\isacharless}{\kern0pt}n{\isachardot}{\kern0pt}\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}id{\isacharunderscore}{\kern0pt}blinfun\ {\isacharminus}{\kern0pt}\ Q{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isacharcircum}{\kern0pt}{\isacharcircum}{\kern0pt}\ i{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isachardollar}{\kern0pt}\ j\ {\isachardollar}{\kern0pt}\ k{\isacharparenright}{\kern0pt}\ {\isasymlonglonglongrightarrow}\ {\isacharparenleft}{\kern0pt}{\isasymSum}i{\isachardot}{\kern0pt}\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}id{\isacharunderscore}{\kern0pt}blinfun\ {\isacharminus}{\kern0pt}\ Q{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isacharcircum}{\kern0pt}{\isacharcircum}{\kern0pt}\ i{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isachardollar}{\kern0pt}\ j\ {\isachardollar}{\kern0pt}\ k{\isachardoublequoteclose}\ \isakeyword{for}\ j\ k\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ tendsto{\isacharunderscore}{\kern0pt}vec{\isacharunderscore}{\kern0pt}nth\ \isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isasymlambda}n{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSum}i{\isacharless}{\kern0pt}n{\isachardot}{\kern0pt}\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}id{\isacharunderscore}{\kern0pt}blinfun\ {\isacharminus}{\kern0pt}\ Q{\isadigit{2}}{\isacharparenright}{\kern0pt}\ {\isacharcircum}{\kern0pt}{\isacharcircum}{\kern0pt}\ i{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymlonglonglongrightarrow}\ {\isacharparenleft}{\kern0pt}{\isasymSum}i{\isachardot}{\kern0pt}\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}id{\isacharunderscore}{\kern0pt}blinfun\ {\isacharminus}{\kern0pt}\ Q{\isadigit{2}}{\isacharparenright}{\kern0pt}\ {\isacharcircum}{\kern0pt}{\isacharcircum}{\kern0pt}\ i{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ bounded{\isacharunderscore}{\kern0pt}linear{\isachardot}{\kern0pt}summable{\isacharbrackleft}{\kern0pt}of\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix{\isacharbrackright}{\kern0pt}\ summable{\isacharunderscore}{\kern0pt}LIMSEQ\ simp\ add{\isacharcolon}{\kern0pt}\ bounded{\isacharunderscore}{\kern0pt}linear{\isacharunderscore}{\kern0pt}blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ i{\isadigit{2}}{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}\ summable{\isacharunderscore}{\kern0pt}inv{\isacharunderscore}{\kern0pt}Q{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ le{\isadigit{2}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isasymlambda}n{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSum}i{\isacharless}{\kern0pt}n{\isachardot}{\kern0pt}\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}id{\isacharunderscore}{\kern0pt}blinfun\ {\isacharminus}{\kern0pt}\ Q{\isadigit{2}}{\isacharparenright}{\kern0pt}\ {\isacharcircum}{\kern0pt}{\isacharcircum}{\kern0pt}\ i{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isachardollar}{\kern0pt}\ j\ {\isachardollar}{\kern0pt}\ k{\isacharparenright}{\kern0pt}\ {\isasymlonglonglongrightarrow}\ {\isacharparenleft}{\kern0pt}{\isasymSum}i{\isachardot}{\kern0pt}\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}id{\isacharunderscore}{\kern0pt}blinfun\ {\isacharminus}{\kern0pt}\ Q{\isadigit{2}}{\isacharparenright}{\kern0pt}\ {\isacharcircum}{\kern0pt}{\isacharcircum}{\kern0pt}\ i{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isachardollar}{\kern0pt}\ j\ {\isachardollar}{\kern0pt}\ k{\isachardoublequoteclose}\ \isakeyword{for}\ j\ k\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ tendsto{\isacharunderscore}{\kern0pt}vec{\isacharunderscore}{\kern0pt}nth\ \isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}{\isasymSum}i{\isachardot}{\kern0pt}\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}id{\isacharunderscore}{\kern0pt}blinfun\ {\isacharminus}{\kern0pt}\ Q{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isacharcircum}{\kern0pt}{\isacharcircum}{\kern0pt}\ i{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardollar}{\kern0pt}\ j\ {\isachardollar}{\kern0pt}\ k{\isacharparenright}{\kern0pt}\ {\isasymle}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}{\isasymSum}i{\isachardot}{\kern0pt}\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}id{\isacharunderscore}{\kern0pt}blinfun\ {\isacharminus}{\kern0pt}\ Q{\isadigit{2}}{\isacharparenright}{\kern0pt}\ {\isacharcircum}{\kern0pt}{\isacharcircum}{\kern0pt}\ i{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isachardollar}{\kern0pt}\ j\ {\isachardollar}{\kern0pt}\ k{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{for}\ j\ k\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}fastforce\ intro{\isacharcolon}{\kern0pt}\ Topological{\isacharunderscore}{\kern0pt}Spaces{\isachardot}{\kern0pt}lim{\isacharunderscore}{\kern0pt}mono{\isacharbrackleft}{\kern0pt}OF\ le{\isacharunderscore}{\kern0pt}n{\isacharunderscore}{\kern0pt}elem\ le{\isadigit{1}}\ le{\isadigit{2}}{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ less{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}vec{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isacharequal}{\kern0pt}\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ {\isacharparenleft}{\kern0pt}inv\isactrlsub L\ Q{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ summable{\isacharunderscore}{\kern0pt}inv{\isacharunderscore}{\kern0pt}Q\ i{\isadigit{2}}{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}\ i{\isadigit{2}}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ bounded{\isacharunderscore}{\kern0pt}linear{\isachardot}{\kern0pt}suminf{\isacharbrackleft}{\kern0pt}OF\ bounded{\isacharunderscore}{\kern0pt}linear{\isacharunderscore}{\kern0pt}blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix{\isacharcomma}{\kern0pt}\ symmetric{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{finally}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ Q{\isadigit{1}}{\isacharunderscore}{\kern0pt}le{\isacharunderscore}{\kern0pt}Q{\isadigit{2}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ {\isacharparenleft}{\kern0pt}inv\isactrlsub L\ Q{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isasymle}\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ {\isacharparenleft}{\kern0pt}inv\isactrlsub L\ Q{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isacommand{{\isachardot}{\kern0pt}}\isamarkupfalse%
-\isanewline
-\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isacharasterisk}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isadigit{0}}\ {\isasymle}\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}inv\isactrlsub L\ Q{\isadigit{1}}{\isacharparenright}{\kern0pt}\ o\isactrlsub L\ R{\isadigit{1}}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ {\isachardoublequoteopen}{\isadigit{0}}\ {\isasymle}\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}inv\isactrlsub L\ Q{\isadigit{2}}{\isacharparenright}{\kern0pt}\ o\isactrlsub L\ R{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assms\ is{\isacharunderscore}{\kern0pt}splitting{\isacharunderscore}{\kern0pt}blin{\isacharunderscore}{\kern0pt}def{\isacharprime}{\kern0pt}\ \isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}comp\ intro{\isacharcolon}{\kern0pt}\ nonneg{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}mult{\isacharparenright}{\kern0pt}\isanewline
-\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isadigit{0}}\ {\isasymle}\ {\isacharparenleft}{\kern0pt}id{\isacharunderscore}{\kern0pt}blinfun\ {\isacharminus}{\kern0pt}\ l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isasymP}\isactrlsub {\isadigit{1}}\ d{\isacharparenright}{\kern0pt}\ {\isadigit{1}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ less{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}le{\isacharbrackleft}{\kern0pt}OF\ disc{\isacharunderscore}{\kern0pt}lt{\isacharunderscore}{\kern0pt}one{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ blinfun{\isachardot}{\kern0pt}diff{\isacharunderscore}{\kern0pt}left\ less{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}bfun{\isacharunderscore}{\kern0pt}def\ blinfun{\isachardot}{\kern0pt}scaleR{\isacharunderscore}{\kern0pt}left{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}inv\isactrlsub L\ Q{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}id{\isacharunderscore}{\kern0pt}blinfun\ {\isacharminus}{\kern0pt}\ l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isasymP}\isactrlsub {\isadigit{1}}\ d{\isacharparenright}{\kern0pt}\ {\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isasymle}\ {\isacharparenleft}{\kern0pt}inv\isactrlsub L\ Q{\isadigit{2}}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}id{\isacharunderscore}{\kern0pt}blinfun\ {\isacharminus}{\kern0pt}\ l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isasymP}\isactrlsub {\isadigit{1}}\ d{\isacharparenright}{\kern0pt}\ {\isadigit{1}}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ Q{\isadigit{1}}{\isacharunderscore}{\kern0pt}le{\isacharunderscore}{\kern0pt}Q{\isadigit{2}}\ blinfun{\isachardot}{\kern0pt}diff{\isacharunderscore}{\kern0pt}left\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}diff\ diff{\isacharunderscore}{\kern0pt}ge{\isacharunderscore}{\kern0pt}{\isadigit{0}}{\isacharunderscore}{\kern0pt}iff{\isacharunderscore}{\kern0pt}ge\ nonneg{\isacharunderscore}{\kern0pt}blinfun{\isacharunderscore}{\kern0pt}nonneg{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}inv\isactrlsub L\ Q{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}Q{\isadigit{1}}\ {\isacharminus}{\kern0pt}\ R{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isasymle}\ {\isacharparenleft}{\kern0pt}inv\isactrlsub L\ Q{\isadigit{2}}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}Q{\isadigit{2}}\ {\isacharminus}{\kern0pt}\ R{\isadigit{2}}{\isacharparenright}{\kern0pt}\ {\isadigit{1}}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharcomma}{\kern0pt}\ opaque{\isacharunderscore}{\kern0pt}lifting{\isacharparenright}{\kern0pt}\ assms{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}\ assms{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}\ is{\isacharunderscore}{\kern0pt}splitting{\isacharunderscore}{\kern0pt}blin{\isacharunderscore}{\kern0pt}def{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}inv\isactrlsub L\ Q{\isadigit{1}}\ o\isactrlsub L\ Q{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isadigit{1}}\ {\isacharminus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}inv\isactrlsub L\ Q{\isadigit{1}}\ o\isactrlsub L\ R{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isadigit{1}}\ {\isasymle}\ {\isacharparenleft}{\kern0pt}inv\isactrlsub L\ Q{\isadigit{2}}\ o\isactrlsub L\ Q{\isadigit{2}}{\isacharparenright}{\kern0pt}\ {\isadigit{1}}\ {\isacharminus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}inv\isactrlsub L\ Q{\isadigit{2}}\ o\isactrlsub L\ R{\isadigit{2}}{\isacharparenright}{\kern0pt}\ {\isadigit{1}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ blinfun{\isachardot}{\kern0pt}add{\isacharunderscore}{\kern0pt}left\ blinfun{\isachardot}{\kern0pt}diff{\isacharunderscore}{\kern0pt}right\ blinfun{\isachardot}{\kern0pt}diff{\isacharunderscore}{\kern0pt}left{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}inv\isactrlsub L\ Q{\isadigit{2}}\ o\isactrlsub L\ R{\isadigit{2}}{\isacharparenright}{\kern0pt}\ {\isadigit{1}}\ {\isasymle}\ {\isacharparenleft}{\kern0pt}inv\isactrlsub L\ Q{\isadigit{1}}\ o\isactrlsub L\ R{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isadigit{1}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assms\ \isanewline
-\ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ is{\isacharunderscore}{\kern0pt}splitting{\isacharunderscore}{\kern0pt}blin{\isacharunderscore}{\kern0pt}def{\isacharprime}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isadigit{0}}\ {\isasymle}\ {\isacharparenleft}{\kern0pt}inv\isactrlsub L\ Q{\isadigit{2}}\ o\isactrlsub L\ R{\isadigit{2}}{\isacharparenright}{\kern0pt}\ {\isadigit{1}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isacharasterisk}{\kern0pt}\ \isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}fastforce\ simp{\isacharcolon}{\kern0pt}\ less{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}bfunI\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ nonneg{\isacharunderscore}{\kern0pt}blinfun{\isacharunderscore}{\kern0pt}nonneg{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}norm\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}inv\isactrlsub L\ Q{\isadigit{2}}\ o\isactrlsub L\ R{\isadigit{2}}{\isacharparenright}{\kern0pt}\ {\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isasymle}\ norm\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}inv\isactrlsub L\ Q{\isadigit{1}}\ o\isactrlsub L\ R{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isadigit{1}}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ less{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}bfun{\isacharunderscore}{\kern0pt}def\ norm{\isacharunderscore}{\kern0pt}bfun{\isacharunderscore}{\kern0pt}def{\isacharprime}{\kern0pt}\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ abs{\isacharunderscore}{\kern0pt}ge{\isacharunderscore}{\kern0pt}self\ cSUP{\isacharunderscore}{\kern0pt}mono\ intro{\isacharcolon}{\kern0pt}\ order{\isachardot}{\kern0pt}trans{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}norm\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}inv\isactrlsub L\ Q{\isadigit{2}}\ o\isactrlsub L\ R{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymle}\ norm\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}inv\isactrlsub L\ Q{\isadigit{1}}\ o\isactrlsub L\ R{\isadigit{1}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ norm{\isacharunderscore}{\kern0pt}nonneg{\isacharunderscore}{\kern0pt}blinfun{\isacharunderscore}{\kern0pt}one\ {\isacharasterisk}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Gauss Seidel Splitting%
-}
-\isamarkuptrue%
-%
-\isamarkupsubsubsection{Definition of Upper and Lower Triangular Matrices%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{definition}\isamarkupfalse%
-\ {\isachardoublequoteopen}P{\isacharunderscore}{\kern0pt}dec\ d\ {\isasymequiv}\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ {\isacharparenleft}{\kern0pt}{\isasymP}\isactrlsub {\isadigit{1}}\ {\isacharparenleft}{\kern0pt}mk{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\ d{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ {\isachardoublequoteopen}P{\isacharunderscore}{\kern0pt}upper\ d\ {\isasymequiv}\ {\isacharparenleft}{\kern0pt}{\isasymchi}\ i\ j{\isachardot}{\kern0pt}\ if\ i\ {\isasymle}\ j\ then\ P{\isacharunderscore}{\kern0pt}dec\ d\ {\isachardollar}{\kern0pt}\ i\ {\isachardollar}{\kern0pt}\ j\ else\ {\isadigit{0}}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ {\isachardoublequoteopen}P{\isacharunderscore}{\kern0pt}lower\ d\ {\isasymequiv}\ {\isacharparenleft}{\kern0pt}{\isasymchi}\ i\ j{\isachardot}{\kern0pt}\ if\ j\ {\isacharless}{\kern0pt}\ i\ then\ P{\isacharunderscore}{\kern0pt}dec\ d\ {\isachardollar}{\kern0pt}\ i\ {\isachardollar}{\kern0pt}\ j\ else\ {\isadigit{0}}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymP}\isactrlsub U\ d\ {\isacharequal}{\kern0pt}\ matrix{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}blinfun\ {\isacharparenleft}{\kern0pt}P{\isacharunderscore}{\kern0pt}upper\ d{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymP}\isactrlsub L\ d\ {\isacharequal}{\kern0pt}\ matrix{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}blinfun\ {\isacharparenleft}{\kern0pt}P{\isacharunderscore}{\kern0pt}lower\ d{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ P{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}elem{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}P{\isacharunderscore}{\kern0pt}dec\ d\ {\isachardollar}{\kern0pt}\ i\ {\isachardollar}{\kern0pt}\ j\ {\isacharequal}{\kern0pt}\ pmf\ {\isacharparenleft}{\kern0pt}K\ {\isacharparenleft}{\kern0pt}i{\isacharcomma}{\kern0pt}\ d\ i{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ j{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}def\ matrix{\isacharunderscore}{\kern0pt}def\ {\isasymP}\isactrlsub {\isadigit{1}}{\isachardot}{\kern0pt}rep{\isacharunderscore}{\kern0pt}eq\ K{\isacharunderscore}{\kern0pt}st{\isacharunderscore}{\kern0pt}def\ \ P{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}def\ push{\isacharunderscore}{\kern0pt}exp{\isachardot}{\kern0pt}rep{\isacharunderscore}{\kern0pt}eq\ vec{\isacharunderscore}{\kern0pt}lambda{\isacharunderscore}{\kern0pt}beta\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}subst\ pmf{\isacharunderscore}{\kern0pt}expectation{\isacharunderscore}{\kern0pt}bind{\isacharbrackleft}{\kern0pt}of\ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}d\ i{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\ \isanewline
-\ \ \ \ {\isacharparenleft}{\kern0pt}auto\ split{\isacharcolon}{\kern0pt}\ if{\isacharunderscore}{\kern0pt}splits\ simp{\isacharcolon}{\kern0pt}\ mk{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det{\isacharunderscore}{\kern0pt}def\ axis{\isacharunderscore}{\kern0pt}def\ vec{\isacharunderscore}{\kern0pt}lambda{\isacharunderscore}{\kern0pt}inverse\ integral{\isacharunderscore}{\kern0pt}measure{\isacharunderscore}{\kern0pt}pmf{\isacharbrackleft}{\kern0pt}of\ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}j{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\ \ \isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ nonneg{\isacharunderscore}{\kern0pt}{\isasymP}\isactrlsub U{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}nonneg{\isacharunderscore}{\kern0pt}blinfun\ {\isacharparenleft}{\kern0pt}{\isasymP}\isactrlsub U\ d{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ {\isasymP}\isactrlsub U{\isacharunderscore}{\kern0pt}def\ Finite{\isacharunderscore}{\kern0pt}Cartesian{\isacharunderscore}{\kern0pt}Product{\isachardot}{\kern0pt}less{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}vec{\isacharunderscore}{\kern0pt}def\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}inv\ P{\isacharunderscore}{\kern0pt}upper{\isacharunderscore}{\kern0pt}def\ P{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}elem\ \isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ nonneg{\isacharunderscore}{\kern0pt}P{\isacharunderscore}{\kern0pt}dec{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isadigit{0}}\ {\isasymle}\ P{\isacharunderscore}{\kern0pt}dec\ d{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ Finite{\isacharunderscore}{\kern0pt}Cartesian{\isacharunderscore}{\kern0pt}Product{\isachardot}{\kern0pt}less{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}vec{\isacharunderscore}{\kern0pt}def\ P{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}elem{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ nonneg{\isacharunderscore}{\kern0pt}P{\isacharunderscore}{\kern0pt}upper{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isadigit{0}}\ {\isasymle}\ P{\isacharunderscore}{\kern0pt}upper\ d{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ nonneg{\isacharunderscore}{\kern0pt}P{\isacharunderscore}{\kern0pt}dec\ \isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ Finite{\isacharunderscore}{\kern0pt}Cartesian{\isacharunderscore}{\kern0pt}Product{\isachardot}{\kern0pt}less{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}vec{\isacharunderscore}{\kern0pt}def\ P{\isacharunderscore}{\kern0pt}upper{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ nonneg{\isacharunderscore}{\kern0pt}P{\isacharunderscore}{\kern0pt}lower{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isadigit{0}}\ {\isasymle}\ P{\isacharunderscore}{\kern0pt}lower\ d{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ nonneg{\isacharunderscore}{\kern0pt}P{\isacharunderscore}{\kern0pt}dec\ \isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ Finite{\isacharunderscore}{\kern0pt}Cartesian{\isacharunderscore}{\kern0pt}Product{\isachardot}{\kern0pt}less{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}vec{\isacharunderscore}{\kern0pt}def\ P{\isacharunderscore}{\kern0pt}lower{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ nonneg{\isacharunderscore}{\kern0pt}{\isasymP}\isactrlsub L{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}nonneg{\isacharunderscore}{\kern0pt}blinfun\ {\isacharparenleft}{\kern0pt}{\isasymP}\isactrlsub L\ d{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ {\isasymP}\isactrlsub L{\isacharunderscore}{\kern0pt}def\ Finite{\isacharunderscore}{\kern0pt}Cartesian{\isacharunderscore}{\kern0pt}Product{\isachardot}{\kern0pt}less{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}vec{\isacharunderscore}{\kern0pt}def\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}inv\ P{\isacharunderscore}{\kern0pt}lower{\isacharunderscore}{\kern0pt}def\ P{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}elem\ \isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ nonneg{\isacharunderscore}{\kern0pt}{\isasymP}\isactrlsub {\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}nonneg{\isacharunderscore}{\kern0pt}blinfun\ {\isacharparenleft}{\kern0pt}{\isasymP}\isactrlsub {\isadigit{1}}\ d{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}def\ matrix{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ Finite{\isacharunderscore}{\kern0pt}Cartesian{\isacharunderscore}{\kern0pt}Product{\isachardot}{\kern0pt}less{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}vec{\isacharunderscore}{\kern0pt}def\ axis{\isacharunderscore}{\kern0pt}def\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isasymP}\isactrlsub {\isadigit{1}}{\isacharunderscore}{\kern0pt}pos\ less{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}bfunD{\isacharbrackleft}{\kern0pt}of\ {\isadigit{0}}{\isacharcomma}{\kern0pt}\ simplified{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ norm{\isacharunderscore}{\kern0pt}{\isasymP}\isactrlsub L{\isacharunderscore}{\kern0pt}le{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}norm\ {\isacharparenleft}{\kern0pt}{\isasymP}\isactrlsub L\ d{\isacharparenright}{\kern0pt}\ {\isasymle}\ norm\ {\isacharparenleft}{\kern0pt}{\isasymP}\isactrlsub {\isadigit{1}}\ {\isacharparenleft}{\kern0pt}mk{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\ d{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ nonneg{\isacharunderscore}{\kern0pt}{\isasymP}\isactrlsub {\isadigit{1}}\ \isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}fastforce\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ matrix{\isacharunderscore}{\kern0pt}le{\isacharunderscore}{\kern0pt}norm{\isacharunderscore}{\kern0pt}mono\ simp{\isacharcolon}{\kern0pt}\ Finite{\isacharunderscore}{\kern0pt}Cartesian{\isacharunderscore}{\kern0pt}Product{\isachardot}{\kern0pt}less{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}vec{\isacharunderscore}{\kern0pt}def\ P{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}def\ P{\isacharunderscore}{\kern0pt}lower{\isacharunderscore}{\kern0pt}def\ {\isasymP}\isactrlsub L{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ norm{\isacharunderscore}{\kern0pt}{\isasymP}\isactrlsub L{\isacharunderscore}{\kern0pt}le{\isacharunderscore}{\kern0pt}one{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}norm\ {\isacharparenleft}{\kern0pt}{\isasymP}\isactrlsub L\ d{\isacharparenright}{\kern0pt}\ {\isasymle}\ {\isadigit{1}}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ norm{\isacharunderscore}{\kern0pt}{\isasymP}\isactrlsub L{\isacharunderscore}{\kern0pt}le\ norm{\isacharunderscore}{\kern0pt}{\isasymP}\isactrlsub {\isadigit{1}}\ \isacommand{by}\isamarkupfalse%
-\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ norm{\isacharunderscore}{\kern0pt}{\isasymP}\isactrlsub L{\isacharunderscore}{\kern0pt}less{\isacharunderscore}{\kern0pt}one{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}norm\ {\isacharparenleft}{\kern0pt}l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isasymP}\isactrlsub L\ d{\isacharparenright}{\kern0pt}\ {\isacharless}{\kern0pt}\ {\isadigit{1}}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ order{\isachardot}{\kern0pt}strict{\isacharunderscore}{\kern0pt}trans{\isadigit{1}}{\isacharbrackleft}{\kern0pt}OF\ mult{\isacharunderscore}{\kern0pt}left{\isacharunderscore}{\kern0pt}le\ disc{\isacharunderscore}{\kern0pt}lt{\isacharunderscore}{\kern0pt}one{\isacharbrackright}{\kern0pt}\ zero{\isacharunderscore}{\kern0pt}le{\isacharunderscore}{\kern0pt}disc\ norm{\isacharunderscore}{\kern0pt}{\isasymP}\isactrlsub L{\isacharunderscore}{\kern0pt}le{\isacharunderscore}{\kern0pt}one\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ {\isasymP}\isactrlsub L{\isacharunderscore}{\kern0pt}le{\isacharunderscore}{\kern0pt}{\isasymP}\isactrlsub {\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isadigit{0}}\ {\isasymle}\ v\ {\isasymLongrightarrow}\ {\isasymP}\isactrlsub L\ d\ v\ {\isasymle}\ {\isasymP}\isactrlsub {\isadigit{1}}\ {\isacharparenleft}{\kern0pt}mk{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\ d{\isacharparenright}{\kern0pt}\ v{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isadigit{0}}\ {\isasymle}\ v{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}P{\isacharunderscore}{\kern0pt}lower\ d\ {\isasymle}\ P{\isacharunderscore}{\kern0pt}dec\ d{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ nonneg{\isacharunderscore}{\kern0pt}P{\isacharunderscore}{\kern0pt}dec\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ P{\isacharunderscore}{\kern0pt}lower{\isacharunderscore}{\kern0pt}def\ less{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}vec{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ P{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}def\ {\isasymP}\isactrlsub L{\isacharunderscore}{\kern0pt}def\ blinfun{\isacharunderscore}{\kern0pt}apply{\isacharunderscore}{\kern0pt}mono\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}inv\ nonneg{\isacharunderscore}{\kern0pt}{\isasymP}\isactrlsub L{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ {\isasymP}\isactrlsub U{\isacharunderscore}{\kern0pt}le{\isacharunderscore}{\kern0pt}{\isasymP}\isactrlsub {\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isadigit{0}}\ {\isasymle}\ v\ {\isasymLongrightarrow}\ {\isasymP}\isactrlsub U\ d\ v\ {\isasymle}\ {\isasymP}\isactrlsub {\isadigit{1}}\ {\isacharparenleft}{\kern0pt}mk{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\ d{\isacharparenright}{\kern0pt}\ v{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isadigit{0}}\ {\isasymle}\ v{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}P{\isacharunderscore}{\kern0pt}upper\ d\ {\isasymle}\ P{\isacharunderscore}{\kern0pt}dec\ d{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ nonneg{\isacharunderscore}{\kern0pt}P{\isacharunderscore}{\kern0pt}dec\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ P{\isacharunderscore}{\kern0pt}upper{\isacharunderscore}{\kern0pt}def\ less{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}vec{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ P{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}def\ {\isasymP}\isactrlsub U{\isacharunderscore}{\kern0pt}def\ blinfun{\isacharunderscore}{\kern0pt}apply{\isacharunderscore}{\kern0pt}mono\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}inv\ nonneg{\isacharunderscore}{\kern0pt}{\isasymP}\isactrlsub U{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ row{\isacharunderscore}{\kern0pt}P{\isacharunderscore}{\kern0pt}upper{\isacharunderscore}{\kern0pt}indep{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}d\ s\ {\isacharequal}{\kern0pt}\ d{\isacharprime}{\kern0pt}\ s\ {\isasymLongrightarrow}\ row\ s\ {\isacharparenleft}{\kern0pt}P{\isacharunderscore}{\kern0pt}upper\ d{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ row\ s\ {\isacharparenleft}{\kern0pt}P{\isacharunderscore}{\kern0pt}upper\ d{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ row{\isacharunderscore}{\kern0pt}def\ P{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}elem\ P{\isacharunderscore}{\kern0pt}upper{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ row{\isacharunderscore}{\kern0pt}P{\isacharunderscore}{\kern0pt}lower{\isacharunderscore}{\kern0pt}indep{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}d\ s\ {\isacharequal}{\kern0pt}\ d{\isacharprime}{\kern0pt}\ s\ {\isasymLongrightarrow}\ row\ s\ {\isacharparenleft}{\kern0pt}P{\isacharunderscore}{\kern0pt}lower\ d{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ row\ s\ {\isacharparenleft}{\kern0pt}P{\isacharunderscore}{\kern0pt}lower\ d{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ row{\isacharunderscore}{\kern0pt}def\ P{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}elem\ P{\isacharunderscore}{\kern0pt}lower{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ triangular{\isacharunderscore}{\kern0pt}mat{\isacharunderscore}{\kern0pt}P{\isacharunderscore}{\kern0pt}upper{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}upper{\isacharunderscore}{\kern0pt}triangular{\isacharunderscore}{\kern0pt}mat\ {\isacharparenleft}{\kern0pt}P{\isacharunderscore}{\kern0pt}upper\ d{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ upper{\isacharunderscore}{\kern0pt}triangular{\isacharunderscore}{\kern0pt}mat{\isacharunderscore}{\kern0pt}def\ P{\isacharunderscore}{\kern0pt}upper{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ slt{\isacharunderscore}{\kern0pt}P{\isacharunderscore}{\kern0pt}lower{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}strict{\isacharunderscore}{\kern0pt}lower{\isacharunderscore}{\kern0pt}triangular{\isacharunderscore}{\kern0pt}mat\ {\isacharparenleft}{\kern0pt}P{\isacharunderscore}{\kern0pt}lower\ d{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ strict{\isacharunderscore}{\kern0pt}lower{\isacharunderscore}{\kern0pt}triangular{\isacharunderscore}{\kern0pt}mat{\isacharunderscore}{\kern0pt}def\ P{\isacharunderscore}{\kern0pt}lower{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ lt{\isacharunderscore}{\kern0pt}P{\isacharunderscore}{\kern0pt}lower{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}lower{\isacharunderscore}{\kern0pt}triangular{\isacharunderscore}{\kern0pt}mat\ {\isacharparenleft}{\kern0pt}P{\isacharunderscore}{\kern0pt}lower\ d{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ lower{\isacharunderscore}{\kern0pt}triangular{\isacharunderscore}{\kern0pt}mat{\isacharunderscore}{\kern0pt}def\ P{\isacharunderscore}{\kern0pt}lower{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsubsection{Gauss Seidel is a Regular Splitting%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{definition}\isamarkupfalse%
-\ {\isachardoublequoteopen}Q{\isacharunderscore}{\kern0pt}GS\ d\ {\isacharequal}{\kern0pt}\ id{\isacharunderscore}{\kern0pt}blinfun\ {\isacharminus}{\kern0pt}\ l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isasymP}\isactrlsub L\ d{\isachardoublequoteclose}\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ {\isachardoublequoteopen}R{\isacharunderscore}{\kern0pt}GS\ d\ {\isacharequal}{\kern0pt}\ l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isasymP}\isactrlsub U\ d{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ splitting{\isacharunderscore}{\kern0pt}gauss{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}is{\isacharunderscore}{\kern0pt}splitting{\isacharunderscore}{\kern0pt}blin\ {\isacharparenleft}{\kern0pt}id{\isacharunderscore}{\kern0pt}blinfun\ {\isacharminus}{\kern0pt}\ l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isasymP}\isactrlsub {\isadigit{1}}\ {\isacharparenleft}{\kern0pt}mk{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\ d{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}Q{\isacharunderscore}{\kern0pt}GS\ d{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}R{\isacharunderscore}{\kern0pt}GS\ d{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ is{\isacharunderscore}{\kern0pt}splitting{\isacharunderscore}{\kern0pt}blin{\isacharunderscore}{\kern0pt}def{\isacharprime}{\kern0pt}\isanewline
-\isacommand{proof}\isamarkupfalse%
-\ safe\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}nonneg{\isacharunderscore}{\kern0pt}blinfun\ {\isacharparenleft}{\kern0pt}R{\isacharunderscore}{\kern0pt}GS\ d{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ R{\isacharunderscore}{\kern0pt}GS{\isacharunderscore}{\kern0pt}def\ {\isasymP}\isactrlsub U{\isacharunderscore}{\kern0pt}def\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}scaleR\ Finite{\isacharunderscore}{\kern0pt}Cartesian{\isacharunderscore}{\kern0pt}Product{\isachardot}{\kern0pt}less{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}vec{\isacharunderscore}{\kern0pt}def\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}inv\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ nonneg{\isacharunderscore}{\kern0pt}P{\isacharunderscore}{\kern0pt}upper\ \isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ mult{\isacharunderscore}{\kern0pt}nonneg{\isacharunderscore}{\kern0pt}nonneg{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymP}\isactrlsub L\ d\ {\isacharplus}{\kern0pt}\ {\isasymP}\isactrlsub U\ d\ {\isacharequal}{\kern0pt}\ {\isasymP}\isactrlsub {\isadigit{1}}\ {\isacharparenleft}{\kern0pt}mk{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\ d{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{for}\ d\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymP}\isactrlsub L\ d\ {\isacharplus}{\kern0pt}\ {\isasymP}\isactrlsub U\ d\ \ {\isacharequal}{\kern0pt}\ matrix{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}blinfun\ {\isacharparenleft}{\kern0pt}{\isasymchi}\ i\ j{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ {\isacharparenleft}{\kern0pt}{\isasymP}\isactrlsub {\isadigit{1}}\ {\isacharparenleft}{\kern0pt}mk{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\ d{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isachardollar}{\kern0pt}\ i\ {\isachardollar}{\kern0pt}\ j{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ {\isasymP}\isactrlsub L{\isacharunderscore}{\kern0pt}def\ {\isasymP}\isactrlsub U{\isacharunderscore}{\kern0pt}def\ P{\isacharunderscore}{\kern0pt}lower{\isacharunderscore}{\kern0pt}def\ P{\isacharunderscore}{\kern0pt}upper{\isacharunderscore}{\kern0pt}def\ P{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}def\ matrix{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}blinfun{\isacharunderscore}{\kern0pt}add{\isacharbrackleft}{\kern0pt}symmetric{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ vec{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}iff\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ arg{\isacharunderscore}{\kern0pt}cong{\isacharbrackleft}{\kern0pt}of\ {\isacharunderscore}{\kern0pt}\ {\isacharunderscore}{\kern0pt}\ matrix{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}blinfun{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymP}\isactrlsub {\isadigit{1}}\ {\isacharparenleft}{\kern0pt}mk{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\ d{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ matrix{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}blinfun{\isacharunderscore}{\kern0pt}inv{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{finally}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymP}\isactrlsub L\ d\ {\isacharplus}{\kern0pt}\ {\isasymP}\isactrlsub U\ d\ {\isacharequal}{\kern0pt}\ {\isasymP}\isactrlsub {\isadigit{1}}\ {\isacharparenleft}{\kern0pt}mk{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\ d{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isacommand{{\isachardot}{\kern0pt}}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}id{\isacharunderscore}{\kern0pt}blinfun\ {\isacharminus}{\kern0pt}\ l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isasymP}\isactrlsub {\isadigit{1}}\ {\isacharparenleft}{\kern0pt}mk{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\ d{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ Q{\isacharunderscore}{\kern0pt}GS\ d\ {\isacharminus}{\kern0pt}\ R{\isacharunderscore}{\kern0pt}GS\ d{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ Q{\isacharunderscore}{\kern0pt}GS{\isacharunderscore}{\kern0pt}def\ R{\isacharunderscore}{\kern0pt}GS{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ algebra{\isacharunderscore}{\kern0pt}simps\ scaleR{\isacharunderscore}{\kern0pt}add{\isacharunderscore}{\kern0pt}right{\isacharbrackleft}{\kern0pt}symmetric{\isacharbrackright}{\kern0pt}\ simp\ del{\isacharcolon}{\kern0pt}\ scaleR{\isacharunderscore}{\kern0pt}add{\isacharunderscore}{\kern0pt}right{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ n{\isacharunderscore}{\kern0pt}le{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}norm\ {\isacharparenleft}{\kern0pt}l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isasymP}\isactrlsub L\ d{\isacharparenright}{\kern0pt}\ {\isacharless}{\kern0pt}\ {\isadigit{1}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ mult{\isacharunderscore}{\kern0pt}left{\isacharunderscore}{\kern0pt}le{\isacharbrackleft}{\kern0pt}OF\ norm{\isacharunderscore}{\kern0pt}{\isasymP}\isactrlsub L{\isacharunderscore}{\kern0pt}le{\isacharunderscore}{\kern0pt}one{\isacharbrackleft}{\kern0pt}of\ d{\isacharbrackright}{\kern0pt}\ zero{\isacharunderscore}{\kern0pt}le{\isacharunderscore}{\kern0pt}disc{\isacharbrackright}{\kern0pt}\ order{\isachardot}{\kern0pt}strict{\isacharunderscore}{\kern0pt}trans{\isadigit{1}}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ intro{\isacharcolon}{\kern0pt}\ disc{\isacharunderscore}{\kern0pt}lt{\isacharunderscore}{\kern0pt}one{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}invertible\isactrlsub L\ {\isacharparenleft}{\kern0pt}Q{\isacharunderscore}{\kern0pt}GS\ d{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ Q{\isacharunderscore}{\kern0pt}GS{\isacharunderscore}{\kern0pt}def\ invertible\isactrlsub L{\isacharunderscore}{\kern0pt}inf{\isacharunderscore}{\kern0pt}sum{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}inv\isactrlsub L\ {\isacharparenleft}{\kern0pt}Q{\isacharunderscore}{\kern0pt}GS\ d{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSum}i{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isasymP}\isactrlsub L\ d{\isacharparenright}{\kern0pt}\ {\isacharcircum}{\kern0pt}{\isacharcircum}{\kern0pt}\ i{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ inv\isactrlsub L{\isacharunderscore}{\kern0pt}inf{\isacharunderscore}{\kern0pt}sum\ n{\isacharunderscore}{\kern0pt}le\ \isacommand{unfolding}\isamarkupfalse%
-\ Q{\isacharunderscore}{\kern0pt}GS{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isacharasterisk}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ {\isacharparenleft}{\kern0pt}inv\isactrlsub L\ {\isacharparenleft}{\kern0pt}Q{\isacharunderscore}{\kern0pt}GS\ d{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isachardollar}{\kern0pt}\ i\ {\isachardollar}{\kern0pt}\ j\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSum}k{\isachardot}{\kern0pt}\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isasymP}\isactrlsub L\ d{\isacharparenright}{\kern0pt}\ {\isacharcircum}{\kern0pt}{\isacharcircum}{\kern0pt}\ k{\isacharparenright}{\kern0pt}\ {\isachardollar}{\kern0pt}\ i\ {\isachardollar}{\kern0pt}\ j{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{for}\ i\ j\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ summable{\isacharunderscore}{\kern0pt}inv{\isacharunderscore}{\kern0pt}Q{\isacharbrackleft}{\kern0pt}of\ {\isachardoublequoteopen}Q{\isacharunderscore}{\kern0pt}GS\ d{\isachardoublequoteclose}{\isacharbrackright}{\kern0pt}\ norm{\isacharunderscore}{\kern0pt}{\isasymP}\isactrlsub L{\isacharunderscore}{\kern0pt}less{\isacharunderscore}{\kern0pt}one\ \isanewline
-\ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ Q{\isacharunderscore}{\kern0pt}GS{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}subst\ bounded{\isacharunderscore}{\kern0pt}linear{\isachardot}{\kern0pt}suminf{\isacharbrackleft}{\kern0pt}symmetric{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\ \isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}auto\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ bounded{\isacharunderscore}{\kern0pt}linear{\isacharunderscore}{\kern0pt}compose{\isacharbrackleft}{\kern0pt}OF\ bounded{\isacharunderscore}{\kern0pt}linear{\isacharunderscore}{\kern0pt}vec{\isacharunderscore}{\kern0pt}nth{\isacharbrackright}{\kern0pt}\ bounded{\isacharunderscore}{\kern0pt}linear{\isacharunderscore}{\kern0pt}compose{\isacharbrackleft}{\kern0pt}OF\ bounded{\isacharunderscore}{\kern0pt}linear{\isacharunderscore}{\kern0pt}blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isadigit{0}}\ {\isasymle}\ l{\isacharcircum}{\kern0pt}i\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ matpow\ {\isacharparenleft}{\kern0pt}P{\isacharunderscore}{\kern0pt}lower\ d{\isacharparenright}{\kern0pt}\ i{\isachardoublequoteclose}\ \isakeyword{for}\ i\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ nonneg{\isacharunderscore}{\kern0pt}matpow{\isacharbrackleft}{\kern0pt}OF\ nonneg{\isacharunderscore}{\kern0pt}P{\isacharunderscore}{\kern0pt}lower{\isacharbrackright}{\kern0pt}\ \isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}meson\ scaleR{\isacharunderscore}{\kern0pt}nonneg{\isacharunderscore}{\kern0pt}nonneg\ zero{\isacharunderscore}{\kern0pt}le{\isacharunderscore}{\kern0pt}disc\ zero{\isacharunderscore}{\kern0pt}le{\isacharunderscore}{\kern0pt}power{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isadigit{0}}\ {\isasymle}\ {\isacharparenleft}{\kern0pt}{\isasymSum}k{\isachardot}{\kern0pt}\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isasymP}\isactrlsub L\ d{\isacharparenright}{\kern0pt}\ {\isacharcircum}{\kern0pt}{\isacharcircum}{\kern0pt}\ k{\isacharparenright}{\kern0pt}\ {\isachardollar}{\kern0pt}\ i\ {\isachardollar}{\kern0pt}\ j{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{for}\ i\ j\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}intro\ suminf{\isacharunderscore}{\kern0pt}nonneg{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}summable\ {\isacharparenleft}{\kern0pt}{\isasymlambda}k{\isachardot}{\kern0pt}\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isasymP}\isactrlsub L\ d{\isacharparenright}{\kern0pt}\ {\isacharcircum}{\kern0pt}{\isacharcircum}{\kern0pt}\ k{\isacharparenright}{\kern0pt}\ {\isachardollar}{\kern0pt}\ i\ {\isachardollar}{\kern0pt}\ j{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ summable{\isacharunderscore}{\kern0pt}inv{\isacharunderscore}{\kern0pt}Q{\isacharbrackleft}{\kern0pt}of\ {\isachardoublequoteopen}Q{\isacharunderscore}{\kern0pt}GS\ d{\isachardoublequoteclose}{\isacharbrackright}{\kern0pt}\ norm{\isacharunderscore}{\kern0pt}{\isasymP}\isactrlsub L{\isacharunderscore}{\kern0pt}less{\isacharunderscore}{\kern0pt}one\ \isanewline
-\ \ \ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ Q{\isacharunderscore}{\kern0pt}GS{\isacharunderscore}{\kern0pt}def\ \isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}fastforce\isanewline
-\ \ \ \ \ \ \ \ \ \ simp{\isacharcolon}{\kern0pt}\ \isanewline
-\ \ \ \ \ \ \ \ \ \ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}matpow\ nonneg{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}nonneg\ blincomp{\isacharunderscore}{\kern0pt}scaleR{\isacharunderscore}{\kern0pt}right\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}scaleR\isanewline
-\ \ \ \ \ \ \ \ \ \ intro{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ bounded{\isacharunderscore}{\kern0pt}linear{\isachardot}{\kern0pt}summable{\isacharbrackleft}{\kern0pt}of\ {\isacharunderscore}{\kern0pt}\ {\isachardoublequoteopen}{\isasymlambda}i{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isasymP}\isactrlsub L\ d{\isacharparenright}{\kern0pt}\ {\isacharcircum}{\kern0pt}{\isacharcircum}{\kern0pt}\ i{\isachardoublequoteclose}{\isacharbrackright}{\kern0pt}\ \isanewline
-\ \ \ \ \ \ \ \ \ \ bounded{\isacharunderscore}{\kern0pt}linear{\isacharunderscore}{\kern0pt}compose{\isacharbrackleft}{\kern0pt}OF\ bounded{\isacharunderscore}{\kern0pt}linear{\isacharunderscore}{\kern0pt}vec{\isacharunderscore}{\kern0pt}nth{\isacharbrackright}{\kern0pt}\ \isanewline
-\ \ \ \ \ \ \ \ \ \ bounded{\isacharunderscore}{\kern0pt}linear{\isacharunderscore}{\kern0pt}compose{\isacharbrackleft}{\kern0pt}OF\ bounded{\isacharunderscore}{\kern0pt}linear{\isacharunderscore}{\kern0pt}blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymAnd}n{\isachardot}{\kern0pt}\ {\isadigit{0}}\ {\isasymle}\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isasymP}\isactrlsub L\ d{\isacharparenright}{\kern0pt}\ {\isacharcircum}{\kern0pt}{\isacharcircum}{\kern0pt}\ n{\isacharparenright}{\kern0pt}\ {\isachardollar}{\kern0pt}\ i\ {\isachardollar}{\kern0pt}\ j{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ nonneg{\isacharunderscore}{\kern0pt}matpow{\isacharbrackleft}{\kern0pt}OF\ nonneg{\isacharunderscore}{\kern0pt}P{\isacharunderscore}{\kern0pt}lower{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ {\isasymP}\isactrlsub L{\isacharunderscore}{\kern0pt}def\ nonneg{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}nonneg\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}scaleR\ matpow{\isacharunderscore}{\kern0pt}scaleR\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}matpow{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}nonneg{\isacharunderscore}{\kern0pt}blinfun\ {\isacharparenleft}{\kern0pt}inv\isactrlsub L\ {\isacharparenleft}{\kern0pt}Q{\isacharunderscore}{\kern0pt}GS\ d{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ {\isacharasterisk}{\kern0pt}\ Finite{\isacharunderscore}{\kern0pt}Cartesian{\isacharunderscore}{\kern0pt}Product{\isachardot}{\kern0pt}less{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}vec{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{abbreviation}\isamarkupfalse%
-\ {\isachardoublequoteopen}r{\isacharunderscore}{\kern0pt}det\isactrlsub b\ d\ {\isasymequiv}\ r{\isacharunderscore}{\kern0pt}dec\isactrlsub b\ {\isacharparenleft}{\kern0pt}mk{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\ d{\isacharparenright}{\kern0pt}\ {\isachardoublequoteclose}\isanewline
-\isacommand{abbreviation}\isamarkupfalse%
-\ {\isachardoublequoteopen}r{\isacharunderscore}{\kern0pt}vec\ d\ {\isasymequiv}\ {\isasymchi}\ i{\isachardot}{\kern0pt}\ r{\isacharunderscore}{\kern0pt}dec\isactrlsub b\ {\isacharparenleft}{\kern0pt}mk{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\ d{\isacharparenright}{\kern0pt}\ i{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{abbreviation}\isamarkupfalse%
-\ {\isachardoublequoteopen}Q{\isacharunderscore}{\kern0pt}mat\ d\ {\isasymequiv}\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ {\isacharparenleft}{\kern0pt}Q{\isacharunderscore}{\kern0pt}GS\ d{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isacommand{abbreviation}\isamarkupfalse%
-\ {\isachardoublequoteopen}R{\isacharunderscore}{\kern0pt}mat\ d\ {\isasymequiv}\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ {\isacharparenleft}{\kern0pt}R{\isacharunderscore}{\kern0pt}GS\ d{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ Q{\isacharunderscore}{\kern0pt}mat{\isacharunderscore}{\kern0pt}def{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}Q{\isacharunderscore}{\kern0pt}mat\ d\ {\isacharequal}{\kern0pt}\ mat\ {\isadigit{1}}\ {\isacharminus}{\kern0pt}\ l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ P{\isacharunderscore}{\kern0pt}lower\ d{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ Q{\isacharunderscore}{\kern0pt}GS{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ {\isasymP}\isactrlsub L{\isacharunderscore}{\kern0pt}def\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}diff\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}id\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}scaleR{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ R{\isacharunderscore}{\kern0pt}mat{\isacharunderscore}{\kern0pt}def{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}R{\isacharunderscore}{\kern0pt}mat\ d\ {\isacharequal}{\kern0pt}\ l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ P{\isacharunderscore}{\kern0pt}upper\ d{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ R{\isacharunderscore}{\kern0pt}GS{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ {\isasymP}\isactrlsub U{\isacharunderscore}{\kern0pt}def\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}scaleR{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ triangular{\isacharunderscore}{\kern0pt}mat{\isacharunderscore}{\kern0pt}R{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}upper{\isacharunderscore}{\kern0pt}triangular{\isacharunderscore}{\kern0pt}mat\ {\isacharparenleft}{\kern0pt}R{\isacharunderscore}{\kern0pt}mat\ d{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ triangular{\isacharunderscore}{\kern0pt}mat{\isacharunderscore}{\kern0pt}P{\isacharunderscore}{\kern0pt}upper\isanewline
-\ \ \isacommand{unfolding}\isamarkupfalse%
-\ upper{\isacharunderscore}{\kern0pt}triangular{\isacharunderscore}{\kern0pt}mat{\isacharunderscore}{\kern0pt}def\ R{\isacharunderscore}{\kern0pt}mat{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ {\isachardoublequoteopen}GS{\isacharunderscore}{\kern0pt}inv\ d\ v\ {\isasymequiv}\ matrix{\isacharunderscore}{\kern0pt}inv\ {\isacharparenleft}{\kern0pt}Q{\isacharunderscore}{\kern0pt}mat\ d{\isacharparenright}{\kern0pt}\ {\isacharasterisk}{\kern0pt}v\ {\isacharparenleft}{\kern0pt}r{\isacharunderscore}{\kern0pt}vec\ d\ {\isacharplus}{\kern0pt}\ R{\isacharunderscore}{\kern0pt}mat\ d\ {\isacharasterisk}{\kern0pt}v\ v{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}%
-\begin{isamarkuptext}%
-\isa{Q{\isacharunderscore}{\kern0pt}mat} can be expressed as an infinite sum of \isa{P{\isacharunderscore}{\kern0pt}lower}. 
-  It is therefore lower triangular.%
-\end{isamarkuptext}\isamarkuptrue%
-\isacommand{lemma}\isamarkupfalse%
-\ inv{\isacharunderscore}{\kern0pt}Q{\isacharunderscore}{\kern0pt}mat{\isacharunderscore}{\kern0pt}suminf{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}matrix{\isacharunderscore}{\kern0pt}inv\ {\isacharparenleft}{\kern0pt}Q{\isacharunderscore}{\kern0pt}mat\ d{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSum}k{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}matpow\ {\isacharparenleft}{\kern0pt}l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isacharparenleft}{\kern0pt}P{\isacharunderscore}{\kern0pt}lower\ d{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ k{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}matrix{\isacharunderscore}{\kern0pt}inv\ {\isacharparenleft}{\kern0pt}Q{\isacharunderscore}{\kern0pt}mat\ d{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ {\isacharparenleft}{\kern0pt}inv\isactrlsub L\ {\isacharparenleft}{\kern0pt}Q{\isacharunderscore}{\kern0pt}GS\ d{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}inverse{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}\ is{\isacharunderscore}{\kern0pt}splitting{\isacharunderscore}{\kern0pt}blin{\isacharunderscore}{\kern0pt}def{\isacharprime}{\kern0pt}\ splitting{\isacharunderscore}{\kern0pt}gauss\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isacharequal}{\kern0pt}\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ {\isacharparenleft}{\kern0pt}{\isasymSum}i{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isasymP}\isactrlsub L\ d{\isacharparenright}{\kern0pt}{\isacharcircum}{\kern0pt}{\isacharcircum}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ norm{\isacharunderscore}{\kern0pt}{\isasymP}\isactrlsub L{\isacharunderscore}{\kern0pt}less{\isacharunderscore}{\kern0pt}one\ \isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ Q{\isacharunderscore}{\kern0pt}GS{\isacharunderscore}{\kern0pt}def\ inv\isactrlsub L{\isacharunderscore}{\kern0pt}inf{\isacharunderscore}{\kern0pt}sum{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSum}k{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isasymP}\isactrlsub L\ d{\isacharparenright}{\kern0pt}{\isacharcircum}{\kern0pt}{\isacharcircum}{\kern0pt}k{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ summable{\isacharunderscore}{\kern0pt}inv{\isacharunderscore}{\kern0pt}Q{\isacharbrackleft}{\kern0pt}of\ {\isachardoublequoteopen}Q{\isacharunderscore}{\kern0pt}GS\ d{\isachardoublequoteclose}{\isacharbrackright}{\kern0pt}\ norm{\isacharunderscore}{\kern0pt}{\isasymP}\isactrlsub L{\isacharunderscore}{\kern0pt}less{\isacharunderscore}{\kern0pt}one\ bounded{\isacharunderscore}{\kern0pt}linear{\isacharunderscore}{\kern0pt}blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\isanewline
-\ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ Q{\isacharunderscore}{\kern0pt}GS{\isacharunderscore}{\kern0pt}def\ row{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}subst\ bounded{\isacharunderscore}{\kern0pt}linear{\isachardot}{\kern0pt}suminf{\isacharparenright}{\kern0pt}\ auto\isanewline
-\ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSum}k{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}matpow\ {\isacharparenleft}{\kern0pt}l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isacharparenleft}{\kern0pt}P{\isacharunderscore}{\kern0pt}lower\ d{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ k{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}scaleR\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}matpow\ {\isasymP}\isactrlsub L{\isacharunderscore}{\kern0pt}def\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}inv{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{finally}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isacommand{{\isachardot}{\kern0pt}}\isamarkupfalse%
-\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ lt{\isacharunderscore}{\kern0pt}Q{\isacharunderscore}{\kern0pt}inv{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}lower{\isacharunderscore}{\kern0pt}triangular{\isacharunderscore}{\kern0pt}mat\ {\isacharparenleft}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}inv\ {\isacharparenleft}{\kern0pt}Q{\isacharunderscore}{\kern0pt}mat\ d{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ inv{\isacharunderscore}{\kern0pt}Q{\isacharunderscore}{\kern0pt}mat{\isacharunderscore}{\kern0pt}suminf\isanewline
-\ \ \isacommand{using}\isamarkupfalse%
-\ summable{\isacharunderscore}{\kern0pt}inv{\isacharunderscore}{\kern0pt}Q{\isacharbrackleft}{\kern0pt}of\ {\isachardoublequoteopen}Q{\isacharunderscore}{\kern0pt}GS\ d{\isachardoublequoteclose}{\isacharbrackright}{\kern0pt}\ norm{\isacharunderscore}{\kern0pt}{\isasymP}\isactrlsub L{\isacharunderscore}{\kern0pt}less{\isacharunderscore}{\kern0pt}one\ summable{\isacharunderscore}{\kern0pt}blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix{\isacharbrackleft}{\kern0pt}of\ {\isachardoublequoteopen}{\isasymlambda}i{\isachardot}{\kern0pt}\ \ {\isacharparenleft}{\kern0pt}l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isasymP}\isactrlsub L\ d{\isacharparenright}{\kern0pt}{\isacharcircum}{\kern0pt}{\isacharcircum}{\kern0pt}i{\isachardoublequoteclose}{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}intro\ lower{\isacharunderscore}{\kern0pt}triangular{\isacharunderscore}{\kern0pt}suminf\ lower{\isacharunderscore}{\kern0pt}triangular{\isacharunderscore}{\kern0pt}pow{\isacharparenright}{\kern0pt}\ \isanewline
-\ \ \ \ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ lower{\isacharunderscore}{\kern0pt}triangular{\isacharunderscore}{\kern0pt}mat{\isacharunderscore}{\kern0pt}def\ P{\isacharunderscore}{\kern0pt}lower{\isacharunderscore}{\kern0pt}def\ Q{\isacharunderscore}{\kern0pt}GS{\isacharunderscore}{\kern0pt}def\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}scaleR\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}matpow\ {\isasymP}\isactrlsub L{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\begin{isamarkuptext}%
-Each row of the matrix \isa{Q{\isacharunderscore}{\kern0pt}mat\ d} only depends on \isa{d}'s actions in lower states.%
-\end{isamarkuptext}\isamarkuptrue%
-\isacommand{lemma}\isamarkupfalse%
-\ inv{\isacharunderscore}{\kern0pt}Q{\isacharunderscore}{\kern0pt}mat{\isacharunderscore}{\kern0pt}indep{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}{\isasymAnd}i{\isachardot}{\kern0pt}\ i\ {\isasymle}\ s\ {\isasymLongrightarrow}\ d\ i\ {\isacharequal}{\kern0pt}\ d{\isacharprime}{\kern0pt}\ i{\isachardoublequoteclose}\ {\isachardoublequoteopen}i\ {\isasymle}\ s{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ \ {\isachardoublequoteopen}row\ i\ {\isacharparenleft}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}inv\ {\isacharparenleft}{\kern0pt}Q{\isacharunderscore}{\kern0pt}mat\ d{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ row\ i\ {\isacharparenleft}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}inv\ {\isacharparenleft}{\kern0pt}Q{\isacharunderscore}{\kern0pt}mat\ d{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}row\ i\ {\isacharparenleft}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}inv\ {\isacharparenleft}{\kern0pt}Q{\isacharunderscore}{\kern0pt}mat\ d{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ row\ i\ {\isacharparenleft}{\kern0pt}blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ {\isacharparenleft}{\kern0pt}inv\isactrlsub L\ {\isacharparenleft}{\kern0pt}Q{\isacharunderscore}{\kern0pt}GS\ d{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}inverse{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}\ is{\isacharunderscore}{\kern0pt}splitting{\isacharunderscore}{\kern0pt}blin{\isacharunderscore}{\kern0pt}def{\isacharprime}{\kern0pt}\ splitting{\isacharunderscore}{\kern0pt}gauss\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isacharequal}{\kern0pt}\ row\ i\ {\isacharparenleft}{\kern0pt}blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ {\isacharparenleft}{\kern0pt}{\isasymSum}i{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isasymP}\isactrlsub L\ d{\isacharparenright}{\kern0pt}{\isacharcircum}{\kern0pt}{\isacharcircum}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ norm{\isacharunderscore}{\kern0pt}{\isasymP}\isactrlsub L{\isacharunderscore}{\kern0pt}less{\isacharunderscore}{\kern0pt}one\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ Q{\isacharunderscore}{\kern0pt}GS{\isacharunderscore}{\kern0pt}def\ inv\isactrlsub L{\isacharunderscore}{\kern0pt}inf{\isacharunderscore}{\kern0pt}sum{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSum}k{\isachardot}{\kern0pt}\ row\ i\ {\isacharparenleft}{\kern0pt}blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isasymP}\isactrlsub L\ d{\isacharparenright}{\kern0pt}{\isacharcircum}{\kern0pt}{\isacharcircum}{\kern0pt}k{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ summable{\isacharunderscore}{\kern0pt}inv{\isacharunderscore}{\kern0pt}Q{\isacharbrackleft}{\kern0pt}of\ {\isachardoublequoteopen}Q{\isacharunderscore}{\kern0pt}GS\ d{\isachardoublequoteclose}{\isacharbrackright}{\kern0pt}\ norm{\isacharunderscore}{\kern0pt}{\isasymP}\isactrlsub L{\isacharunderscore}{\kern0pt}less{\isacharunderscore}{\kern0pt}one\isanewline
-\ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ Q{\isacharunderscore}{\kern0pt}GS{\isacharunderscore}{\kern0pt}def\ row{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}subst\ bounded{\isacharunderscore}{\kern0pt}linear{\isachardot}{\kern0pt}suminf{\isacharbrackleft}{\kern0pt}OF\ bounded{\isacharunderscore}{\kern0pt}linear{\isacharunderscore}{\kern0pt}compose{\isacharbrackleft}{\kern0pt}OF\ {\isacharunderscore}{\kern0pt}\ bounded{\isacharunderscore}{\kern0pt}linear{\isacharunderscore}{\kern0pt}blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix{\isacharbrackright}{\kern0pt}{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\ auto\isanewline
-\ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSum}k{\isachardot}{\kern0pt}\ row\ i\ {\isacharparenleft}{\kern0pt}matpow\ {\isacharparenleft}{\kern0pt}l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isacharparenleft}{\kern0pt}P{\isacharunderscore}{\kern0pt}lower\ d{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ k{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}matpow\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}scaleR\ {\isasymP}\isactrlsub L{\isacharunderscore}{\kern0pt}def\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}inv{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSum}k{\isachardot}{\kern0pt}\ l{\isacharcircum}{\kern0pt}k\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ row\ i\ {\isacharparenleft}{\kern0pt}matpow\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}P{\isacharunderscore}{\kern0pt}lower\ d{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ k{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}subst\ matpow{\isacharunderscore}{\kern0pt}scaleR{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ row{\isacharunderscore}{\kern0pt}def\ scaleR{\isacharunderscore}{\kern0pt}vec{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\ \isanewline
-\ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSum}k{\isachardot}{\kern0pt}\ l{\isacharcircum}{\kern0pt}k\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ row\ i\ {\isacharparenleft}{\kern0pt}matpow\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}P{\isacharunderscore}{\kern0pt}lower\ d{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ k{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assms\ \isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}subst\ lower{\isacharunderscore}{\kern0pt}triangular{\isacharunderscore}{\kern0pt}pow{\isacharunderscore}{\kern0pt}eq{\isacharbrackleft}{\kern0pt}of\ {\isachardoublequoteopen}P{\isacharunderscore}{\kern0pt}lower\ d{\isacharprime}{\kern0pt}{\isachardoublequoteclose}{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ P{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}elem\ lt{\isacharunderscore}{\kern0pt}P{\isacharunderscore}{\kern0pt}lower\ row{\isacharunderscore}{\kern0pt}P{\isacharunderscore}{\kern0pt}lower{\isacharunderscore}{\kern0pt}indep{\isacharbrackleft}{\kern0pt}of\ d{\isacharprime}{\kern0pt}\ {\isacharunderscore}{\kern0pt}\ d{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSum}k{\isachardot}{\kern0pt}\ row\ i\ {\isacharparenleft}{\kern0pt}matpow\ {\isacharparenleft}{\kern0pt}l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isacharparenleft}{\kern0pt}P{\isacharunderscore}{\kern0pt}lower\ d{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ k{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}subst\ matpow{\isacharunderscore}{\kern0pt}scaleR{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ row{\isacharunderscore}{\kern0pt}def\ scaleR{\isacharunderscore}{\kern0pt}vec{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSum}k{\isachardot}{\kern0pt}\ row\ i\ {\isacharparenleft}{\kern0pt}blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isasymP}\isactrlsub L\ d{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharcircum}{\kern0pt}{\isacharcircum}{\kern0pt}k{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ {\isasymP}\isactrlsub L{\isacharunderscore}{\kern0pt}def\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}inv\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}matpow\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}scaleR{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isacharequal}{\kern0pt}\ row\ i\ {\isacharparenleft}{\kern0pt}blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ {\isacharparenleft}{\kern0pt}{\isasymSum}i{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isasymP}\isactrlsub L\ d{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharcircum}{\kern0pt}{\isacharcircum}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ summable{\isacharunderscore}{\kern0pt}inv{\isacharunderscore}{\kern0pt}Q{\isacharbrackleft}{\kern0pt}of\ {\isachardoublequoteopen}Q{\isacharunderscore}{\kern0pt}GS\ d{\isacharprime}{\kern0pt}{\isachardoublequoteclose}{\isacharbrackright}{\kern0pt}\ norm{\isacharunderscore}{\kern0pt}{\isasymP}\isactrlsub L{\isacharunderscore}{\kern0pt}less{\isacharunderscore}{\kern0pt}one\ \isanewline
-\ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ Q{\isacharunderscore}{\kern0pt}GS{\isacharunderscore}{\kern0pt}def\ row{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ bounded{\isacharunderscore}{\kern0pt}linear{\isachardot}{\kern0pt}suminf{\isacharbrackleft}{\kern0pt}symmetric{\isacharbrackright}{\kern0pt}\ \isanewline
-\ \ \ \ \ \ \ \ bounded{\isacharunderscore}{\kern0pt}linear{\isacharunderscore}{\kern0pt}compose{\isacharbrackleft}{\kern0pt}OF\ {\isacharunderscore}{\kern0pt}\ bounded{\isacharunderscore}{\kern0pt}linear{\isacharunderscore}{\kern0pt}blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isacharequal}{\kern0pt}\ row\ i\ {\isacharparenleft}{\kern0pt}blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ {\isacharparenleft}{\kern0pt}inv\isactrlsub L\ {\isacharparenleft}{\kern0pt}Q{\isacharunderscore}{\kern0pt}GS\ d{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ Q{\isacharunderscore}{\kern0pt}GS{\isacharunderscore}{\kern0pt}def\ inv\isactrlsub L{\isacharunderscore}{\kern0pt}inf{\isacharunderscore}{\kern0pt}sum\ norm{\isacharunderscore}{\kern0pt}{\isasymP}\isactrlsub L{\isacharunderscore}{\kern0pt}less{\isacharunderscore}{\kern0pt}one{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isacharequal}{\kern0pt}\ row\ i\ {\isacharparenleft}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}inv\ {\isacharparenleft}{\kern0pt}Q{\isacharunderscore}{\kern0pt}mat\ d{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}inverse{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}\ is{\isacharunderscore}{\kern0pt}splitting{\isacharunderscore}{\kern0pt}blin{\isacharunderscore}{\kern0pt}def{\isacharprime}{\kern0pt}\ splitting{\isacharunderscore}{\kern0pt}gauss{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{finally}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isacommand{{\isachardot}{\kern0pt}}\isamarkupfalse%
-\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\begin{isamarkuptext}%
-As a result, also \isa{GS{\isacharunderscore}{\kern0pt}inv} is independent of lower actions.%
-\end{isamarkuptext}\isamarkuptrue%
-\isacommand{lemma}\isamarkupfalse%
-\ GS{\isacharunderscore}{\kern0pt}indep{\isacharunderscore}{\kern0pt}high{\isacharunderscore}{\kern0pt}states{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}{\isasymAnd}s{\isacharprime}{\kern0pt}{\isachardot}{\kern0pt}\ s{\isacharprime}{\kern0pt}\ {\isasymle}\ s\ {\isasymLongrightarrow}\ d\ s{\isacharprime}{\kern0pt}\ {\isacharequal}{\kern0pt}\ d{\isacharprime}{\kern0pt}\ s{\isacharprime}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}GS{\isacharunderscore}{\kern0pt}inv\ d\ v\ {\isachardollar}{\kern0pt}\ s\ {\isacharequal}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}inv\ d{\isacharprime}{\kern0pt}\ v\ {\isachardollar}{\kern0pt}\ s{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}row\ i\ {\isacharparenleft}{\kern0pt}P{\isacharunderscore}{\kern0pt}upper\ d{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ row\ i\ {\isacharparenleft}{\kern0pt}P{\isacharunderscore}{\kern0pt}upper\ d{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{if}\ {\isachardoublequoteopen}i\ {\isasymle}\ s{\isachardoublequoteclose}\ \isakeyword{for}\ i\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assms\ that\ row{\isacharunderscore}{\kern0pt}P{\isacharunderscore}{\kern0pt}upper{\isacharunderscore}{\kern0pt}indep\ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ R{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}upto{\isacharunderscore}{\kern0pt}s{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}row\ i\ {\isacharparenleft}{\kern0pt}R{\isacharunderscore}{\kern0pt}mat\ d{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ row\ i\ {\isacharparenleft}{\kern0pt}R{\isacharunderscore}{\kern0pt}mat\ d{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{if}\ {\isachardoublequoteopen}i\ {\isasymle}\ s{\isachardoublequoteclose}\ \isakeyword{for}\ i\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ that\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ row{\isacharunderscore}{\kern0pt}def\ R{\isacharunderscore}{\kern0pt}mat{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ Qr{\isacharunderscore}{\kern0pt}eq{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}inv\ {\isacharparenleft}{\kern0pt}Q{\isacharunderscore}{\kern0pt}mat\ d{\isacharparenright}{\kern0pt}\ {\isacharasterisk}{\kern0pt}v\ r{\isacharunderscore}{\kern0pt}vec\ d{\isacharparenright}{\kern0pt}\ {\isachardollar}{\kern0pt}\ s\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}inv\ {\isacharparenleft}{\kern0pt}Q{\isacharunderscore}{\kern0pt}mat\ d{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharasterisk}{\kern0pt}v\ r{\isacharunderscore}{\kern0pt}vec\ d{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isachardollar}{\kern0pt}\ s{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}inv\ {\isacharparenleft}{\kern0pt}Q{\isacharunderscore}{\kern0pt}mat\ d{\isacharparenright}{\kern0pt}\ {\isacharasterisk}{\kern0pt}v\ r{\isacharunderscore}{\kern0pt}vec\ d{\isacharparenright}{\kern0pt}\ {\isachardollar}{\kern0pt}\ s\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSum}j{\isasymin}UNIV{\isachardot}{\kern0pt}\ matrix{\isacharunderscore}{\kern0pt}inv\ {\isacharparenleft}{\kern0pt}Q{\isacharunderscore}{\kern0pt}mat\ d{\isacharparenright}{\kern0pt}\ {\isachardollar}{\kern0pt}\ s\ {\isachardollar}{\kern0pt}\ j\ {\isacharasterisk}{\kern0pt}\ r{\isacharunderscore}{\kern0pt}vec\ d\ {\isachardollar}{\kern0pt}\ j{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \ \ \ \isanewline
-\ \ \ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ matrix{\isacharunderscore}{\kern0pt}vector{\isacharunderscore}{\kern0pt}mult{\isacharunderscore}{\kern0pt}def\ \isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSum}j{\isasymin}UNIV{\isachardot}{\kern0pt}\ if\ s\ {\isacharless}{\kern0pt}\ j\ then\ {\isadigit{0}}\ else\ matrix{\isacharunderscore}{\kern0pt}inv\ {\isacharparenleft}{\kern0pt}Q{\isacharunderscore}{\kern0pt}mat\ d{\isacharparenright}{\kern0pt}\ {\isachardollar}{\kern0pt}\ s\ {\isachardollar}{\kern0pt}\ j\ {\isacharasterisk}{\kern0pt}\ r{\isacharunderscore}{\kern0pt}vec\ d\ {\isachardollar}{\kern0pt}\ j{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ lt{\isacharunderscore}{\kern0pt}Q{\isacharunderscore}{\kern0pt}inv\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ sum{\isachardot}{\kern0pt}cong\ simp{\isacharcolon}{\kern0pt}\ lower{\isacharunderscore}{\kern0pt}triangular{\isacharunderscore}{\kern0pt}mat{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSum}j{\isasymin}UNIV{\isachardot}{\kern0pt}\ if\ s\ {\isacharless}{\kern0pt}\ j\ then\ {\isadigit{0}}\ else\ matrix{\isacharunderscore}{\kern0pt}inv\ {\isacharparenleft}{\kern0pt}Q{\isacharunderscore}{\kern0pt}mat\ d{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isachardollar}{\kern0pt}\ s\ {\isachardollar}{\kern0pt}\ j\ {\isacharasterisk}{\kern0pt}\ r{\isacharunderscore}{\kern0pt}vec\ d\ {\isachardollar}{\kern0pt}\ j{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ inv{\isacharunderscore}{\kern0pt}Q{\isacharunderscore}{\kern0pt}mat{\isacharunderscore}{\kern0pt}indep\ assms\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}fastforce\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ sum{\isachardot}{\kern0pt}cong\ simp{\isacharcolon}{\kern0pt}\ row{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}inv\ {\isacharparenleft}{\kern0pt}Q{\isacharunderscore}{\kern0pt}mat\ d{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharasterisk}{\kern0pt}v\ r{\isacharunderscore}{\kern0pt}vec\ d{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isachardollar}{\kern0pt}\ s{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ lt{\isacharunderscore}{\kern0pt}Q{\isacharunderscore}{\kern0pt}inv\ \isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ matrix{\isacharunderscore}{\kern0pt}vector{\isacharunderscore}{\kern0pt}mult{\isacharunderscore}{\kern0pt}def\ assms\ lower{\isacharunderscore}{\kern0pt}triangular{\isacharunderscore}{\kern0pt}mat{\isacharunderscore}{\kern0pt}def\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ sum{\isachardot}{\kern0pt}cong{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{finally}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isacommand{{\isachardot}{\kern0pt}}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ QR{\isacharunderscore}{\kern0pt}eq{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}row\ s\ {\isacharparenleft}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}inv\ {\isacharparenleft}{\kern0pt}Q{\isacharunderscore}{\kern0pt}mat\ d{\isacharparenright}{\kern0pt}\ {\isacharasterisk}{\kern0pt}{\isacharasterisk}{\kern0pt}\ R{\isacharunderscore}{\kern0pt}mat\ d{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ row\ s\ {\isacharparenleft}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}inv\ {\isacharparenleft}{\kern0pt}Q{\isacharunderscore}{\kern0pt}mat\ d{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharasterisk}{\kern0pt}{\isacharasterisk}{\kern0pt}\ R{\isacharunderscore}{\kern0pt}mat\ d{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\ \isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}matrix{\isacharunderscore}{\kern0pt}inv\ {\isacharparenleft}{\kern0pt}Q{\isacharunderscore}{\kern0pt}mat\ d{\isacharparenright}{\kern0pt}\ {\isachardollar}{\kern0pt}\ s\ {\isachardollar}{\kern0pt}\ k\ {\isacharasterisk}{\kern0pt}\ R{\isacharunderscore}{\kern0pt}mat\ d\ {\isachardollar}{\kern0pt}\ k\ {\isachardollar}{\kern0pt}\ j\ {\isacharequal}{\kern0pt}\ matrix{\isacharunderscore}{\kern0pt}inv\ {\isacharparenleft}{\kern0pt}Q{\isacharunderscore}{\kern0pt}mat\ d{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isachardollar}{\kern0pt}\ s\ {\isachardollar}{\kern0pt}\ k\ {\isacharasterisk}{\kern0pt}\ R{\isacharunderscore}{\kern0pt}mat\ d{\isacharprime}{\kern0pt}\ {\isachardollar}{\kern0pt}\ k\ {\isachardollar}{\kern0pt}\ j{\isachardoublequoteclose}\ \isakeyword{for}\ k\ j\isanewline
-\ \ \ \ \isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\ \isanewline
-\ \ \ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}matrix{\isacharunderscore}{\kern0pt}inv\ {\isacharparenleft}{\kern0pt}Q{\isacharunderscore}{\kern0pt}mat\ d{\isacharparenright}{\kern0pt}\ {\isachardollar}{\kern0pt}\ s\ {\isachardollar}{\kern0pt}\ k\ {\isacharasterisk}{\kern0pt}\ R{\isacharunderscore}{\kern0pt}mat\ d\ {\isachardollar}{\kern0pt}\ k\ {\isachardollar}{\kern0pt}\ j\ {\isacharequal}{\kern0pt}\ \isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}if\ s\ {\isacharless}{\kern0pt}\ k\ then\ {\isadigit{0}}\ else\ matrix{\isacharunderscore}{\kern0pt}inv\ {\isacharparenleft}{\kern0pt}Q{\isacharunderscore}{\kern0pt}mat\ d{\isacharparenright}{\kern0pt}\ {\isachardollar}{\kern0pt}\ s\ {\isachardollar}{\kern0pt}\ k\ {\isacharasterisk}{\kern0pt}\ R{\isacharunderscore}{\kern0pt}mat\ d\ {\isachardollar}{\kern0pt}\ k\ {\isachardollar}{\kern0pt}\ j{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ lower{\isacharunderscore}{\kern0pt}triangular{\isacharunderscore}{\kern0pt}mat{\isacharunderscore}{\kern0pt}def\ lt{\isacharunderscore}{\kern0pt}Q{\isacharunderscore}{\kern0pt}inv\ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}if\ s\ {\isacharless}{\kern0pt}\ k\ then\ {\isadigit{0}}\ else\ matrix{\isacharunderscore}{\kern0pt}inv\ {\isacharparenleft}{\kern0pt}Q{\isacharunderscore}{\kern0pt}mat\ d{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isachardollar}{\kern0pt}\ s\ {\isachardollar}{\kern0pt}\ k\ {\isacharasterisk}{\kern0pt}\ R{\isacharunderscore}{\kern0pt}mat\ d\ {\isachardollar}{\kern0pt}\ k\ {\isachardollar}{\kern0pt}\ j{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharcomma}{\kern0pt}\ lifting{\isacharparenright}{\kern0pt}\ Finite{\isacharunderscore}{\kern0pt}Cartesian{\isacharunderscore}{\kern0pt}Product{\isachardot}{\kern0pt}row{\isacharunderscore}{\kern0pt}def\ assms\ inv{\isacharunderscore}{\kern0pt}Q{\isacharunderscore}{\kern0pt}mat{\isacharunderscore}{\kern0pt}indep\ order{\isacharunderscore}{\kern0pt}refl\ vec{\isacharunderscore}{\kern0pt}lambda{\isacharunderscore}{\kern0pt}eta{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}if\ s\ {\isacharless}{\kern0pt}\ k\ {\isasymor}\ j\ {\isacharless}{\kern0pt}\ k\ then\ {\isadigit{0}}\ else\ {\isacharparenleft}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}inv\ {\isacharparenleft}{\kern0pt}Q{\isacharunderscore}{\kern0pt}mat\ d{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isachardollar}{\kern0pt}\ s\ {\isachardollar}{\kern0pt}\ k\ {\isacharasterisk}{\kern0pt}\ R{\isacharunderscore}{\kern0pt}mat\ d\ {\isachardollar}{\kern0pt}\ k\ {\isachardollar}{\kern0pt}\ j{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ triangular{\isacharunderscore}{\kern0pt}mat{\isacharunderscore}{\kern0pt}R\isanewline
-\ \ \ \ \ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ upper{\isacharunderscore}{\kern0pt}triangular{\isacharunderscore}{\kern0pt}mat{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ split{\isacharcolon}{\kern0pt}\ if{\isacharunderscore}{\kern0pt}splits{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}if\ s\ {\isacharless}{\kern0pt}\ k\ {\isasymor}\ j\ {\isacharless}{\kern0pt}\ k\ then\ {\isadigit{0}}\ else\ {\isacharparenleft}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}inv\ {\isacharparenleft}{\kern0pt}Q{\isacharunderscore}{\kern0pt}mat\ d{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isachardollar}{\kern0pt}\ s\ {\isachardollar}{\kern0pt}\ k\ {\isacharasterisk}{\kern0pt}\ R{\isacharunderscore}{\kern0pt}mat\ d{\isacharprime}{\kern0pt}\ {\isachardollar}{\kern0pt}\ k\ {\isachardollar}{\kern0pt}\ j{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ R{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}upto{\isacharunderscore}{\kern0pt}s\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ row{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isacharequal}{\kern0pt}\ matrix{\isacharunderscore}{\kern0pt}inv\ {\isacharparenleft}{\kern0pt}Q{\isacharunderscore}{\kern0pt}mat\ d{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isachardollar}{\kern0pt}\ s\ {\isachardollar}{\kern0pt}\ k\ {\isacharasterisk}{\kern0pt}\ R{\isacharunderscore}{\kern0pt}mat\ d{\isacharprime}{\kern0pt}\ {\isachardollar}{\kern0pt}\ k\ {\isachardollar}{\kern0pt}\ j{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ lower{\isacharunderscore}{\kern0pt}triangular{\isacharunderscore}{\kern0pt}mat{\isacharunderscore}{\kern0pt}def\ lt{\isacharunderscore}{\kern0pt}Q{\isacharunderscore}{\kern0pt}inv\ mult{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}zero\ triangular{\isacharunderscore}{\kern0pt}mat{\isacharunderscore}{\kern0pt}R\ upper{\isacharunderscore}{\kern0pt}triangular{\isacharunderscore}{\kern0pt}mat{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{finally}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isacommand{{\isachardot}{\kern0pt}}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ row{\isacharunderscore}{\kern0pt}def\ matrix{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}mult{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ QR{\isacharunderscore}{\kern0pt}eq\ Qr{\isacharunderscore}{\kern0pt}eq\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp\ add{\isacharcolon}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}inv{\isacharunderscore}{\kern0pt}def\ vec{\isachardot}{\kern0pt}add\ row{\isacharunderscore}{\kern0pt}def\ matrix{\isacharunderscore}{\kern0pt}vector{\isacharunderscore}{\kern0pt}mul{\isacharunderscore}{\kern0pt}assoc\ matrix{\isacharunderscore}{\kern0pt}vector{\isacharunderscore}{\kern0pt}mult{\isacharunderscore}{\kern0pt}code{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\begin{isamarkuptext}%
-This recursive definition mimics the computation of the GS iteration.%
-\end{isamarkuptext}\isamarkuptrue%
-\isacommand{lemma}\isamarkupfalse%
-\ GS{\isacharunderscore}{\kern0pt}inv{\isacharunderscore}{\kern0pt}rec{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}GS{\isacharunderscore}{\kern0pt}inv\ d\ v\ {\isacharequal}{\kern0pt}\ r{\isacharunderscore}{\kern0pt}vec\ d\ {\isacharplus}{\kern0pt}\ l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isacharparenleft}{\kern0pt}P{\isacharunderscore}{\kern0pt}upper\ d\ {\isacharasterisk}{\kern0pt}v\ v\ {\isacharplus}{\kern0pt}\ P{\isacharunderscore}{\kern0pt}lower\ d\ {\isacharasterisk}{\kern0pt}v\ {\isacharparenleft}{\kern0pt}GS{\isacharunderscore}{\kern0pt}inv\ d\ v{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}Q{\isacharunderscore}{\kern0pt}mat\ d\ {\isacharasterisk}{\kern0pt}v\ {\isacharparenleft}{\kern0pt}GS{\isacharunderscore}{\kern0pt}inv\ d\ v{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ r{\isacharunderscore}{\kern0pt}vec\ d\ {\isacharplus}{\kern0pt}\ R{\isacharunderscore}{\kern0pt}mat\ d\ {\isacharasterisk}{\kern0pt}v\ v{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ splitting{\isacharunderscore}{\kern0pt}gauss{\isacharbrackleft}{\kern0pt}of\ d{\isacharbrackright}{\kern0pt}\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}inverse{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ GS{\isacharunderscore}{\kern0pt}inv{\isacharunderscore}{\kern0pt}def\ matrix{\isacharunderscore}{\kern0pt}vector{\isacharunderscore}{\kern0pt}mul{\isacharunderscore}{\kern0pt}assoc\ is{\isacharunderscore}{\kern0pt}splitting{\isacharunderscore}{\kern0pt}blin{\isacharunderscore}{\kern0pt}def{\isacharprime}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}subst\ matrix{\isacharunderscore}{\kern0pt}inv{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ auto\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\ \isanewline
-\ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ Q{\isacharunderscore}{\kern0pt}mat{\isacharunderscore}{\kern0pt}def\ R{\isacharunderscore}{\kern0pt}mat{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ algebra{\isacharunderscore}{\kern0pt}simps\ scaleR{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}vector{\isacharunderscore}{\kern0pt}assoc{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ is{\isacharunderscore}{\kern0pt}am{\isacharunderscore}{\kern0pt}GS{\isacharunderscore}{\kern0pt}inv{\isacharunderscore}{\kern0pt}extend{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}{\isasymAnd}s{\isachardot}{\kern0pt}\ s\ {\isacharless}{\kern0pt}\ k\ {\isasymLongrightarrow}\ is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max\ {\isacharparenleft}{\kern0pt}{\isasymlambda}d{\isachardot}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}inv\ d\ v\ {\isachardollar}{\kern0pt}\ s{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymlambda}d{\isachardot}{\kern0pt}\ d\ {\isasymin}\ D\isactrlsub D{\isacharparenright}{\kern0pt}\ d{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isakeyword{and}\ {\isachardoublequoteopen}is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max\ {\isacharparenleft}{\kern0pt}{\isasymlambda}a{\isachardot}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}inv\ {\isacharparenleft}{\kern0pt}d\ {\isacharparenleft}{\kern0pt}k\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ a{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ v\ {\isachardollar}{\kern0pt}\ k{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymlambda}a{\isachardot}{\kern0pt}\ a\ {\isasymin}\ A\ k{\isacharparenright}{\kern0pt}\ a{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isakeyword{and}\ {\isachardoublequoteopen}s\ {\isasymle}\ k{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isakeyword{and}\ {\isachardoublequoteopen}d\ {\isasymin}\ D\isactrlsub D{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max\ {\isacharparenleft}{\kern0pt}{\isasymlambda}d{\isachardot}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}inv\ d\ v\ {\isachardollar}{\kern0pt}\ s{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymlambda}d{\isachardot}{\kern0pt}\ d\ {\isasymin}\ D\isactrlsub D{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}d\ {\isacharparenleft}{\kern0pt}k\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ a{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ am{\isacharunderscore}{\kern0pt}k{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max\ {\isacharparenleft}{\kern0pt}{\isasymlambda}d{\isachardot}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}inv\ d\ v\ {\isachardollar}{\kern0pt}\ k{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymlambda}d{\isachardot}{\kern0pt}\ d\ {\isasymin}\ D\isactrlsub D{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}d\ {\isacharparenleft}{\kern0pt}k\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ a{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}rule\ is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}linorderI{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{fix}\isamarkupfalse%
-\ y\isanewline
-\ \ \ \ \isacommand{assume}\isamarkupfalse%
-\ {\isachardoublequoteopen}y\ {\isasymin}\ D\isactrlsub D{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}GS{\isacharunderscore}{\kern0pt}inv\ y\ v\ {\isachardollar}{\kern0pt}\ k\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}r{\isacharunderscore}{\kern0pt}vec\ y\ {\isacharplus}{\kern0pt}\ l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isacharparenleft}{\kern0pt}P{\isacharunderscore}{\kern0pt}upper\ y\ {\isacharasterisk}{\kern0pt}v\ v\ {\isacharplus}{\kern0pt}\ P{\isacharunderscore}{\kern0pt}lower\ y\ {\isacharasterisk}{\kern0pt}v\ {\isacharparenleft}{\kern0pt}GS{\isacharunderscore}{\kern0pt}inv\ y\ v{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isachardollar}{\kern0pt}\ k{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ GS{\isacharunderscore}{\kern0pt}inv{\isacharunderscore}{\kern0pt}rec\ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isacharequal}{\kern0pt}\ r\ {\isacharparenleft}{\kern0pt}k{\isacharcomma}{\kern0pt}\ y\ k{\isacharparenright}{\kern0pt}\ {\isacharplus}{\kern0pt}\ l\ {\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}P{\isacharunderscore}{\kern0pt}upper\ y\ {\isacharasterisk}{\kern0pt}v\ v{\isacharparenright}{\kern0pt}\ {\isachardollar}{\kern0pt}\ k\ {\isacharplus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}P{\isacharunderscore}{\kern0pt}lower\ y\ {\isacharasterisk}{\kern0pt}v\ GS{\isacharunderscore}{\kern0pt}inv\ y\ v{\isacharparenright}{\kern0pt}\ {\isachardollar}{\kern0pt}\ k{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isasymle}\ r\ {\isacharparenleft}{\kern0pt}k{\isacharcomma}{\kern0pt}\ {\isacharparenleft}{\kern0pt}d{\isacharparenleft}{\kern0pt}k\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ y\ k{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ k{\isacharparenright}{\kern0pt}\ {\isacharplus}{\kern0pt}\ l\ {\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}P{\isacharunderscore}{\kern0pt}upper\ {\isacharparenleft}{\kern0pt}d{\isacharparenleft}{\kern0pt}k\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ y\ k{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharasterisk}{\kern0pt}v\ v{\isacharparenright}{\kern0pt}\ {\isachardollar}{\kern0pt}\ k\ {\isacharplus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}P{\isacharunderscore}{\kern0pt}lower\ {\isacharparenleft}{\kern0pt}d{\isacharparenleft}{\kern0pt}k\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ y\ k{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharasterisk}{\kern0pt}v\ GS{\isacharunderscore}{\kern0pt}inv\ {\isacharparenleft}{\kern0pt}d{\isacharparenleft}{\kern0pt}k\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ y\ k{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ v{\isacharparenright}{\kern0pt}\ {\isachardollar}{\kern0pt}\ k{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}rule\ add{\isacharunderscore}{\kern0pt}mono{\isacharcomma}{\kern0pt}\ goal{\isacharunderscore}{\kern0pt}cases{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{case}\isamarkupfalse%
-\ {\isadigit{2}}\isanewline
-\ \ \ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}case\isanewline
-\ \ \ \ \ \ \isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}intro\ mult{\isacharunderscore}{\kern0pt}left{\isacharunderscore}{\kern0pt}mono\ add{\isacharunderscore}{\kern0pt}mono{\isacharcomma}{\kern0pt}\ goal{\isacharunderscore}{\kern0pt}cases{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{case}\isamarkupfalse%
-\ {\isadigit{1}}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}case\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ matrix{\isacharunderscore}{\kern0pt}vector{\isacharunderscore}{\kern0pt}mult{\isacharunderscore}{\kern0pt}def\ P{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}elem\ fun{\isacharunderscore}{\kern0pt}upd{\isacharunderscore}{\kern0pt}same\ P{\isacharunderscore}{\kern0pt}upper{\isacharunderscore}{\kern0pt}def\ cong{\isacharcolon}{\kern0pt}\ if{\isacharunderscore}{\kern0pt}cong{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ \isacommand{case}\isamarkupfalse%
-\ {\isadigit{2}}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}case\ \isanewline
-\ \ \ \ \ \ \ \ \isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}P{\isacharunderscore}{\kern0pt}lower\ y\ {\isacharasterisk}{\kern0pt}v\ GS{\isacharunderscore}{\kern0pt}inv\ y\ v{\isacharparenright}{\kern0pt}\ {\isachardollar}{\kern0pt}\ k\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}P{\isacharunderscore}{\kern0pt}lower\ {\isacharparenleft}{\kern0pt}d{\isacharparenleft}{\kern0pt}k\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ y\ k{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharasterisk}{\kern0pt}v\ GS{\isacharunderscore}{\kern0pt}inv\ y\ v{\isacharparenright}{\kern0pt}\ {\isachardollar}{\kern0pt}\ k{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ matrix{\isacharunderscore}{\kern0pt}vector{\isacharunderscore}{\kern0pt}mult{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ P{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}elem\ fun{\isacharunderscore}{\kern0pt}upd{\isacharunderscore}{\kern0pt}same\ P{\isacharunderscore}{\kern0pt}lower{\isacharunderscore}{\kern0pt}def\ cong{\isacharcolon}{\kern0pt}\ if{\isacharunderscore}{\kern0pt}cong{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSum}j{\isasymin}UNIV{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}if\ j\ {\isacharless}{\kern0pt}\ k\ then\ pmf\ {\isacharparenleft}{\kern0pt}K\ {\isacharparenleft}{\kern0pt}k{\isacharcomma}{\kern0pt}\ y\ k{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ j\ \ {\isacharasterisk}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}inv\ y\ v\ {\isachardollar}{\kern0pt}\ j\ \ else\ {\isadigit{0}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ matrix{\isacharunderscore}{\kern0pt}vector{\isacharunderscore}{\kern0pt}mult{\isacharunderscore}{\kern0pt}def\ P{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}elem\ P{\isacharunderscore}{\kern0pt}lower{\isacharunderscore}{\kern0pt}def\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ sum{\isachardot}{\kern0pt}cong{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isasymle}\ {\isacharparenleft}{\kern0pt}{\isasymSum}j{\isasymin}UNIV{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}if\ j\ {\isacharless}{\kern0pt}\ k\ then\ pmf\ {\isacharparenleft}{\kern0pt}K\ {\isacharparenleft}{\kern0pt}k{\isacharcomma}{\kern0pt}\ y\ k{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ j\ \ {\isacharasterisk}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}inv\ d\ v\ {\isachardollar}{\kern0pt}\ j\ \ else\ {\isadigit{0}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assms\ {\isacartoucheopen}y{\isasymin}D\isactrlsub D{\isacartoucheclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}fastforce\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ sum{\isacharunderscore}{\kern0pt}mono\ mult{\isacharunderscore}{\kern0pt}left{\isacharunderscore}{\kern0pt}mono\ dest{\isacharcolon}{\kern0pt}\ is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}linorderD{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isacharequal}{\kern0pt}\ \ {\isacharparenleft}{\kern0pt}{\isasymSum}j{\isasymin}UNIV{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}if\ j\ {\isacharless}{\kern0pt}\ k\ then\ pmf\ {\isacharparenleft}{\kern0pt}K\ {\isacharparenleft}{\kern0pt}k{\isacharcomma}{\kern0pt}\ y\ k{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ j\ \ {\isacharasterisk}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}inv\ {\isacharparenleft}{\kern0pt}d{\isacharparenleft}{\kern0pt}k\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ y\ k{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ v\ {\isachardollar}{\kern0pt}\ j\ else\ {\isadigit{0}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \ \isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ GS{\isacharunderscore}{\kern0pt}indep{\isacharunderscore}{\kern0pt}high{\isacharunderscore}{\kern0pt}states{\isacharbrackleft}{\kern0pt}of\ {\isacharunderscore}{\kern0pt}\ {\isachardoublequoteopen}d{\isacharparenleft}{\kern0pt}k\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ y\ k{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ d{\isacharcomma}{\kern0pt}\ symmetric{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}fastforce\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ sum{\isachardot}{\kern0pt}cong\ dest{\isacharcolon}{\kern0pt}\ leD{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isacharequal}{\kern0pt}\ \ {\isacharparenleft}{\kern0pt}P{\isacharunderscore}{\kern0pt}lower\ {\isacharparenleft}{\kern0pt}d{\isacharparenleft}{\kern0pt}k\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ y\ k{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharasterisk}{\kern0pt}v\ GS{\isacharunderscore}{\kern0pt}inv\ {\isacharparenleft}{\kern0pt}d{\isacharparenleft}{\kern0pt}k\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ y\ k{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ v{\isacharparenright}{\kern0pt}\ {\isachardollar}{\kern0pt}\ k{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ matrix{\isacharunderscore}{\kern0pt}vector{\isacharunderscore}{\kern0pt}mult{\isacharunderscore}{\kern0pt}def\ P{\isacharunderscore}{\kern0pt}lower{\isacharunderscore}{\kern0pt}def\ P{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}elem\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}fastforce\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ sum{\isachardot}{\kern0pt}cong{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{finally}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isacommand{{\isachardot}{\kern0pt}}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \isacommand{qed}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \isacommand{qed}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}r{\isacharunderscore}{\kern0pt}vec\ {\isacharparenleft}{\kern0pt}d{\isacharparenleft}{\kern0pt}k\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ y\ k{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharplus}{\kern0pt}\ l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}P{\isacharunderscore}{\kern0pt}upper\ {\isacharparenleft}{\kern0pt}d{\isacharparenleft}{\kern0pt}k\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ y\ k{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharasterisk}{\kern0pt}v\ v{\isacharparenright}{\kern0pt}\ {\isacharplus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}P{\isacharunderscore}{\kern0pt}lower\ {\isacharparenleft}{\kern0pt}d{\isacharparenleft}{\kern0pt}k\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ y\ k{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharasterisk}{\kern0pt}v\ GS{\isacharunderscore}{\kern0pt}inv\ {\isacharparenleft}{\kern0pt}d{\isacharparenleft}{\kern0pt}k\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ y\ k{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ v{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isachardollar}{\kern0pt}\ k{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isacharequal}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}inv\ {\isacharparenleft}{\kern0pt}d{\isacharparenleft}{\kern0pt}k\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ y\ k{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ v\ {\isachardollar}{\kern0pt}\ k{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ GS{\isacharunderscore}{\kern0pt}inv{\isacharunderscore}{\kern0pt}rec\ \isacommand{by}\isamarkupfalse%
-\ presburger\isanewline
-\ \ \ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isasymle}\ GS{\isacharunderscore}{\kern0pt}inv\ {\isacharparenleft}{\kern0pt}d{\isacharparenleft}{\kern0pt}k\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ a{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ v\ {\isachardollar}{\kern0pt}\ k{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}linorderD{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isacharbrackleft}{\kern0pt}OF\ assms{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isacharbrackright}{\kern0pt}\ {\isacartoucheopen}y\ {\isasymin}\ D\isactrlsub D{\isacartoucheclose}\ is{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \ \ \isacommand{finally}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}GS{\isacharunderscore}{\kern0pt}inv\ y\ v\ {\isachardollar}{\kern0pt}\ k\ {\isasymle}\ GS{\isacharunderscore}{\kern0pt}inv\ {\isacharparenleft}{\kern0pt}d{\isacharparenleft}{\kern0pt}k\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ a{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ v\ {\isachardollar}{\kern0pt}\ k{\isachardoublequoteclose}\isacommand{{\isachardot}{\kern0pt}}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}d{\isacharparenleft}{\kern0pt}k\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isasymin}\ D\isactrlsub D{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assms\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ is{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det{\isacharunderscore}{\kern0pt}def\ is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}linorder{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}cases\ {\isachardoublequoteopen}s\ {\isacharless}{\kern0pt}\ k{\isachardoublequoteclose}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{case}\isamarkupfalse%
-\ True\isanewline
-\ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ am{\isacharunderscore}{\kern0pt}k\ assms{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}{\isacharbrackleft}{\kern0pt}OF\ True{\isacharbrackright}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}indep{\isacharunderscore}{\kern0pt}high{\isacharunderscore}{\kern0pt}states{\isacharbrackleft}{\kern0pt}of\ s\ {\isachardoublequoteopen}d\ {\isacharparenleft}{\kern0pt}k\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ a{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ d{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}fastforce\ dest{\isacharcolon}{\kern0pt}\ is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}linorderD\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}linorderI{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{case}\isamarkupfalse%
-\ False\isanewline
-\ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assms\ am{\isacharunderscore}{\kern0pt}k\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}GS{\isacharunderscore}{\kern0pt}le{\isacharcolon}{\kern0pt}\ \isanewline
-\ \ {\isachardoublequoteopen}{\isasymexists}d{\isachardot}{\kern0pt}\ {\isasymforall}s{\isasymle}k{\isachardot}{\kern0pt}\ is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max\ {\isacharparenleft}{\kern0pt}{\isasymlambda}d{\isachardot}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}inv\ d\ v\ {\isachardollar}{\kern0pt}\ s{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymlambda}d{\isachardot}{\kern0pt}\ d\ {\isasymin}\ D\isactrlsub D{\isacharparenright}{\kern0pt}\ d{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}induction\ k\ rule{\isacharcolon}{\kern0pt}\ less{\isacharunderscore}{\kern0pt}induct{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{case}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}less\ x{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}case\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}cases\ {\isachardoublequoteopen}{\isasymexists}y{\isachardot}{\kern0pt}\ y\ {\isacharless}{\kern0pt}\ x{\isachardoublequoteclose}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{case}\isamarkupfalse%
-\ True\isanewline
-\ \ \ \ \isacommand{define}\isamarkupfalse%
-\ y\ \isakeyword{where}\ {\isachardoublequoteopen}y\ {\isacharequal}{\kern0pt}\ Max\ {\isacharbraceleft}{\kern0pt}y{\isachardot}{\kern0pt}\ y\ {\isacharless}{\kern0pt}\ x{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\ \ \isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}y\ {\isacharless}{\kern0pt}\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ Max{\isacharunderscore}{\kern0pt}in\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ True\ y{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{obtain}\isamarkupfalse%
-\ d{\isacharunderscore}{\kern0pt}opt\ \isakeyword{where}\ d{\isacharunderscore}{\kern0pt}opt{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max\ {\isacharparenleft}{\kern0pt}{\isasymlambda}d{\isachardot}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}inv\ d\ v\ {\isachardollar}{\kern0pt}\ s{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymlambda}d{\isachardot}{\kern0pt}\ d\ {\isasymin}\ D\isactrlsub D{\isacharparenright}{\kern0pt}\ d{\isacharunderscore}{\kern0pt}opt{\isachardoublequoteclose}\ \isakeyword{if}\ {\isachardoublequoteopen}s\ {\isasymle}\ y{\isachardoublequoteclose}\ \isakeyword{for}\ s\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isacartoucheopen}y\ {\isacharless}{\kern0pt}\ x{\isacartoucheclose}\ less\ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\isanewline
-\ \ \ \ \isacommand{define}\isamarkupfalse%
-\ d{\isacharunderscore}{\kern0pt}act\ \isakeyword{where}\ d{\isacharunderscore}{\kern0pt}act{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}d{\isacharunderscore}{\kern0pt}act\ a\ {\isacharequal}{\kern0pt}\ d{\isacharunderscore}{\kern0pt}opt{\isacharparenleft}{\kern0pt}x\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ a{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{for}\ a\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ le{\isacharunderscore}{\kern0pt}y{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}a\ {\isacharless}{\kern0pt}\ x\ {\isasymLongrightarrow}\ a\ {\isasymle}\ y{\isachardoublequoteclose}\ \isakeyword{for}\ a\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ y{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}GS{\isacharunderscore}{\kern0pt}inv\ d\ v\ {\isacharequal}{\kern0pt}\ r{\isacharunderscore}{\kern0pt}vec\ d\ {\isacharplus}{\kern0pt}\ l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isacharparenleft}{\kern0pt}P{\isacharunderscore}{\kern0pt}upper\ d\ {\isacharasterisk}{\kern0pt}v\ v\ {\isacharplus}{\kern0pt}\ P{\isacharunderscore}{\kern0pt}lower\ d\ {\isacharasterisk}{\kern0pt}v\ {\isacharparenleft}{\kern0pt}GS{\isacharunderscore}{\kern0pt}inv\ d\ v{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{for}\ d\isanewline
-\ \ \ \ \isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}Q{\isacharunderscore}{\kern0pt}mat\ d\ {\isacharasterisk}{\kern0pt}v\ {\isacharparenleft}{\kern0pt}GS{\isacharunderscore}{\kern0pt}inv\ d\ v{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}R{\isacharunderscore}{\kern0pt}mat\ d\ {\isacharasterisk}{\kern0pt}v\ v\ {\isacharplus}{\kern0pt}\ r{\isacharunderscore}{\kern0pt}vec\ d{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ GS{\isacharunderscore}{\kern0pt}inv{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ splitting{\isacharunderscore}{\kern0pt}gauss{\isacharbrackleft}{\kern0pt}unfolded\ is{\isacharunderscore}{\kern0pt}splitting{\isacharunderscore}{\kern0pt}blin{\isacharunderscore}{\kern0pt}def{\isacharprime}{\kern0pt}{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ matrix{\isacharunderscore}{\kern0pt}vector{\isacharunderscore}{\kern0pt}mul{\isacharunderscore}{\kern0pt}assoc\ matrix{\isacharunderscore}{\kern0pt}inv{\isacharunderscore}{\kern0pt}right{\isacharbrackleft}{\kern0pt}OF\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}inverse{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ Q{\isacharunderscore}{\kern0pt}mat{\isacharunderscore}{\kern0pt}def\ R{\isacharunderscore}{\kern0pt}mat{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ scaleR{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}vector{\isacharunderscore}{\kern0pt}assoc\ algebra{\isacharunderscore}{\kern0pt}simps{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isasymSqunion}d\ {\isasymin}\ D\isactrlsub D{\isachardot}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}inv\ d\ v\ {\isachardollar}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSqunion}d\ {\isasymin}\ D\isactrlsub D{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}r{\isacharunderscore}{\kern0pt}vec\ d\ {\isacharplus}{\kern0pt}\ l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isacharparenleft}{\kern0pt}P{\isacharunderscore}{\kern0pt}upper\ d\ {\isacharasterisk}{\kern0pt}v\ v\ {\isacharplus}{\kern0pt}\ P{\isacharunderscore}{\kern0pt}lower\ d\ {\isacharasterisk}{\kern0pt}v\ {\isacharparenleft}{\kern0pt}GS{\isacharunderscore}{\kern0pt}inv\ d\ v{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isachardollar}{\kern0pt}\ x{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isadigit{1}}\ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSqunion}a\ {\isasymin}\ A\ x{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}r{\isacharunderscore}{\kern0pt}vec\ {\isacharparenleft}{\kern0pt}d{\isacharunderscore}{\kern0pt}act\ a{\isacharparenright}{\kern0pt}\ {\isacharplus}{\kern0pt}\ l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isacharparenleft}{\kern0pt}P{\isacharunderscore}{\kern0pt}upper\ {\isacharparenleft}{\kern0pt}d{\isacharunderscore}{\kern0pt}act\ a{\isacharparenright}{\kern0pt}\ {\isacharasterisk}{\kern0pt}v\ v\ {\isacharplus}{\kern0pt}\ P{\isacharunderscore}{\kern0pt}lower\ {\isacharparenleft}{\kern0pt}d{\isacharunderscore}{\kern0pt}act\ a{\isacharparenright}{\kern0pt}\ {\isacharasterisk}{\kern0pt}v\ {\isacharparenleft}{\kern0pt}GS{\isacharunderscore}{\kern0pt}inv\ {\isacharparenleft}{\kern0pt}d{\isacharunderscore}{\kern0pt}act\ a{\isacharparenright}{\kern0pt}\ v{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isachardollar}{\kern0pt}\ x{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}rule\ antisym{\isacharcomma}{\kern0pt}\ rule\ cSUP{\isacharunderscore}{\kern0pt}mono{\isacharcomma}{\kern0pt}\ goal{\isacharunderscore}{\kern0pt}cases{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{case}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}{\isadigit{3}}\ n{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}P{\isacharunderscore}{\kern0pt}upper\ n\ {\isacharasterisk}{\kern0pt}v\ v{\isacharparenright}{\kern0pt}\ {\isachardollar}{\kern0pt}\ x\ {\isasymle}\ {\isacharparenleft}{\kern0pt}P{\isacharunderscore}{\kern0pt}upper\ {\isacharparenleft}{\kern0pt}d{\isacharunderscore}{\kern0pt}opt{\isacharparenleft}{\kern0pt}x\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ n\ x{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharasterisk}{\kern0pt}v\ v{\isacharparenright}{\kern0pt}\ {\isachardollar}{\kern0pt}\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ P{\isacharunderscore}{\kern0pt}upper{\isacharunderscore}{\kern0pt}def\ matrix{\isacharunderscore}{\kern0pt}vector{\isacharunderscore}{\kern0pt}mult{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ P{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}elem\ cong{\isacharcolon}{\kern0pt}\ if{\isacharunderscore}{\kern0pt}cong{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isanewline
-\ \ \ \ \ \ \isacommand{{\isacharbraceleft}{\kern0pt}}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymAnd}j{\isachardot}{\kern0pt}\ j\ {\isacharless}{\kern0pt}\ x\ {\isasymLongrightarrow}\ GS{\isacharunderscore}{\kern0pt}inv\ n\ v\ {\isachardollar}{\kern0pt}\ j\ {\isasymle}\ GS{\isacharunderscore}{\kern0pt}inv\ {\isacharparenleft}{\kern0pt}d{\isacharunderscore}{\kern0pt}opt{\isacharparenleft}{\kern0pt}x\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ n\ x{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ v\ {\isachardollar}{\kern0pt}\ j{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ d{\isacharunderscore}{\kern0pt}opt{\isacharbrackleft}{\kern0pt}OF\ le{\isacharunderscore}{\kern0pt}y{\isacharbrackright}{\kern0pt}\ {\isadigit{3}}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}subst\ GS{\isacharunderscore}{\kern0pt}indep{\isacharunderscore}{\kern0pt}high{\isacharunderscore}{\kern0pt}states{\isacharbrackleft}{\kern0pt}of\ {\isacharunderscore}{\kern0pt}\ {\isachardoublequoteopen}d{\isacharunderscore}{\kern0pt}opt{\isacharparenleft}{\kern0pt}x\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ n\ x{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ d{\isacharunderscore}{\kern0pt}opt{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}linorder{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}P{\isacharunderscore}{\kern0pt}lower\ n\ {\isacharasterisk}{\kern0pt}v\ GS{\isacharunderscore}{\kern0pt}inv\ n\ v{\isacharparenright}{\kern0pt}\ {\isachardollar}{\kern0pt}\ x\ {\isasymle}\ {\isacharparenleft}{\kern0pt}P{\isacharunderscore}{\kern0pt}lower\ {\isacharparenleft}{\kern0pt}d{\isacharunderscore}{\kern0pt}opt{\isacharparenleft}{\kern0pt}x\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ n\ x{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharasterisk}{\kern0pt}v\ GS{\isacharunderscore}{\kern0pt}inv\ {\isacharparenleft}{\kern0pt}d{\isacharunderscore}{\kern0pt}opt{\isacharparenleft}{\kern0pt}x\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ n\ x{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ v{\isacharparenright}{\kern0pt}\ {\isachardollar}{\kern0pt}\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ matrix{\isacharunderscore}{\kern0pt}vector{\isacharunderscore}{\kern0pt}mult{\isacharunderscore}{\kern0pt}def\ P{\isacharunderscore}{\kern0pt}lower{\isacharunderscore}{\kern0pt}def\ P{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}elem\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}fastforce\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ mult{\isacharunderscore}{\kern0pt}left{\isacharunderscore}{\kern0pt}mono\ sum{\isacharunderscore}{\kern0pt}mono{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{{\isacharbraceright}{\kern0pt}}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}case\isanewline
-\ \ \ \ \ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ d{\isacharunderscore}{\kern0pt}act\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ bexI{\isacharbrackleft}{\kern0pt}of\ {\isacharunderscore}{\kern0pt}\ {\isachardoublequoteopen}n\ x{\isachardoublequoteclose}{\isacharbrackright}{\kern0pt}\ mult{\isacharunderscore}{\kern0pt}left{\isacharunderscore}{\kern0pt}mono\ add{\isacharunderscore}{\kern0pt}mono\ simp{\isacharcolon}{\kern0pt}\ is{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \isacommand{case}\isamarkupfalse%
-\ {\isadigit{4}}\isanewline
-\ \ \ \ \ \ \isacommand{then}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}case\isanewline
-\ \ \ \ \ \ \isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}rule\ cSUP{\isacharunderscore}{\kern0pt}mono{\isacharcomma}{\kern0pt}\ goal{\isacharunderscore}{\kern0pt}cases{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{case}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}{\isadigit{3}}\ n{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{then}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}case\ \isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ d{\isacharunderscore}{\kern0pt}opt\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}fastforce\ simp{\isacharcolon}{\kern0pt}\ d{\isacharunderscore}{\kern0pt}act\ is{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det{\isacharunderscore}{\kern0pt}def\ is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}linorder\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ bexI{\isacharbrackleft}{\kern0pt}of\ {\isacharunderscore}{\kern0pt}\ {\isachardoublequoteopen}d{\isacharunderscore}{\kern0pt}act\ n{\isachardoublequoteclose}{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{qed}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ A{\isacharunderscore}{\kern0pt}ne{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{qed}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSqunion}a\ {\isasymin}\ A\ x{\isachardot}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}inv\ {\isacharparenleft}{\kern0pt}d{\isacharunderscore}{\kern0pt}act\ a{\isacharparenright}{\kern0pt}\ v\ {\isachardollar}{\kern0pt}\ x{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isadigit{1}}\ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \isacommand{finally}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isacharasterisk}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isasymSqunion}d\ {\isasymin}\ D\isactrlsub D{\isachardot}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}inv\ d\ v\ {\isachardollar}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSqunion}a\ {\isasymin}\ A\ x{\isachardot}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}inv\ {\isacharparenleft}{\kern0pt}d{\isacharunderscore}{\kern0pt}act\ a{\isacharparenright}{\kern0pt}\ v\ {\isachardollar}{\kern0pt}\ x{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isacommand{{\isachardot}{\kern0pt}}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{then}\isamarkupfalse%
-\ \isacommand{obtain}\isamarkupfalse%
-\ a{\isacharunderscore}{\kern0pt}opt\ \isakeyword{where}\ a{\isacharunderscore}{\kern0pt}opt{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max\ {\isacharparenleft}{\kern0pt}{\isasymlambda}a{\isachardot}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}inv\ {\isacharparenleft}{\kern0pt}d{\isacharunderscore}{\kern0pt}act\ a{\isacharparenright}{\kern0pt}\ v\ {\isachardollar}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymlambda}a{\isachardot}{\kern0pt}\ a\ {\isasymin}\ A\ x{\isacharparenright}{\kern0pt}\ a{\isacharunderscore}{\kern0pt}opt{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ A{\isacharunderscore}{\kern0pt}ne\ finite\ finite{\isacharunderscore}{\kern0pt}is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isasymSqunion}d\ {\isasymin}\ D\isactrlsub D{\isachardot}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}inv\ d\ v\ {\isachardollar}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}inv\ {\isacharparenleft}{\kern0pt}d{\isacharunderscore}{\kern0pt}act\ a{\isacharunderscore}{\kern0pt}opt{\isacharparenright}{\kern0pt}\ v\ {\isachardollar}{\kern0pt}\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharasterisk}{\kern0pt}\ arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}SUP{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ am{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}opt{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max\ {\isacharparenleft}{\kern0pt}{\isasymlambda}d{\isachardot}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}inv\ d\ v\ {\isachardollar}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymlambda}d{\isachardot}{\kern0pt}\ d\ {\isasymin}\ D\isactrlsub D{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}d{\isacharunderscore}{\kern0pt}act\ a{\isacharunderscore}{\kern0pt}opt{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ a{\isacharunderscore}{\kern0pt}opt\ d{\isacharunderscore}{\kern0pt}opt\ d{\isacharunderscore}{\kern0pt}act\ \isacommand{unfolding}\isamarkupfalse%
-\ is{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}fastforce\ dest{\isacharcolon}{\kern0pt}\ is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}linorderD{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ SUP{\isacharunderscore}{\kern0pt}is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max\ {\isacharparenleft}{\kern0pt}{\isasymlambda}d{\isachardot}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}inv\ d\ v\ {\isachardollar}{\kern0pt}\ x{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymlambda}d{\isachardot}{\kern0pt}\ d\ {\isasymin}\ D\isactrlsub D{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}d{\isacharunderscore}{\kern0pt}act\ a{\isacharunderscore}{\kern0pt}opt{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{if}\ {\isachardoublequoteopen}x{\isacharprime}{\kern0pt}\ {\isacharless}{\kern0pt}\ x{\isachardoublequoteclose}\ \isakeyword{for}\ x{\isacharprime}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}s{\isacharprime}{\kern0pt}\ {\isasymle}\ x{\isacharprime}{\kern0pt}\ {\isasymLongrightarrow}\ d{\isacharunderscore}{\kern0pt}act\ a{\isacharunderscore}{\kern0pt}opt\ s{\isacharprime}{\kern0pt}\ {\isacharequal}{\kern0pt}\ d{\isacharunderscore}{\kern0pt}opt\ s{\isacharprime}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{for}\ s{\isacharprime}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ d{\isacharunderscore}{\kern0pt}act\ that\ is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}linorderD{\isacharbrackleft}{\kern0pt}OF\ d{\isacharunderscore}{\kern0pt}opt{\isacharbrackleft}{\kern0pt}OF\ le{\isacharunderscore}{\kern0pt}y{\isacharbrackleft}{\kern0pt}OF\ that{\isacharbrackright}{\kern0pt}{\isacharbrackright}{\kern0pt}{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ am{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}opt\ is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}linorderD{\isacharbrackleft}{\kern0pt}OF\ d{\isacharunderscore}{\kern0pt}opt{\isacharbrackleft}{\kern0pt}OF\ le{\isacharunderscore}{\kern0pt}y{\isacharbrackleft}{\kern0pt}OF\ that{\isacharbrackright}{\kern0pt}{\isacharbrackright}{\kern0pt}{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}indep{\isacharunderscore}{\kern0pt}high{\isacharunderscore}{\kern0pt}states{\isacharbrackleft}{\kern0pt}of\ {\isacharunderscore}{\kern0pt}\ {\isachardoublequoteopen}d{\isacharunderscore}{\kern0pt}act\ a{\isacharunderscore}{\kern0pt}opt{\isachardoublequoteclose}\ d{\isacharunderscore}{\kern0pt}opt{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ am{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}opt\ antisym{\isacharunderscore}{\kern0pt}conv{\isadigit{1}}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{case}\isamarkupfalse%
-\ False\isanewline
-\ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ finite{\isacharunderscore}{\kern0pt}is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharbrackleft}{\kern0pt}OF\ finite{\isacharunderscore}{\kern0pt}D\isactrlsub D{\isacharbrackright}{\kern0pt}\ \isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}fastforce\ simp{\isacharcolon}{\kern0pt}\ arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}def\ someI{\isacharunderscore}{\kern0pt}ex\ dest{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ le{\isacharunderscore}{\kern0pt}neq{\isacharunderscore}{\kern0pt}trans{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ ex{\isacharunderscore}{\kern0pt}is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}GS{\isacharcolon}{\kern0pt}\ \isanewline
-\ \ {\isachardoublequoteopen}{\isasymexists}d{\isachardot}{\kern0pt}\ {\isasymforall}s{\isachardot}{\kern0pt}\ is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max\ {\isacharparenleft}{\kern0pt}{\isasymlambda}d{\isachardot}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}inv\ d\ v\ {\isachardollar}{\kern0pt}\ s{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymlambda}d{\isachardot}{\kern0pt}\ d\ {\isasymin}\ D\isactrlsub D{\isacharparenright}{\kern0pt}\ d{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}GS{\isacharunderscore}{\kern0pt}le{\isacharbrackleft}{\kern0pt}of\ {\isachardoublequoteopen}Max\ UNIV{\isachardoublequoteclose}{\isacharbrackright}{\kern0pt}\ \ \isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{function}\isamarkupfalse%
-\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}fun\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}fun\ v\ s\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSqunion}a\ {\isasymin}\ A\ s{\isachardot}{\kern0pt}\ r\ {\isacharparenleft}{\kern0pt}s{\isacharcomma}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isacharplus}{\kern0pt}\ l\ {\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}\isanewline
-\ \ {\isacharparenleft}{\kern0pt}{\isasymSum}s{\isacharprime}{\kern0pt}\ {\isacharless}{\kern0pt}\ s{\isachardot}{\kern0pt}\ pmf\ {\isacharparenleft}{\kern0pt}K\ {\isacharparenleft}{\kern0pt}s{\isacharcomma}{\kern0pt}a{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ s{\isacharprime}{\kern0pt}\ {\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}fun\ v\ s{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharplus}{\kern0pt}\ \isanewline
-\ \ {\isacharparenleft}{\kern0pt}{\isasymSum}s{\isacharprime}{\kern0pt}\ {\isasymin}\ {\isacharbraceleft}{\kern0pt}s{\isacharprime}{\kern0pt}{\isachardot}{\kern0pt}\ s\ {\isasymle}\ s{\isacharprime}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ pmf\ {\isacharparenleft}{\kern0pt}K\ {\isacharparenleft}{\kern0pt}s{\isacharcomma}{\kern0pt}a{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ s{\isacharprime}{\kern0pt}\ {\isacharasterisk}{\kern0pt}\ v\ s{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isacommand{termination}\isamarkupfalse%
-\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}relation\ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}{\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}y{\isacharparenright}{\kern0pt}{\isachardot}{\kern0pt}\ snd\ x\ {\isacharless}{\kern0pt}\ snd\ y{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}{\isacharcomma}{\kern0pt}\ rule\ wfI{\isacharunderscore}{\kern0pt}min{\isacharcomma}{\kern0pt}\ goal{\isacharunderscore}{\kern0pt}cases{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{case}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}\ x\ Q{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\ {\isachardoublequoteopen}\ x\ {\isasymin}\ Q{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isacharasterisk}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}u{\isachardot}{\kern0pt}\ {\isasymexists}a{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}a{\isacharcomma}{\kern0pt}\ u{\isacharparenright}{\kern0pt}\ {\isasymin}\ Q{\isacharbraceright}{\kern0pt}\ {\isasymnoteq}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}mono{\isacharunderscore}{\kern0pt}tags{\isacharcomma}{\kern0pt}\ lifting{\isacharparenright}{\kern0pt}\ {\isacartoucheopen}x{\isasymin}Q{\isacartoucheclose}\ prod{\isachardot}{\kern0pt}collapse\ Collect{\isacharunderscore}{\kern0pt}empty{\isacharunderscore}{\kern0pt}eq{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymexists}a\ x{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}a{\isacharcomma}{\kern0pt}x{\isacharparenright}{\kern0pt}{\isasymin}Q\ {\isasymand}\ x\ {\isacharequal}{\kern0pt}\ Min\ {\isacharparenleft}{\kern0pt}snd\ {\isacharbackquote}{\kern0pt}\ Q{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ image{\isacharunderscore}{\kern0pt}iff{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}mono{\isacharunderscore}{\kern0pt}tags{\isacharcomma}{\kern0pt}\ lifting{\isacharparenright}{\kern0pt}\ equals{\isadigit{0}}D\ Min{\isacharunderscore}{\kern0pt}in{\isacharbrackleft}{\kern0pt}OF\ finite{\isacharbrackright}{\kern0pt}\ prod{\isachardot}{\kern0pt}collapse\ image{\isacharunderscore}{\kern0pt}iff{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{then}\isamarkupfalse%
-\ \isacommand{obtain}\isamarkupfalse%
-\ x\ \isakeyword{where}\ {\isachardoublequoteopen}x\ {\isasymin}\ Q{\isachardoublequoteclose}\ {\isachardoublequoteopen}snd\ x\ {\isacharequal}{\kern0pt}\ Min\ {\isacharbraceleft}{\kern0pt}snd\ x{\isacharbar}{\kern0pt}\ x{\isachardot}{\kern0pt}\ x\ {\isasymin}\ Q{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ Setcompr{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}image\ snd{\isacharunderscore}{\kern0pt}conv{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}case\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isacharasterisk}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}intro\ bexI{\isacharbrackleft}{\kern0pt}of\ {\isacharunderscore}{\kern0pt}\ x{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\ auto\isanewline
-\isacommand{qed}\isamarkupfalse%
-\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{declare}\isamarkupfalse%
-\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}fun{\isachardot}{\kern0pt}simps{\isacharbrackleft}{\kern0pt}simp\ del{\isacharbrackright}{\kern0pt}\ \isanewline
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ {\isachardoublequoteopen}GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}elem\ v\ s\ a\ {\isacharequal}{\kern0pt}\ r\ {\isacharparenleft}{\kern0pt}s{\isacharcomma}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isacharplus}{\kern0pt}\ l\ {\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}\isanewline
-\ \ {\isacharparenleft}{\kern0pt}{\isasymSum}s{\isacharprime}{\kern0pt}\ {\isacharless}{\kern0pt}\ s{\isachardot}{\kern0pt}\ pmf\ {\isacharparenleft}{\kern0pt}K\ {\isacharparenleft}{\kern0pt}s{\isacharcomma}{\kern0pt}a{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ s{\isacharprime}{\kern0pt}\ {\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}fun\ v\ s{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharplus}{\kern0pt}\ \isanewline
-\ \ {\isacharparenleft}{\kern0pt}{\isasymSum}s{\isacharprime}{\kern0pt}\ {\isasymin}\ {\isacharbraceleft}{\kern0pt}s{\isacharprime}{\kern0pt}{\isachardot}{\kern0pt}\ s\ {\isasymle}\ s{\isacharprime}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ pmf\ {\isacharparenleft}{\kern0pt}K\ {\isacharparenleft}{\kern0pt}s{\isacharcomma}{\kern0pt}a{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ s{\isacharprime}{\kern0pt}\ {\isacharasterisk}{\kern0pt}\ v\ s{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}fun{\isacharunderscore}{\kern0pt}elem{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}fun\ v\ s\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSqunion}a\ {\isasymin}\ A\ s{\isachardot}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}elem\ v\ s\ a{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}elem{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \isacommand{using}\isamarkupfalse%
-\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}fun{\isachardot}{\kern0pt}simps\ \isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ blast%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ {\isachardoublequoteopen}GS{\isacharunderscore}{\kern0pt}rec\ v\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymchi}\ s{\isachardot}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}fun\ {\isacharparenleft}{\kern0pt}vec{\isacharunderscore}{\kern0pt}nth\ v{\isacharparenright}{\kern0pt}\ s{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}def{\isacharprime}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}GS{\isacharunderscore}{\kern0pt}rec\ v\ {\isachardollar}{\kern0pt}\ s\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSqunion}a\ {\isasymin}\ A\ s{\isachardot}{\kern0pt}\ r\ {\isacharparenleft}{\kern0pt}s{\isacharcomma}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isacharplus}{\kern0pt}\ l\ {\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}\isanewline
-\ \ {\isacharparenleft}{\kern0pt}{\isasymSum}s{\isacharprime}{\kern0pt}\ {\isacharless}{\kern0pt}\ s{\isachardot}{\kern0pt}\ pmf\ {\isacharparenleft}{\kern0pt}K\ {\isacharparenleft}{\kern0pt}s{\isacharcomma}{\kern0pt}a{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ s{\isacharprime}{\kern0pt}\ {\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}GS{\isacharunderscore}{\kern0pt}rec\ v\ {\isachardollar}{\kern0pt}\ s{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharplus}{\kern0pt}\ \isanewline
-\ \ {\isacharparenleft}{\kern0pt}{\isasymSum}s{\isacharprime}{\kern0pt}\ {\isasymin}\ {\isacharbraceleft}{\kern0pt}s{\isacharprime}{\kern0pt}{\isachardot}{\kern0pt}\ s\ {\isasymle}\ s{\isacharprime}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ pmf\ {\isacharparenleft}{\kern0pt}K\ {\isacharparenleft}{\kern0pt}s{\isacharcomma}{\kern0pt}a{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ s{\isacharprime}{\kern0pt}\ {\isacharasterisk}{\kern0pt}\ v\ {\isachardollar}{\kern0pt}\ s{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}fun{\isachardot}{\kern0pt}simps{\isacharbrackleft}{\kern0pt}of\ {\isacharunderscore}{\kern0pt}\ s{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}eq{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}GS{\isacharunderscore}{\kern0pt}rec\ v\ {\isachardollar}{\kern0pt}\ s\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSqunion}a\ {\isasymin}\ A\ s{\isachardot}{\kern0pt}\ r\ {\isacharparenleft}{\kern0pt}s{\isacharcomma}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isacharplus}{\kern0pt}\ l\ {\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}\isanewline
-\ \ {\isacharparenleft}{\kern0pt}P{\isacharunderscore}{\kern0pt}lower\ {\isacharparenleft}{\kern0pt}d{\isacharparenleft}{\kern0pt}s\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ a{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharasterisk}{\kern0pt}v\ {\isacharparenleft}{\kern0pt}GS{\isacharunderscore}{\kern0pt}rec\ v{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isachardollar}{\kern0pt}\ s\ {\isacharplus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}P{\isacharunderscore}{\kern0pt}upper\ {\isacharparenleft}{\kern0pt}d{\isacharparenleft}{\kern0pt}s\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ a{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharasterisk}{\kern0pt}v\ v{\isacharparenright}{\kern0pt}\ {\isachardollar}{\kern0pt}\ s{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}def{\isacharprime}{\kern0pt}{\isacharbrackleft}{\kern0pt}of\ v\ s{\isacharbrackright}{\kern0pt}\ P{\isacharunderscore}{\kern0pt}lower{\isacharunderscore}{\kern0pt}def\ P{\isacharunderscore}{\kern0pt}upper{\isacharunderscore}{\kern0pt}def\ P{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}elem\ matrix{\isacharunderscore}{\kern0pt}vector{\isacharunderscore}{\kern0pt}mult{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ if{\isacharunderscore}{\kern0pt}distrib{\isacharbrackleft}{\kern0pt}\isakeyword{where}\ f\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}{\isasymlambda}x{\isachardot}{\kern0pt}\ x\ {\isacharasterisk}{\kern0pt}\ {\isacharunderscore}{\kern0pt}\ {\isachardollar}{\kern0pt}\ {\isacharunderscore}{\kern0pt}{\isachardoublequoteclose}{\isacharbrackright}{\kern0pt}\ sum{\isachardot}{\kern0pt}If{\isacharunderscore}{\kern0pt}cases\ lessThan{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isacommand{definition}\isamarkupfalse%
-\ {\isachardoublequoteopen}GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}step\ d\ v\ {\isasymequiv}\ r{\isacharunderscore}{\kern0pt}vec\ d\ {\isacharplus}{\kern0pt}\ l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isacharparenleft}{\kern0pt}P{\isacharunderscore}{\kern0pt}lower\ d\ {\isacharasterisk}{\kern0pt}v\ GS{\isacharunderscore}{\kern0pt}rec\ v\ {\isacharplus}{\kern0pt}\ P{\isacharunderscore}{\kern0pt}upper\ d\ {\isacharasterisk}{\kern0pt}v\ v{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}eq{\isacharprime}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}GS{\isacharunderscore}{\kern0pt}rec\ v\ {\isachardollar}{\kern0pt}\ s\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSqunion}a\ {\isasymin}\ A\ s{\isachardot}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}step\ {\isacharparenleft}{\kern0pt}d{\isacharparenleft}{\kern0pt}s{\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ a{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ v\ {\isachardollar}{\kern0pt}\ s{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}eq\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}step{\isacharunderscore}{\kern0pt}def\ \isacommand{by}\isamarkupfalse%
-\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}vec{\isacharcolon}{\kern0pt}\isanewline
-\ \ {\isachardoublequoteopen}GS{\isacharunderscore}{\kern0pt}rec\ v\ {\isachardollar}{\kern0pt}\ s\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSqunion}d{\isasymin}D\isactrlsub D{\isachardot}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}step\ d\ v\ {\isachardollar}{\kern0pt}\ s{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{obtain}\isamarkupfalse%
-\ d\ \isakeyword{where}\ d{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max\ {\isacharparenleft}{\kern0pt}{\isasymlambda}d{\isachardot}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}step\ d\ v\ {\isachardollar}{\kern0pt}\ s{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymlambda}d{\isachardot}{\kern0pt}\ d\ {\isasymin}D\isactrlsub D{\isacharparenright}{\kern0pt}\ d{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ finite{\isacharunderscore}{\kern0pt}is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharbrackleft}{\kern0pt}OF\ finite{\isacharcomma}{\kern0pt}\ of\ {\isachardoublequoteopen}D\isactrlsub D{\isachardoublequoteclose}\ {\isacharbrackright}{\kern0pt}\ ex{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}GS{\isacharunderscore}{\kern0pt}rec\ v\ {\isachardollar}{\kern0pt}\ s\ {\isacharequal}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}step\ d\ v\ {\isachardollar}{\kern0pt}\ s{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}eq{\isacharprime}{\kern0pt}{\isacharbrackleft}{\kern0pt}of\ {\isacharunderscore}{\kern0pt}\ {\isacharunderscore}{\kern0pt}\ d{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}intro\ antisym\ cSUP{\isacharunderscore}{\kern0pt}least{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymAnd}x{\isachardot}{\kern0pt}\ x\ {\isasymin}\ A\ s\ {\isasymLongrightarrow}\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}step\ {\isacharparenleft}{\kern0pt}d{\isacharparenleft}{\kern0pt}s\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ x{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ v\ {\isachardollar}{\kern0pt}\ s\ {\isasymle}\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}step\ d\ v\ {\isachardollar}{\kern0pt}\ s{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ A{\isacharunderscore}{\kern0pt}ne\ d\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}intro\ is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}linorderD{\isacharbrackleft}{\kern0pt}OF\ d{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ is{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det{\isacharunderscore}{\kern0pt}def\ is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}linorder{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}step\ d\ v\ {\isachardollar}{\kern0pt}\ s\ {\isasymle}\ {\isacharparenleft}{\kern0pt}{\isasymSqunion}a{\isasymin}A\ s{\isachardot}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}step\ {\isacharparenleft}{\kern0pt}d{\isacharparenleft}{\kern0pt}s\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ a{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ v\ {\isachardollar}{\kern0pt}\ s{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ d\ \isacommand{unfolding}\isamarkupfalse%
-\ is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}linorder\ is{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det{\isacharunderscore}{\kern0pt}def\ fun{\isacharunderscore}{\kern0pt}upd{\isacharunderscore}{\kern0pt}triv\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ cSUP{\isacharunderscore}{\kern0pt}upper{\isadigit{2}}{\isacharbrackleft}{\kern0pt}of\ {\isacharunderscore}{\kern0pt}\ {\isacharunderscore}{\kern0pt}\ {\isachardoublequoteopen}d\ s{\isachardoublequoteclose}{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ A{\isacharunderscore}{\kern0pt}ne{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ d\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}subst\ arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}SUP{\isacharbrackleft}{\kern0pt}symmetric{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\ auto\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{lift{\isacharunderscore}{\kern0pt}definition}\isamarkupfalse%
-\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}fun\isactrlsub b\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}s\ {\isasymRightarrow}\isactrlsub b\ real{\isacharparenright}{\kern0pt}\ {\isasymRightarrow}\ {\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}s\ {\isasymRightarrow}\isactrlsub b\ real{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{is}\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}fun\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ {\isachardoublequoteopen}GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}fun{\isacharunderscore}{\kern0pt}inner\ {\isacharparenleft}{\kern0pt}v\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isacharprime}{\kern0pt}s\ {\isasymRightarrow}\isactrlsub b\ real{\isacharparenright}{\kern0pt}\ s\ a\ {\isasymequiv}\ r\ {\isacharparenleft}{\kern0pt}s{\isacharcomma}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isacharplus}{\kern0pt}\ l\ {\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}\isanewline
-\ \ {\isacharparenleft}{\kern0pt}{\isasymSum}s{\isacharprime}{\kern0pt}\ {\isacharless}{\kern0pt}\ s{\isachardot}{\kern0pt}\ pmf\ {\isacharparenleft}{\kern0pt}K\ {\isacharparenleft}{\kern0pt}s{\isacharcomma}{\kern0pt}a{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ s{\isacharprime}{\kern0pt}\ {\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}fun\isactrlsub b\ v\ s{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharplus}{\kern0pt}\ \isanewline
-\ \ {\isacharparenleft}{\kern0pt}{\isasymSum}s{\isacharprime}{\kern0pt}\ {\isasymin}\ {\isacharbraceleft}{\kern0pt}s{\isacharprime}{\kern0pt}{\isachardot}{\kern0pt}\ s\ {\isasymle}\ s{\isacharprime}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ pmf\ {\isacharparenleft}{\kern0pt}K\ {\isacharparenleft}{\kern0pt}s{\isacharcomma}{\kern0pt}a{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ s{\isacharprime}{\kern0pt}\ {\isacharasterisk}{\kern0pt}\ v\ s{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}iter\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}iter\ v\ s\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSqunion}a\ {\isasymin}\ A\ s{\isachardot}{\kern0pt}\ r\ {\isacharparenleft}{\kern0pt}s{\isacharcomma}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isacharplus}{\kern0pt}\ l\ {\isacharasterisk}{\kern0pt}\isanewline
-\ \ {\isacharparenleft}{\kern0pt}{\isasymSum}s{\isacharprime}{\kern0pt}\ {\isasymin}\ UNIV{\isachardot}{\kern0pt}\ pmf\ {\isacharparenleft}{\kern0pt}K\ {\isacharparenleft}{\kern0pt}s{\isacharcomma}{\kern0pt}a{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ s{\isacharprime}{\kern0pt}\ {\isacharasterisk}{\kern0pt}\ v\ s{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}fun{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}GS{\isacharunderscore}{\kern0pt}iter{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}{\isasymforall}s{\isacharprime}{\kern0pt}\ {\isacharless}{\kern0pt}\ s{\isachardot}{\kern0pt}\ v{\isacharunderscore}{\kern0pt}next\ s{\isacharprime}{\kern0pt}\ {\isacharequal}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}fun\ v\ s{\isacharprime}{\kern0pt}{\isachardoublequoteclose}\ {\isachardoublequoteopen}{\isasymforall}s{\isacharprime}{\kern0pt}\ {\isasymin}\ {\isacharbraceleft}{\kern0pt}s{\isacharprime}{\kern0pt}{\isachardot}{\kern0pt}\ s\ {\isasymle}\ s{\isacharprime}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ v{\isacharunderscore}{\kern0pt}next\ s{\isacharprime}{\kern0pt}\ {\isacharequal}{\kern0pt}\ v\ s{\isacharprime}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}fun\ v\ s\ {\isacharequal}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}iter\ v{\isacharunderscore}{\kern0pt}next\ s{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}s{\isacharprime}{\kern0pt}{\isachardot}{\kern0pt}\ s{\isacharprime}{\kern0pt}\ {\isacharless}{\kern0pt}\ s{\isacharbraceright}{\kern0pt}\ {\isasymunion}\ {\isacharbraceleft}{\kern0pt}s{\isacharprime}{\kern0pt}{\isachardot}{\kern0pt}\ s\ {\isasymle}\ s{\isacharprime}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ UNIV{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isacharasterisk}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isasymSum}s{\isacharprime}{\kern0pt}{\isacharless}{\kern0pt}s{\isachardot}{\kern0pt}\ f\ s{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharplus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSum}s{\isacharprime}{\kern0pt}{\isasymin}Collect\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}{\isasymle}{\isacharparenright}{\kern0pt}\ s{\isacharparenright}{\kern0pt}{\isachardot}{\kern0pt}\ f\ s{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSum}s{\isacharprime}{\kern0pt}\ {\isasymin}\ UNIV{\isachardot}{\kern0pt}\ f\ s{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{for}\ f\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}subst\ sum{\isachardot}{\kern0pt}union{\isacharunderscore}{\kern0pt}disjoint{\isacharbrackleft}{\kern0pt}symmetric{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}auto\ simp\ add{\isacharcolon}{\kern0pt}\ lessThan{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}fun\ v\ s\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSqunion}a{\isasymin}A\ s{\isachardot}{\kern0pt}\ r\ {\isacharparenleft}{\kern0pt}s{\isacharcomma}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isacharplus}{\kern0pt}\ l\ {\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}{\isasymSum}s{\isacharprime}{\kern0pt}{\isacharless}{\kern0pt}s{\isachardot}{\kern0pt}\ pmf\ {\isacharparenleft}{\kern0pt}K\ {\isacharparenleft}{\kern0pt}s{\isacharcomma}{\kern0pt}\ a{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ s{\isacharprime}{\kern0pt}\ {\isacharasterisk}{\kern0pt}\ v{\isacharunderscore}{\kern0pt}next\ s{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharplus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSum}s{\isacharprime}{\kern0pt}{\isasymin}Collect\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}{\isasymle}{\isacharparenright}{\kern0pt}\ s{\isacharparenright}{\kern0pt}{\isachardot}{\kern0pt}\ pmf\ {\isacharparenleft}{\kern0pt}K\ {\isacharparenleft}{\kern0pt}s{\isacharcomma}{\kern0pt}\ a{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ s{\isacharprime}{\kern0pt}\ {\isacharasterisk}{\kern0pt}\ v\ s{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assms\ \isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}subst\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}fun{\isachardot}{\kern0pt}simps{\isacharparenright}{\kern0pt}\ auto\isanewline
-\ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSqunion}a{\isasymin}A\ s{\isachardot}{\kern0pt}\ r\ {\isacharparenleft}{\kern0pt}s{\isacharcomma}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isacharplus}{\kern0pt}\ l\ {\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}{\isasymSum}s{\isacharprime}{\kern0pt}{\isacharless}{\kern0pt}s{\isachardot}{\kern0pt}\ pmf\ {\isacharparenleft}{\kern0pt}K\ {\isacharparenleft}{\kern0pt}s{\isacharcomma}{\kern0pt}\ a{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ s{\isacharprime}{\kern0pt}\ {\isacharasterisk}{\kern0pt}\ v{\isacharunderscore}{\kern0pt}next\ s{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharplus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSum}s{\isacharprime}{\kern0pt}{\isasymin}Collect\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}{\isasymle}{\isacharparenright}{\kern0pt}\ s{\isacharparenright}{\kern0pt}{\isachardot}{\kern0pt}\ pmf\ {\isacharparenleft}{\kern0pt}K\ {\isacharparenleft}{\kern0pt}s{\isacharcomma}{\kern0pt}\ a{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ s{\isacharprime}{\kern0pt}\ {\isacharasterisk}{\kern0pt}\ v{\isacharunderscore}{\kern0pt}next\ s{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assms\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isacharequal}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}iter\ v{\isacharunderscore}{\kern0pt}next\ s{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ {\isacharasterisk}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}iter{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{finally}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\ \isacommand{{\isachardot}{\kern0pt}}\isamarkupfalse%
-\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ foldl{\isacharunderscore}{\kern0pt}upd{\isacharunderscore}{\kern0pt}notin{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymnotin}\ set\ X\ {\isasymLongrightarrow}\ foldl\ {\isacharparenleft}{\kern0pt}{\isasymlambda}f\ y{\isachardot}{\kern0pt}\ f{\isacharparenleft}{\kern0pt}y\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ g\ f\ y{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ c\ X\ x\ {\isacharequal}{\kern0pt}\ c\ x{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}induction\ X\ arbitrary{\isacharcolon}{\kern0pt}\ c{\isacharparenright}{\kern0pt}\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ foldl{\isacharunderscore}{\kern0pt}upd{\isacharunderscore}{\kern0pt}notin{\isacharprime}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymnotin}\ set\ Y\ {\isasymLongrightarrow}\ foldl\ {\isacharparenleft}{\kern0pt}{\isasymlambda}f\ y{\isachardot}{\kern0pt}\ f{\isacharparenleft}{\kern0pt}y\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ g\ f\ y{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ c\ {\isacharparenleft}{\kern0pt}X{\isacharat}{\kern0pt}Y{\isacharparenright}{\kern0pt}\ x\ {\isacharequal}{\kern0pt}\ foldl\ {\isacharparenleft}{\kern0pt}{\isasymlambda}f\ y{\isachardot}{\kern0pt}\ f{\isacharparenleft}{\kern0pt}y\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ g\ f\ y{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ c\ X\ x{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}induction\ X\ arbitrary{\isacharcolon}{\kern0pt}\ x\ c\ Y{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}auto\ simp\ add{\isacharcolon}{\kern0pt}\ foldl{\isacharunderscore}{\kern0pt}upd{\isacharunderscore}{\kern0pt}notin{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ sorted{\isacharunderscore}{\kern0pt}list{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}set{\isacharunderscore}{\kern0pt}split{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}finite\ X{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}sorted{\isacharunderscore}{\kern0pt}list{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}set\ X\ {\isacharequal}{\kern0pt}\ sorted{\isacharunderscore}{\kern0pt}list{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}set\ {\isacharbraceleft}{\kern0pt}x\ {\isasymin}\ X{\isachardot}{\kern0pt}\ x\ {\isacharless}{\kern0pt}\ y{\isacharbraceright}{\kern0pt}\ {\isacharat}{\kern0pt}\ sorted{\isacharunderscore}{\kern0pt}list{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}set\ {\isacharbraceleft}{\kern0pt}x\ {\isasymin}\ X{\isachardot}{\kern0pt}\ y\ {\isasymle}\ x{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ assms\isanewline
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}induction\ {\isachardoublequoteopen}card\ X{\isachardoublequoteclose}\ arbitrary{\isacharcolon}{\kern0pt}\ X{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{case}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}Suc\ n\ X{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}sorted{\isacharunderscore}{\kern0pt}list{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}set\ X\ {\isacharequal}{\kern0pt}\ Min\ X\ {\isacharhash}{\kern0pt}\ sorted{\isacharunderscore}{\kern0pt}list{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}set\ {\isacharparenleft}{\kern0pt}X\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}Min\ X{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ Suc\ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ intro{\isacharcolon}{\kern0pt}\ sorted{\isacharunderscore}{\kern0pt}list{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}set{\isacharunderscore}{\kern0pt}nonempty{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isacharequal}{\kern0pt}\ Min\ X\ {\isacharhash}{\kern0pt}\ sorted{\isacharunderscore}{\kern0pt}list{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}set\ {\isacharbraceleft}{\kern0pt}x\ {\isasymin}\ {\isacharparenleft}{\kern0pt}X\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}Min\ X{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardot}{\kern0pt}\ x\ {\isacharless}{\kern0pt}\ y{\isacharbraceright}{\kern0pt}\ {\isacharat}{\kern0pt}\ sorted{\isacharunderscore}{\kern0pt}list{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}set\ {\isacharbraceleft}{\kern0pt}x\ {\isasymin}\ {\isacharparenleft}{\kern0pt}X\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}Min\ X{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardot}{\kern0pt}\ y\ {\isasymle}\ x{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ Suc\ card{\isachardot}{\kern0pt}remove{\isacharbrackleft}{\kern0pt}OF\ Suc{\isacharparenleft}{\kern0pt}{\isadigit{3}}{\isacharparenright}{\kern0pt}\ Min{\isacharunderscore}{\kern0pt}in{\isacharbrackright}{\kern0pt}\ card{\isachardot}{\kern0pt}empty\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}fastforce\ simp{\isacharcolon}{\kern0pt}\ Suc{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharplus}{\kern0pt}\isanewline
-\ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isacharequal}{\kern0pt}\ sorted{\isacharunderscore}{\kern0pt}list{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}set\ {\isacharparenleft}{\kern0pt}{\isacharbraceleft}{\kern0pt}x\ {\isasymin}\ X{\isachardot}{\kern0pt}\ x\ {\isacharless}{\kern0pt}\ y{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharat}{\kern0pt}\ sorted{\isacharunderscore}{\kern0pt}list{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}set\ {\isacharbraceleft}{\kern0pt}x\ {\isasymin}\ X{\isachardot}{\kern0pt}\ y\ {\isasymle}\ x{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}cases\ {\isachardoublequoteopen}Min\ X\ {\isacharless}{\kern0pt}\ y{\isachardoublequoteclose}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{case}\isamarkupfalse%
-\ True\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ Min{\isacharunderscore}{\kern0pt}eq{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}Min\ X\ {\isacharequal}{\kern0pt}\ Min\ {\isacharbraceleft}{\kern0pt}x\ {\isasymin}\ X{\isachardot}{\kern0pt}\ x\ {\isacharless}{\kern0pt}\ y{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ True\ Suc\ Min{\isacharunderscore}{\kern0pt}in\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}subst\ eq{\isacharunderscore}{\kern0pt}Min{\isacharunderscore}{\kern0pt}iff{\isacharparenright}{\kern0pt}\ fastforce{\isacharplus}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}x\ {\isasymin}\ {\isacharparenleft}{\kern0pt}X\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}Min\ X{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardot}{\kern0pt}\ x\ {\isacharless}{\kern0pt}\ y{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}x\ {\isasymin}\ X{\isachardot}{\kern0pt}\ x\ {\isacharless}{\kern0pt}\ y{\isacharbraceright}{\kern0pt}\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}Min\ {\isacharbraceleft}{\kern0pt}x\ {\isasymin}\ X{\isachardot}{\kern0pt}\ x\ {\isacharless}{\kern0pt}\ y{\isacharbraceright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ Min{\isacharunderscore}{\kern0pt}eq\ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}Min\ X\ {\isacharhash}{\kern0pt}\ sorted{\isacharunderscore}{\kern0pt}list{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}set\ {\isacharbraceleft}{\kern0pt}x\ {\isasymin}\ {\isacharparenleft}{\kern0pt}X\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}Min\ X{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardot}{\kern0pt}\ x\ {\isacharless}{\kern0pt}\ y{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ \isanewline
-\ \ \ \ \ \ Min\ {\isacharbraceleft}{\kern0pt}x\ {\isasymin}\ X{\isachardot}{\kern0pt}\ x\ {\isacharless}{\kern0pt}\ y{\isacharbraceright}{\kern0pt}\ {\isacharhash}{\kern0pt}\ sorted{\isacharunderscore}{\kern0pt}list{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}set\ {\isacharparenleft}{\kern0pt}{\isacharbraceleft}{\kern0pt}x\ {\isasymin}\ X{\isachardot}{\kern0pt}\ x\ {\isacharless}{\kern0pt}\ y{\isacharbraceright}{\kern0pt}\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}Min\ {\isacharbraceleft}{\kern0pt}x\ {\isasymin}\ X{\isachardot}{\kern0pt}\ x\ {\isacharless}{\kern0pt}\ y{\isacharbraceright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ Min{\isacharunderscore}{\kern0pt}eq\ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isacharequal}{\kern0pt}\ sorted{\isacharunderscore}{\kern0pt}list{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}set\ {\isacharparenleft}{\kern0pt}{\isacharbraceleft}{\kern0pt}x\ {\isasymin}\ X{\isachardot}{\kern0pt}\ x\ {\isacharless}{\kern0pt}\ y{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ Suc\ True\ Min{\isacharunderscore}{\kern0pt}in\ Min{\isacharunderscore}{\kern0pt}eq\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}subst\ sorted{\isacharunderscore}{\kern0pt}list{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}set{\isacharunderscore}{\kern0pt}nonempty{\isacharbrackleft}{\kern0pt}symmetric{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\ fastforce{\isacharplus}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{finally}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}Min\ X\ {\isacharhash}{\kern0pt}\ sorted{\isacharunderscore}{\kern0pt}list{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}set\ {\isacharbraceleft}{\kern0pt}x\ {\isasymin}\ {\isacharparenleft}{\kern0pt}X\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}Min\ X{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardot}{\kern0pt}\ x\ {\isacharless}{\kern0pt}\ y{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ sorted{\isacharunderscore}{\kern0pt}list{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}set\ {\isacharparenleft}{\kern0pt}{\isacharbraceleft}{\kern0pt}x\ {\isasymin}\ X{\isachardot}{\kern0pt}\ x\ {\isacharless}{\kern0pt}\ y{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isacommand{{\isachardot}{\kern0pt}}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}Min\ X\ {\isacharhash}{\kern0pt}\ sorted{\isacharunderscore}{\kern0pt}list{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}set\ {\isacharbraceleft}{\kern0pt}x\ {\isasymin}\ {\isacharparenleft}{\kern0pt}X\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}Min\ X{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardot}{\kern0pt}\ x\ {\isacharless}{\kern0pt}\ y{\isacharbraceright}{\kern0pt}\ {\isacharat}{\kern0pt}\ sorted{\isacharunderscore}{\kern0pt}list{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}set\ {\isacharbraceleft}{\kern0pt}x\ {\isasymin}\ {\isacharparenleft}{\kern0pt}X\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}Min\ X{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardot}{\kern0pt}\ y\ {\isasymle}\ x{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ \isanewline
-\ \ \ \ \ \ sorted{\isacharunderscore}{\kern0pt}list{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}set\ {\isacharparenleft}{\kern0pt}{\isacharbraceleft}{\kern0pt}x\ {\isasymin}\ X{\isachardot}{\kern0pt}\ x\ {\isacharless}{\kern0pt}\ y{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharat}{\kern0pt}\ sorted{\isacharunderscore}{\kern0pt}list{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}set\ {\isacharbraceleft}{\kern0pt}x\ {\isasymin}\ {\isacharparenleft}{\kern0pt}X\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}Min\ X{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardot}{\kern0pt}\ y\ {\isasymle}\ x{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \isacommand{then}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ True\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ append{\isacharunderscore}{\kern0pt}Cons{\isacharbrackleft}{\kern0pt}symmetric{\isacharbrackright}{\kern0pt}\ simp\ del{\isacharcolon}{\kern0pt}\ append{\isacharunderscore}{\kern0pt}Cons\ dest{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ leD\ intro{\isacharcolon}{\kern0pt}\ arg{\isacharunderscore}{\kern0pt}cong{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{case}\isamarkupfalse%
-\ False\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ Min{\isacharunderscore}{\kern0pt}eq{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}Min\ X\ {\isacharequal}{\kern0pt}\ Min\ {\isacharbraceleft}{\kern0pt}x\ {\isasymin}\ X{\isachardot}{\kern0pt}\ y\ {\isasymle}\ x{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ False\ Suc\ Min{\isacharunderscore}{\kern0pt}in\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}subst\ eq{\isacharunderscore}{\kern0pt}Min{\isacharunderscore}{\kern0pt}iff{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}fastforce\ simp{\isacharcolon}{\kern0pt}\ linorder{\isacharunderscore}{\kern0pt}class{\isachardot}{\kern0pt}not{\isacharunderscore}{\kern0pt}less{\isacharparenright}{\kern0pt}{\isacharplus}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{2}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}x\ {\isasymin}\ {\isacharparenleft}{\kern0pt}X\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}Min\ X{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardot}{\kern0pt}\ y\ {\isasymle}\ x{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}x\ {\isasymin}\ X{\isachardot}{\kern0pt}\ y\ {\isasymle}\ x{\isacharbraceright}{\kern0pt}\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}Min\ {\isacharbraceleft}{\kern0pt}x\ {\isasymin}\ X{\isachardot}{\kern0pt}\ y\ {\isasymle}\ x{\isacharbraceright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ Min{\isacharunderscore}{\kern0pt}eq\ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}x\ {\isasymin}\ X\ {\isasymLongrightarrow}\ {\isasymnot}\ x\ {\isacharless}{\kern0pt}\ y{\isachardoublequoteclose}\ \isakeyword{for}\ x\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ False\ Min{\isacharunderscore}{\kern0pt}less{\isacharunderscore}{\kern0pt}iff\ Suc{\isacharparenleft}{\kern0pt}{\isadigit{3}}{\isacharparenright}{\kern0pt}\ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}x\ {\isasymin}\ X{\isachardot}{\kern0pt}\ x\ {\isacharless}{\kern0pt}\ y{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\ \isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}Min\ X\ {\isacharhash}{\kern0pt}\ sorted{\isacharunderscore}{\kern0pt}list{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}set\ {\isacharbraceleft}{\kern0pt}x\ {\isasymin}\ X\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}Min\ X{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ x\ {\isacharless}{\kern0pt}\ y{\isacharbraceright}{\kern0pt}\ {\isacharat}{\kern0pt}\ sorted{\isacharunderscore}{\kern0pt}list{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}set\ {\isacharbraceleft}{\kern0pt}x\ {\isasymin}\ X\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}Min\ X{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ y\ {\isasymle}\ x{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ Min\ X\ {\isacharhash}{\kern0pt}\ sorted{\isacharunderscore}{\kern0pt}list{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}set\ {\isacharbraceleft}{\kern0pt}x\ {\isasymin}\ X\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}Min\ X{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ y\ {\isasymle}\ x{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ Suc\ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isacharequal}{\kern0pt}\ Min\ {\isacharbraceleft}{\kern0pt}x\ {\isasymin}\ X{\isachardot}{\kern0pt}\ y\ {\isasymle}\ x{\isacharbraceright}{\kern0pt}\ {\isacharhash}{\kern0pt}\ sorted{\isacharunderscore}{\kern0pt}list{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}set\ {\isacharparenleft}{\kern0pt}{\isacharbraceleft}{\kern0pt}x\ {\isasymin}\ X{\isachardot}{\kern0pt}\ y\ {\isasymle}\ x{\isacharbraceright}{\kern0pt}\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}Min\ {\isacharbraceleft}{\kern0pt}x\ {\isasymin}\ X{\isachardot}{\kern0pt}\ y\ {\isasymle}\ x{\isacharbraceright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ Min{\isacharunderscore}{\kern0pt}eq\ {\isadigit{2}}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isacharequal}{\kern0pt}\ sorted{\isacharunderscore}{\kern0pt}list{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}set\ {\isacharparenleft}{\kern0pt}{\isacharbraceleft}{\kern0pt}x\ {\isasymin}\ X{\isachardot}{\kern0pt}\ y\ {\isasymle}\ x{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \ \ \ \ \ \isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ Suc\ False\ Min{\isacharunderscore}{\kern0pt}in\ Min{\isacharunderscore}{\kern0pt}eq\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}subst\ sorted{\isacharunderscore}{\kern0pt}list{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}set{\isacharunderscore}{\kern0pt}nonempty{\isacharbrackleft}{\kern0pt}symmetric{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\ fastforce{\isacharplus}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isacharequal}{\kern0pt}\ sorted{\isacharunderscore}{\kern0pt}list{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}set\ {\isacharparenleft}{\kern0pt}{\isacharbraceleft}{\kern0pt}x\ {\isasymin}\ X{\isachardot}{\kern0pt}\ x\ {\isacharless}{\kern0pt}\ y{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharat}{\kern0pt}\ sorted{\isacharunderscore}{\kern0pt}list{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}set\ {\isacharparenleft}{\kern0pt}{\isacharbraceleft}{\kern0pt}x\ {\isasymin}\ X{\isachardot}{\kern0pt}\ y\ {\isasymle}\ x{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ {\isacartoucheopen}{\isacharbraceleft}{\kern0pt}x\ {\isasymin}\ X{\isachardot}{\kern0pt}\ x\ {\isacharless}{\kern0pt}\ y{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacartoucheclose}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{finally}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isacommand{{\isachardot}{\kern0pt}}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{finally}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}case\isacommand{{\isachardot}{\kern0pt}}\isamarkupfalse%
-\isanewline
-\isacommand{qed}\isamarkupfalse%
-\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ sorted{\isacharunderscore}{\kern0pt}list{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}set{\isacharunderscore}{\kern0pt}split{\isacharprime}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}finite\ X{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}sorted{\isacharunderscore}{\kern0pt}list{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}set\ X\ {\isacharequal}{\kern0pt}\ sorted{\isacharunderscore}{\kern0pt}list{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}set\ {\isacharbraceleft}{\kern0pt}x\ {\isasymin}\ X{\isachardot}{\kern0pt}\ x\ {\isasymle}\ y{\isacharbraceright}{\kern0pt}\ {\isacharat}{\kern0pt}\ sorted{\isacharunderscore}{\kern0pt}list{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}set\ {\isacharbraceleft}{\kern0pt}x\ {\isasymin}\ X{\isachardot}{\kern0pt}\ y\ {\isacharless}{\kern0pt}\ x{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ sorted{\isacharunderscore}{\kern0pt}list{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}set{\isacharunderscore}{\kern0pt}split{\isacharbrackleft}{\kern0pt}of\ X{\isacharbrackright}{\kern0pt}\isanewline
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}cases\ {\isachardoublequoteopen}{\isasymexists}x\ {\isasymin}\ X{\isachardot}{\kern0pt}\ y\ {\isacharless}{\kern0pt}\ x{\isachardoublequoteclose}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{case}\isamarkupfalse%
-\ True\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}x\ {\isasymin}\ X{\isachardot}{\kern0pt}\ x\ {\isasymle}\ y{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}x\ {\isasymin}\ X{\isachardot}{\kern0pt}\ x\ {\isacharless}{\kern0pt}\ Min\ {\isacharbraceleft}{\kern0pt}x\ {\isasymin}\ X{\isachardot}{\kern0pt}\ y\ {\isacharless}{\kern0pt}\ x{\isacharbraceright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assms\ True\ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}subst\ Min{\isacharunderscore}{\kern0pt}gr{\isacharunderscore}{\kern0pt}iff{\isacharparenright}{\kern0pt}\ auto\isanewline
-\ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}x\ {\isasymin}\ X{\isachardot}{\kern0pt}\ y\ {\isacharless}{\kern0pt}\ x{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}x\ {\isasymin}\ X{\isachardot}{\kern0pt}\ Min\ {\isacharbraceleft}{\kern0pt}x\ {\isasymin}\ X{\isachardot}{\kern0pt}\ y\ {\isacharless}{\kern0pt}\ x{\isacharbraceright}{\kern0pt}\ {\isasymle}\ \ x{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assms\ True\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}subst\ Min{\isacharunderscore}{\kern0pt}le{\isacharunderscore}{\kern0pt}iff{\isacharparenright}{\kern0pt}\ auto\isanewline
-\ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ sorted{\isacharunderscore}{\kern0pt}list{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}set{\isacharunderscore}{\kern0pt}split{\isacharbrackleft}{\kern0pt}OF\ assms{\isacharcomma}{\kern0pt}\ of\ {\isachardoublequoteopen}Min\ {\isacharbraceleft}{\kern0pt}x\ {\isasymin}\ X{\isachardot}{\kern0pt}\ y\ {\isacharless}{\kern0pt}\ x{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{case}\isamarkupfalse%
-\ False\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isacharasterisk}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}x\ {\isasymin}\ X{\isachardot}{\kern0pt}\ y\ {\isacharless}{\kern0pt}\ x{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}x\ {\isasymin}\ X{\isachardot}{\kern0pt}\ x\ {\isasymle}\ y{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ X{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp\ add{\isacharcolon}{\kern0pt}linorder{\isacharunderscore}{\kern0pt}class{\isachardot}{\kern0pt}not{\isacharunderscore}{\kern0pt}less{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ False\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ {\isacharasterisk}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}fun{\isacharunderscore}{\kern0pt}code{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}fun\ v\ s\ {\isacharequal}{\kern0pt}\ foldl\ {\isacharparenleft}{\kern0pt}{\isasymlambda}v\ s{\isachardot}{\kern0pt}\ v{\isacharparenleft}{\kern0pt}s\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}iter\ v\ s{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ v\ {\isacharparenleft}{\kern0pt}sorted{\isacharunderscore}{\kern0pt}list{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}set\ {\isacharbraceleft}{\kern0pt}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}s{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ s{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}induction\ s\ rule{\isacharcolon}{\kern0pt}\ less{\isacharunderscore}{\kern0pt}induct{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{case}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}less\ s{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}foldl\ {\isacharparenleft}{\kern0pt}{\isasymlambda}v\ s{\isachardot}{\kern0pt}\ v{\isacharparenleft}{\kern0pt}s\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}iter\ v\ s{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ v\ {\isacharparenleft}{\kern0pt}sorted{\isacharunderscore}{\kern0pt}list{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}set\ {\isacharbraceleft}{\kern0pt}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}s{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ s\isanewline
-\ \ \ \ {\isacharequal}{\kern0pt}\ foldl\ {\isacharparenleft}{\kern0pt}{\isasymlambda}v\ s{\isachardot}{\kern0pt}\ v{\isacharparenleft}{\kern0pt}s\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}iter\ v\ s{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ v\ {\isacharparenleft}{\kern0pt}sorted{\isacharunderscore}{\kern0pt}list{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}set\ {\isacharbraceleft}{\kern0pt}x\ {\isasymin}\ {\isacharbraceleft}{\kern0pt}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}s{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ x\ {\isacharless}{\kern0pt}\ s{\isacharbraceright}{\kern0pt}\ {\isacharat}{\kern0pt}\ sorted{\isacharunderscore}{\kern0pt}list{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}set\ {\isacharbraceleft}{\kern0pt}x\ {\isasymin}\ {\isacharbraceleft}{\kern0pt}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}s{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ s\ {\isasymle}\ x{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ s{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}subst\ sorted{\isacharunderscore}{\kern0pt}list{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}set{\isacharunderscore}{\kern0pt}split{\isacharbrackleft}{\kern0pt}of\ {\isacharunderscore}{\kern0pt}\ s{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\ auto\isanewline
-\ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isacharequal}{\kern0pt}\ foldl\ {\isacharparenleft}{\kern0pt}{\isasymlambda}v\ s{\isachardot}{\kern0pt}\ v{\isacharparenleft}{\kern0pt}s\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}iter\ v\ s{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ v\ {\isacharparenleft}{\kern0pt}sorted{\isacharunderscore}{\kern0pt}list{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}set\ {\isacharbraceleft}{\kern0pt}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharless}{\kern0pt}s{\isacharbraceright}{\kern0pt}\ {\isacharat}{\kern0pt}\ sorted{\isacharunderscore}{\kern0pt}list{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}set\ {\isacharbraceleft}{\kern0pt}s{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ s{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}x\ {\isasymin}\ {\isacharbraceleft}{\kern0pt}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}s{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ x\ {\isacharless}{\kern0pt}s{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharless}{\kern0pt}s{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}x\ {\isasymin}\ {\isacharbraceleft}{\kern0pt}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}s{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ s\ {\isasymle}\ x{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}s{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isacharequal}{\kern0pt}\ \ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}iter\ {\isacharparenleft}{\kern0pt}foldl\ {\isacharparenleft}{\kern0pt}{\isasymlambda}v\ s{\isachardot}{\kern0pt}\ v{\isacharparenleft}{\kern0pt}s\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}iter\ v\ s{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ v\ {\isacharparenleft}{\kern0pt}sorted{\isacharunderscore}{\kern0pt}list{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}set\ {\isacharbraceleft}{\kern0pt}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharless}{\kern0pt}s{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ s{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isacharequal}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}fun\ v\ s{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}intro\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}fun{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}GS{\isacharunderscore}{\kern0pt}iter{\isacharbrackleft}{\kern0pt}symmetric{\isacharbrackright}{\kern0pt}{\isacharcomma}{\kern0pt}\ safe{\isacharcomma}{\kern0pt}\ goal{\isacharunderscore}{\kern0pt}cases{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{case}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}\ s{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{assume}\isamarkupfalse%
-\ {\isachardoublequoteopen}s{\isacharprime}{\kern0pt}\ {\isacharless}{\kern0pt}\ s{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ {\isacharasterisk}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}Collect\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}{\isacharless}{\kern0pt}{\isacharparenright}{\kern0pt}\ s{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymnoteq}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\ \isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}x\ {\isasymin}\ {\isacharbraceleft}{\kern0pt}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharless}{\kern0pt}s{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ x\ {\isacharless}{\kern0pt}\ Min\ {\isacharparenleft}{\kern0pt}Collect\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}{\isacharless}{\kern0pt}{\isacharparenright}{\kern0pt}\ s{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}s{\isacharprime}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ leI\ {\isadigit{1}}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp\ add{\isacharcolon}{\kern0pt}\ Min{\isacharunderscore}{\kern0pt}gr{\isacharunderscore}{\kern0pt}iff{\isacharbrackleft}{\kern0pt}OF\ finite{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}x\ {\isasymin}\ {\isacharbraceleft}{\kern0pt}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharless}{\kern0pt}s{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ Min\ {\isacharparenleft}{\kern0pt}Collect\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}{\isacharless}{\kern0pt}{\isacharparenright}{\kern0pt}\ s{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymle}\ x{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}s{\isacharprime}{\kern0pt}{\isacharless}{\kern0pt}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharless}{\kern0pt}s{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isacharasterisk}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp\ add{\isacharcolon}{\kern0pt}\ Min{\isacharunderscore}{\kern0pt}le{\isacharunderscore}{\kern0pt}iff{\isacharbrackleft}{\kern0pt}OF\ finite{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}foldl\ {\isacharparenleft}{\kern0pt}{\isasymlambda}v\ s{\isachardot}{\kern0pt}\ v{\isacharparenleft}{\kern0pt}s\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}iter\ v\ s{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ v\ {\isacharparenleft}{\kern0pt}sorted{\isacharunderscore}{\kern0pt}list{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}set\ {\isacharbraceleft}{\kern0pt}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharless}{\kern0pt}s{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ s{\isacharprime}{\kern0pt}\ \isanewline
-\ \ {\isacharequal}{\kern0pt}\ foldl\ {\isacharparenleft}{\kern0pt}{\isasymlambda}v\ s{\isachardot}{\kern0pt}\ v{\isacharparenleft}{\kern0pt}s\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}iter\ v\ s{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ v\ {\isacharparenleft}{\kern0pt}sorted{\isacharunderscore}{\kern0pt}list{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}set\ {\isacharbraceleft}{\kern0pt}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}s{\isacharprime}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharat}{\kern0pt}\ sorted{\isacharunderscore}{\kern0pt}list{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}set\ {\isacharbraceleft}{\kern0pt}s{\isacharprime}{\kern0pt}{\isacharless}{\kern0pt}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharless}{\kern0pt}s{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ s{\isacharprime}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}subst\ sorted{\isacharunderscore}{\kern0pt}list{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}set{\isacharunderscore}{\kern0pt}split{\isacharbrackleft}{\kern0pt}of\ {\isacharunderscore}{\kern0pt}\ {\isachardoublequoteopen}Min{\isacharbraceleft}{\kern0pt}s{\isachardot}{\kern0pt}\ s{\isacharprime}{\kern0pt}\ {\isacharless}{\kern0pt}\ s{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\ auto\isanewline
-\ \ \ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isacharequal}{\kern0pt}\ \ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}fun\ v\ s{\isacharprime}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isadigit{1}}{\isachardoublequoteclose}\ less{\isachardot}{\kern0pt}IH\ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}subst\ foldl{\isacharunderscore}{\kern0pt}upd{\isacharunderscore}{\kern0pt}notin{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}\ fastforce{\isacharplus}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{finally}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}case\isacommand{{\isachardot}{\kern0pt}}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ intro{\isacharcolon}{\kern0pt}\ foldl{\isacharunderscore}{\kern0pt}upd{\isacharunderscore}{\kern0pt}notin{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{finally}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}case\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}fun{\isacharunderscore}{\kern0pt}code{\isacharprime}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}fun\ v\ s\ {\isacharequal}{\kern0pt}\ foldl\ {\isacharparenleft}{\kern0pt}{\isasymlambda}v\ s{\isachardot}{\kern0pt}\ v{\isacharparenleft}{\kern0pt}s\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}iter\ v\ s{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ v\ {\isacharparenleft}{\kern0pt}sorted{\isacharunderscore}{\kern0pt}list{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}set\ UNIV{\isacharparenright}{\kern0pt}\ s{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}cases\ {\isachardoublequoteopen}s\ {\isacharequal}{\kern0pt}\ Max\ UNIV{\isachardoublequoteclose}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{case}\isamarkupfalse%
-\ True\isanewline
-\ \ \isacommand{then}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\ \isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}fun{\isacharunderscore}{\kern0pt}code\ atMost{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{case}\isamarkupfalse%
-\ False\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isacharasterisk}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}Collect\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}{\isacharless}{\kern0pt}{\isacharparenright}{\kern0pt}\ s{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymnoteq}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ not{\isacharunderscore}{\kern0pt}le\ eq{\isacharunderscore}{\kern0pt}Max{\isacharunderscore}{\kern0pt}iff{\isacharbrackleft}{\kern0pt}OF\ finite{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}x{\isachardot}{\kern0pt}\ x\ {\isacharless}{\kern0pt}\ Min\ {\isacharparenleft}{\kern0pt}Collect\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}{\isacharless}{\kern0pt}{\isacharparenright}{\kern0pt}\ s{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}s{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ Min{\isacharunderscore}{\kern0pt}less{\isacharunderscore}{\kern0pt}iff{\isacharbrackleft}{\kern0pt}OF\ finite\ {\isacharasterisk}{\kern0pt}{\isacharbrackright}{\kern0pt}\ intro{\isacharcolon}{\kern0pt}\ leI{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{then}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ sorted{\isacharunderscore}{\kern0pt}list{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}set{\isacharunderscore}{\kern0pt}split{\isacharbrackleft}{\kern0pt}of\ UNIV\ {\isachardoublequoteopen}Min{\isacharbraceleft}{\kern0pt}s{\isacharprime}{\kern0pt}{\isachardot}{\kern0pt}\ s\ {\isacharless}{\kern0pt}\ s{\isacharprime}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}{\isacharcomma}{\kern0pt}\ OF\ finite{\isacharbrackright}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}fun{\isacharunderscore}{\kern0pt}code\ \isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}subst\ foldl{\isacharunderscore}{\kern0pt}upd{\isacharunderscore}{\kern0pt}notin{\isacharprime}{\kern0pt}{\isacharbrackleft}{\kern0pt}of\ s{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\ auto\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}fun{\isacharunderscore}{\kern0pt}code{\isacharprime}{\kern0pt}{\isacharprime}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}fun\ v\ {\isacharequal}{\kern0pt}\ foldl\ {\isacharparenleft}{\kern0pt}{\isasymlambda}v\ s{\isachardot}{\kern0pt}\ v{\isacharparenleft}{\kern0pt}s\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}iter\ v\ s{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ v\ {\isacharparenleft}{\kern0pt}sorted{\isacharunderscore}{\kern0pt}list{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}set\ UNIV{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}fun{\isacharunderscore}{\kern0pt}code{\isacharprime}{\kern0pt}\ \isacommand{by}\isamarkupfalse%
-\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}elem{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}GS{\isacharunderscore}{\kern0pt}rec\ v\ {\isachardollar}{\kern0pt}\ s\ {\isacharequal}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}fun\ {\isacharparenleft}{\kern0pt}vec{\isacharunderscore}{\kern0pt}nth\ v{\isacharparenright}{\kern0pt}\ s{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}step{\isacharunderscore}{\kern0pt}elem{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}step\ d\ v\ {\isachardollar}{\kern0pt}\ s\ {\isacharequal}{\kern0pt}\ r\ {\isacharparenleft}{\kern0pt}s{\isacharcomma}{\kern0pt}\ d\ s{\isacharparenright}{\kern0pt}\ {\isacharplus}{\kern0pt}\ l\ {\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}{\isasymSum}s{\isacharprime}{\kern0pt}\ {\isacharless}{\kern0pt}\ s{\isachardot}{\kern0pt}\ pmf\ {\isacharparenleft}{\kern0pt}K\ {\isacharparenleft}{\kern0pt}s{\isacharcomma}{\kern0pt}\ d\ s{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ s{\isacharprime}{\kern0pt}\ {\isacharasterisk}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}rec\ v\ {\isachardollar}{\kern0pt}\ s{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharplus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSum}s{\isacharprime}{\kern0pt}\ {\isasymin}\ {\isacharbraceleft}{\kern0pt}s{\isacharprime}{\kern0pt}{\isachardot}{\kern0pt}\ s\ {\isasymle}\ s{\isacharprime}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ pmf\ {\isacharparenleft}{\kern0pt}K\ {\isacharparenleft}{\kern0pt}s{\isacharcomma}{\kern0pt}\ d\ s{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ s{\isacharprime}{\kern0pt}\ {\isacharasterisk}{\kern0pt}\ v\ {\isachardollar}{\kern0pt}\ s{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}step{\isacharunderscore}{\kern0pt}def\ P{\isacharunderscore}{\kern0pt}upper{\isacharunderscore}{\kern0pt}def\ P{\isacharunderscore}{\kern0pt}lower{\isacharunderscore}{\kern0pt}def\ lessThan{\isacharunderscore}{\kern0pt}def\ P{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}elem\ matrix{\isacharunderscore}{\kern0pt}vector{\isacharunderscore}{\kern0pt}mult{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ sum{\isachardot}{\kern0pt}If{\isacharunderscore}{\kern0pt}cases\ algebra{\isacharunderscore}{\kern0pt}simps\ if{\isacharunderscore}{\kern0pt}distrib{\isacharbrackleft}{\kern0pt}of\ {\isachardoublequoteopen}{\isasymlambda}x{\isachardot}{\kern0pt}\ {\isacharunderscore}{\kern0pt}\ {\isachardollar}{\kern0pt}\ {\isacharunderscore}{\kern0pt}\ {\isacharasterisk}{\kern0pt}\ x{\isachardoublequoteclose}{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}step{\isacharunderscore}{\kern0pt}act{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}d\ {\isasymin}D\isactrlsub D{\isachardoublequoteclose}\ {\isachardoublequoteopen}is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max\ {\isacharparenleft}{\kern0pt}{\isasymlambda}a{\isachardot}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}step\ {\isacharparenleft}{\kern0pt}d{\isacharprime}{\kern0pt}{\isacharparenleft}{\kern0pt}s\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ a{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ v\ {\isachardollar}{\kern0pt}\ s{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymlambda}a{\isachardot}{\kern0pt}\ a\ {\isasymin}A\ s{\isacharparenright}{\kern0pt}\ a{\isachardoublequoteclose}\ \isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max\ {\isacharparenleft}{\kern0pt}{\isasymlambda}d{\isachardot}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}step\ d\ v\ {\isachardollar}{\kern0pt}\ s{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymlambda}d{\isachardot}{\kern0pt}\ d\ {\isasymin}D\isactrlsub D{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}d{\isacharparenleft}{\kern0pt}s\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ a{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ assms\isanewline
-\ \ \isacommand{unfolding}\isamarkupfalse%
-\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}step{\isacharunderscore}{\kern0pt}elem\ is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}linorder\ is{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}step{\isacharunderscore}{\kern0pt}act{\isacharprime}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}d\ {\isasymin}D\isactrlsub D{\isachardoublequoteclose}\ {\isachardoublequoteopen}is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max\ {\isacharparenleft}{\kern0pt}{\isasymlambda}a{\isachardot}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}step\ {\isacharparenleft}{\kern0pt}d{\isacharprime}{\kern0pt}{\isacharparenleft}{\kern0pt}s\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ a{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ v\ {\isachardollar}{\kern0pt}\ s{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymlambda}a{\isachardot}{\kern0pt}\ a\ {\isasymin}A\ s{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}d\ s{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max\ {\isacharparenleft}{\kern0pt}{\isasymlambda}d{\isachardot}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}step\ d\ v\ {\isachardollar}{\kern0pt}\ s{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymlambda}d{\isachardot}{\kern0pt}\ d\ {\isasymin}D\isactrlsub D{\isacharparenright}{\kern0pt}\ d{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}step{\isacharunderscore}{\kern0pt}act{\isacharbrackleft}{\kern0pt}OF\ assms{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ fastforce%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\isanewline
-\ \ is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}GS{\isacharunderscore}{\kern0pt}rec{\isacharcolon}{\kern0pt}\ \isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}{\isasymAnd}s{\isachardot}{\kern0pt}\ is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max\ {\isacharparenleft}{\kern0pt}{\isasymlambda}d{\isachardot}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}step\ d\ v\ {\isachardollar}{\kern0pt}\ s{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymlambda}d{\isachardot}{\kern0pt}\ d\ {\isasymin}\ D\isactrlsub D{\isacharparenright}{\kern0pt}\ d{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}GS{\isacharunderscore}{\kern0pt}rec\ v\ {\isacharequal}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}step\ d\ v{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}SUP{\isacharbrackleft}{\kern0pt}OF\ assms{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ vec{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}iff\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}vec\ {\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\isanewline
-\ \ is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}GS{\isacharunderscore}{\kern0pt}rec{\isacharprime}{\kern0pt}{\isacharcolon}{\kern0pt}\ \isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max\ {\isacharparenleft}{\kern0pt}{\isasymlambda}d{\isachardot}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}step\ d\ v\ {\isachardollar}{\kern0pt}\ s{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymlambda}d{\isachardot}{\kern0pt}\ d\ {\isasymin}\ D\isactrlsub D{\isacharparenright}{\kern0pt}\ d{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}GS{\isacharunderscore}{\kern0pt}rec\ v\ {\isachardollar}{\kern0pt}\ s\ {\isacharequal}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}step\ d\ v\ {\isachardollar}{\kern0pt}\ s{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ assms\ \isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}vec\ arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}SUP{\isacharbrackleft}{\kern0pt}symmetric{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\isanewline
-\ \ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}GS{\isacharunderscore}{\kern0pt}inv{\isacharcolon}{\kern0pt}\ \isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}{\isasymAnd}s{\isachardot}{\kern0pt}\ is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max\ {\isacharparenleft}{\kern0pt}{\isasymlambda}d{\isachardot}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}step\ d\ v\ {\isachardollar}{\kern0pt}\ s{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymlambda}d{\isachardot}{\kern0pt}\ d\ {\isasymin}\ D\isactrlsub D{\isacharparenright}{\kern0pt}\ d{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}GS{\isacharunderscore}{\kern0pt}rec\ v\ {\isacharequal}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}inv\ d\ v{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}GS{\isacharunderscore}{\kern0pt}rec\ v\ {\isacharequal}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}step\ d\ v{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}GS{\isacharunderscore}{\kern0pt}rec{\isacharbrackleft}{\kern0pt}OF\ assms{\isacharbrackright}{\kern0pt}\ \isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}GS{\isacharunderscore}{\kern0pt}rec\ v\ {\isacharequal}{\kern0pt}\ r{\isacharunderscore}{\kern0pt}vec\ d\ {\isacharplus}{\kern0pt}\ R{\isacharunderscore}{\kern0pt}mat\ d\ {\isacharasterisk}{\kern0pt}v\ v\ {\isacharplus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ P{\isacharunderscore}{\kern0pt}lower\ d{\isacharparenright}{\kern0pt}\ {\isacharasterisk}{\kern0pt}v\ GS{\isacharunderscore}{\kern0pt}rec\ v{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ R{\isacharunderscore}{\kern0pt}mat{\isacharunderscore}{\kern0pt}def\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}step{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ scaleR{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}vector{\isacharunderscore}{\kern0pt}assoc\ algebra{\isacharunderscore}{\kern0pt}simps{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}Q{\isacharunderscore}{\kern0pt}mat\ d\ {\isacharasterisk}{\kern0pt}v\ GS{\isacharunderscore}{\kern0pt}rec\ v\ {\isacharequal}{\kern0pt}\ r{\isacharunderscore}{\kern0pt}vec\ d\ {\isacharplus}{\kern0pt}\ R{\isacharunderscore}{\kern0pt}mat\ d\ {\isacharasterisk}{\kern0pt}v\ v{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ Q{\isacharunderscore}{\kern0pt}mat{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharcomma}{\kern0pt}\ lifting{\isacharparenright}{\kern0pt}\ add{\isacharunderscore}{\kern0pt}diff{\isacharunderscore}{\kern0pt}cancel\ matrix{\isacharunderscore}{\kern0pt}vector{\isacharunderscore}{\kern0pt}mult{\isacharunderscore}{\kern0pt}diff{\isacharunderscore}{\kern0pt}rdistrib\ matrix{\isacharunderscore}{\kern0pt}vector{\isacharunderscore}{\kern0pt}mul{\isacharunderscore}{\kern0pt}lid{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}inv\ {\isacharparenleft}{\kern0pt}Q{\isacharunderscore}{\kern0pt}mat\ d{\isacharparenright}{\kern0pt}\ {\isacharasterisk}{\kern0pt}{\isacharasterisk}{\kern0pt}\ Q{\isacharunderscore}{\kern0pt}mat\ d{\isacharparenright}{\kern0pt}\ {\isacharasterisk}{\kern0pt}v\ GS{\isacharunderscore}{\kern0pt}rec\ v\ {\isacharequal}{\kern0pt}\ matrix{\isacharunderscore}{\kern0pt}inv\ {\isacharparenleft}{\kern0pt}Q{\isacharunderscore}{\kern0pt}mat\ d{\isacharparenright}{\kern0pt}\ {\isacharasterisk}{\kern0pt}v\ {\isacharparenleft}{\kern0pt}r{\isacharunderscore}{\kern0pt}vec\ d\ {\isacharplus}{\kern0pt}\ R{\isacharunderscore}{\kern0pt}mat\ d\ {\isacharasterisk}{\kern0pt}v\ v{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ matrix{\isacharunderscore}{\kern0pt}vector{\isacharunderscore}{\kern0pt}mul{\isacharunderscore}{\kern0pt}assoc{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}GS{\isacharunderscore}{\kern0pt}rec\ v\ {\isacharequal}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}inv\ d\ v{\isachardoublequoteclose}\ \isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ splitting{\isacharunderscore}{\kern0pt}gauss\isanewline
-\ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ GS{\isacharunderscore}{\kern0pt}inv{\isacharunderscore}{\kern0pt}def\ is{\isacharunderscore}{\kern0pt}splitting{\isacharunderscore}{\kern0pt}blin{\isacharunderscore}{\kern0pt}def{\isacharprime}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}subst\ {\isacharparenleft}{\kern0pt}asm{\isacharparenright}{\kern0pt}\ matrix{\isacharunderscore}{\kern0pt}inv{\isacharunderscore}{\kern0pt}left{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}fastforce\ intro{\isacharcolon}{\kern0pt}\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}inverse{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharplus}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\isanewline
-\ \ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}step{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}GS{\isacharunderscore}{\kern0pt}inv{\isacharcolon}{\kern0pt}\ \isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}{\isasymAnd}s{\isachardot}{\kern0pt}\ is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max\ {\isacharparenleft}{\kern0pt}{\isasymlambda}d{\isachardot}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}step\ d\ v\ {\isachardollar}{\kern0pt}\ s{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymlambda}d{\isachardot}{\kern0pt}\ d\ {\isasymin}\ D\isactrlsub D{\isacharparenright}{\kern0pt}\ d{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}step\ d\ v\ {\isacharequal}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}inv\ d\ v{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}GS{\isacharunderscore}{\kern0pt}inv{\isacharbrackleft}{\kern0pt}OF\ assms{\isacharbrackright}{\kern0pt}\ is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}GS{\isacharunderscore}{\kern0pt}rec{\isacharbrackleft}{\kern0pt}OF\ assms{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ strict{\isacharunderscore}{\kern0pt}lower{\isacharunderscore}{\kern0pt}triangular{\isacharunderscore}{\kern0pt}mat{\isacharunderscore}{\kern0pt}mult{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}strict{\isacharunderscore}{\kern0pt}lower{\isacharunderscore}{\kern0pt}triangular{\isacharunderscore}{\kern0pt}mat\ M{\isachardoublequoteclose}\ {\isachardoublequoteopen}{\isasymAnd}i{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ j\ {\isasymLongrightarrow}\ v\ {\isachardollar}{\kern0pt}\ i\ {\isacharequal}{\kern0pt}\ v{\isacharprime}{\kern0pt}\ {\isachardollar}{\kern0pt}\ i{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}M\ {\isacharasterisk}{\kern0pt}v\ v{\isacharparenright}{\kern0pt}\ {\isachardollar}{\kern0pt}\ j\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}M\ {\isacharasterisk}{\kern0pt}v\ v{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isachardollar}{\kern0pt}\ j{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}M\ {\isacharasterisk}{\kern0pt}v\ v{\isacharparenright}{\kern0pt}\ {\isachardollar}{\kern0pt}\ j\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSum}i{\isasymin}UNIV{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}if\ j\ {\isasymle}\ i\ then\ {\isadigit{0}}\ else\ \ M\ {\isachardollar}{\kern0pt}\ j\ {\isachardollar}{\kern0pt}\ i\ {\isacharasterisk}{\kern0pt}\ v\ {\isachardollar}{\kern0pt}\ i{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assms\ \isacommand{unfolding}\isamarkupfalse%
-\ strict{\isacharunderscore}{\kern0pt}lower{\isacharunderscore}{\kern0pt}triangular{\isacharunderscore}{\kern0pt}mat{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ matrix{\isacharunderscore}{\kern0pt}vector{\isacharunderscore}{\kern0pt}mult{\isacharunderscore}{\kern0pt}def\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ sum{\isachardot}{\kern0pt}cong{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSum}i{\isasymin}UNIV{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}if\ j\ {\isasymle}\ i\ then\ {\isadigit{0}}\ else\ \ M\ {\isachardollar}{\kern0pt}\ j\ {\isachardollar}{\kern0pt}\ i\ {\isacharasterisk}{\kern0pt}\ v{\isacharprime}{\kern0pt}\ {\isachardollar}{\kern0pt}\ i{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assms\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ sum{\isachardot}{\kern0pt}cong{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}M\ {\isacharasterisk}{\kern0pt}v\ v{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isachardollar}{\kern0pt}\ j{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assms\ \isacommand{unfolding}\isamarkupfalse%
-\ strict{\isacharunderscore}{\kern0pt}lower{\isacharunderscore}{\kern0pt}triangular{\isacharunderscore}{\kern0pt}mat{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ matrix{\isacharunderscore}{\kern0pt}vector{\isacharunderscore}{\kern0pt}mult{\isacharunderscore}{\kern0pt}def\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ sum{\isachardot}{\kern0pt}cong{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{finally}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isacommand{{\isachardot}{\kern0pt}}\isamarkupfalse%
-\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ Q{\isacharunderscore}{\kern0pt}mat{\isacharunderscore}{\kern0pt}invertible{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}invertible\ {\isacharparenleft}{\kern0pt}Q{\isacharunderscore}{\kern0pt}mat\ d{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}meson\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}inverse{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}\ is{\isacharunderscore}{\kern0pt}splitting{\isacharunderscore}{\kern0pt}blin{\isacharunderscore}{\kern0pt}def{\isacharprime}{\kern0pt}\ splitting{\isacharunderscore}{\kern0pt}gauss{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ GS{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}GS{\isacharunderscore}{\kern0pt}inv{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}{\isasymAnd}s{\isachardot}{\kern0pt}\ s\ {\isasymle}\ k\ {\isasymLongrightarrow}\ is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max\ {\isacharparenleft}{\kern0pt}{\isasymlambda}d{\isachardot}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}step\ d\ v\ {\isachardollar}{\kern0pt}\ s{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymlambda}d{\isachardot}{\kern0pt}\ d\ {\isasymin}\ D\isactrlsub D{\isacharparenright}{\kern0pt}\ d{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}s\ {\isasymle}\ k{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}step\ d\ v\ {\isachardollar}{\kern0pt}\ s\ {\isacharequal}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}inv\ d\ v\ {\isachardollar}{\kern0pt}\ s{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isacharasterisk}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}GS{\isacharunderscore}{\kern0pt}rec\ v\ {\isachardollar}{\kern0pt}\ s\ {\isacharequal}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}step\ d\ v\ {\isachardollar}{\kern0pt}\ s{\isachardoublequoteclose}\ \isakeyword{if}\ {\isachardoublequoteopen}s\ {\isasymle}\ k{\isachardoublequoteclose}\ \isakeyword{for}\ s\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assms\ is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}GS{\isacharunderscore}{\kern0pt}rec{\isacharprime}{\kern0pt}\ that\ \isacommand{by}\isamarkupfalse%
-\ presburger\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}GS{\isacharunderscore}{\kern0pt}rec\ v\ {\isachardollar}{\kern0pt}\ s\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}r{\isacharunderscore}{\kern0pt}vec\ d\ {\isacharplus}{\kern0pt}\ R{\isacharunderscore}{\kern0pt}mat\ d\ {\isacharasterisk}{\kern0pt}v\ v\ {\isacharplus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ P{\isacharunderscore}{\kern0pt}lower\ d{\isacharparenright}{\kern0pt}\ {\isacharasterisk}{\kern0pt}v\ GS{\isacharunderscore}{\kern0pt}rec\ v{\isacharparenright}{\kern0pt}\ {\isachardollar}{\kern0pt}\ s{\isachardoublequoteclose}\ \isakeyword{if}\ {\isachardoublequoteopen}s\ {\isasymle}\ k{\isachardoublequoteclose}\ \isakeyword{for}\ s\isanewline
-\ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ R{\isacharunderscore}{\kern0pt}mat{\isacharunderscore}{\kern0pt}def\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}step{\isacharunderscore}{\kern0pt}def\ \isacommand{using}\isamarkupfalse%
-\ that\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ scaleR{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}vector{\isacharunderscore}{\kern0pt}assoc\ pth{\isacharunderscore}{\kern0pt}{\isadigit{6}}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}Q{\isacharunderscore}{\kern0pt}mat\ d\ {\isacharasterisk}{\kern0pt}v\ GS{\isacharunderscore}{\kern0pt}rec\ v{\isacharparenright}{\kern0pt}\ {\isachardollar}{\kern0pt}\ s\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}r{\isacharunderscore}{\kern0pt}vec\ d\ {\isacharplus}{\kern0pt}\ R{\isacharunderscore}{\kern0pt}mat\ d\ {\isacharasterisk}{\kern0pt}v\ v{\isacharparenright}{\kern0pt}\ {\isachardollar}{\kern0pt}\ s{\isachardoublequoteclose}\ \isakeyword{if}\ {\isachardoublequoteopen}s\ {\isasymle}\ k{\isachardoublequoteclose}\ \isakeyword{for}\ s\isanewline
-\ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ Q{\isacharunderscore}{\kern0pt}mat{\isacharunderscore}{\kern0pt}def\ \isacommand{using}\isamarkupfalse%
-\ that\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ matrix{\isacharunderscore}{\kern0pt}vector{\isacharunderscore}{\kern0pt}mult{\isacharunderscore}{\kern0pt}diff{\isacharunderscore}{\kern0pt}rdistrib{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}inv\ {\isacharparenleft}{\kern0pt}Q{\isacharunderscore}{\kern0pt}mat\ d{\isacharparenright}{\kern0pt}\ {\isacharasterisk}{\kern0pt}v\ {\isacharparenleft}{\kern0pt}Q{\isacharunderscore}{\kern0pt}mat\ d\ {\isacharasterisk}{\kern0pt}v\ GS{\isacharunderscore}{\kern0pt}rec\ v{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isachardollar}{\kern0pt}\ s\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}inv\ {\isacharparenleft}{\kern0pt}Q{\isacharunderscore}{\kern0pt}mat\ d{\isacharparenright}{\kern0pt}\ {\isacharasterisk}{\kern0pt}v\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}r{\isacharunderscore}{\kern0pt}vec\ d\ {\isacharplus}{\kern0pt}\ R{\isacharunderscore}{\kern0pt}mat\ d\ {\isacharasterisk}{\kern0pt}v\ v{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isachardollar}{\kern0pt}\ s{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assms\ lt{\isacharunderscore}{\kern0pt}Q{\isacharunderscore}{\kern0pt}inv\ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ intro{\isacharcolon}{\kern0pt}\ lower{\isacharunderscore}{\kern0pt}triangular{\isacharunderscore}{\kern0pt}mat{\isacharunderscore}{\kern0pt}mult{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}step\ d\ v\ {\isachardollar}{\kern0pt}\ s\ {\isacharequal}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}inv\ d\ v\ {\isachardollar}{\kern0pt}\ s{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ GS{\isacharunderscore}{\kern0pt}inv{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ matrix{\isacharunderscore}{\kern0pt}inv{\isacharunderscore}{\kern0pt}left{\isacharbrackleft}{\kern0pt}OF\ Q{\isacharunderscore}{\kern0pt}mat{\isacharunderscore}{\kern0pt}invertible{\isacharbrackright}{\kern0pt}\ assms\ \ {\isacharasterisk}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ matrix{\isacharunderscore}{\kern0pt}vector{\isacharunderscore}{\kern0pt}mul{\isacharunderscore}{\kern0pt}assoc{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}GS{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}splitting{\isacharprime}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}{\isasymAnd}s{\isachardot}{\kern0pt}\ s\ {\isasymle}\ k\ {\isasymLongrightarrow}\ is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max\ {\isacharparenleft}{\kern0pt}{\isasymlambda}d{\isachardot}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}step\ d\ v\ {\isachardollar}{\kern0pt}\ s{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymlambda}d{\isachardot}{\kern0pt}\ d\ {\isasymin}\ D\isactrlsub D{\isacharparenright}{\kern0pt}\ d{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}s\ {\isasymle}\ k{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max\ {\isacharparenleft}{\kern0pt}{\isasymlambda}d{\isachardot}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}inv\ d\ v\ {\isachardollar}{\kern0pt}\ s{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymlambda}d{\isachardot}{\kern0pt}\ d\ {\isasymin}\ D\isactrlsub D{\isacharparenright}{\kern0pt}\ d{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ assms\isanewline
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}induction\ k\ arbitrary{\isacharcolon}{\kern0pt}\ s\ rule{\isacharcolon}{\kern0pt}\ less{\isacharunderscore}{\kern0pt}induct{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{case}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}less\ x{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ d{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}d\ {\isasymin}\ D\isactrlsub D{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assms{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}\ is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}linorderD\ \isacommand{by}\isamarkupfalse%
-\ fast\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max\ {\isacharparenleft}{\kern0pt}{\isasymlambda}a{\isachardot}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}inv\ {\isacharparenleft}{\kern0pt}d{\isacharparenleft}{\kern0pt}s\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ a{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ v\ {\isachardollar}{\kern0pt}\ s{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymlambda}a{\isachardot}{\kern0pt}\ a\ {\isasymin}\ A\ s{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}d\ s{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{if}\ {\isachardoublequoteopen}s\ {\isasymle}\ x{\isachardoublequoteclose}\ \isakeyword{for}\ s\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max\ {\isacharparenleft}{\kern0pt}{\isasymlambda}a{\isachardot}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}step\ {\isacharparenleft}{\kern0pt}d{\isacharparenleft}{\kern0pt}s\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ a{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ v\ {\isachardollar}{\kern0pt}\ s{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymlambda}a{\isachardot}{\kern0pt}\ a\ {\isasymin}\ A\ s{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}d\ s{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ less{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isacharbrackleft}{\kern0pt}OF\ that{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ is{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det{\isacharunderscore}{\kern0pt}def\ is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}linorder\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ {\isacharasterisk}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max\ {\isacharparenleft}{\kern0pt}{\isasymlambda}a{\isachardot}{\kern0pt}\ r\ {\isacharparenleft}{\kern0pt}s{\isacharcomma}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isacharplus}{\kern0pt}\ l\ {\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}P{\isacharunderscore}{\kern0pt}lower\ {\isacharparenleft}{\kern0pt}d{\isacharparenleft}{\kern0pt}s\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ a{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharasterisk}{\kern0pt}v\ GS{\isacharunderscore}{\kern0pt}rec\ v{\isacharparenright}{\kern0pt}\ {\isachardollar}{\kern0pt}\ s\ {\isacharplus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}P{\isacharunderscore}{\kern0pt}upper\ {\isacharparenleft}{\kern0pt}d{\isacharparenleft}{\kern0pt}s\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ a{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharasterisk}{\kern0pt}v\ v{\isacharparenright}{\kern0pt}\ {\isachardollar}{\kern0pt}\ s{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymlambda}a{\isachardot}{\kern0pt}\ a\ {\isasymin}\ A\ s{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}d\ s{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}step{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max\ {\isacharparenleft}{\kern0pt}{\isasymlambda}a{\isachardot}{\kern0pt}\ r\ {\isacharparenleft}{\kern0pt}s{\isacharcomma}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isacharplus}{\kern0pt}\ l\ {\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}P{\isacharunderscore}{\kern0pt}lower\ {\isacharparenleft}{\kern0pt}d{\isacharparenleft}{\kern0pt}s\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ a{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharasterisk}{\kern0pt}v\ GS{\isacharunderscore}{\kern0pt}inv\ {\isacharparenleft}{\kern0pt}d{\isacharparenleft}{\kern0pt}s\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ a{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ v{\isacharparenright}{\kern0pt}\ {\isachardollar}{\kern0pt}\ s\ {\isacharplus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}P{\isacharunderscore}{\kern0pt}upper\ {\isacharparenleft}{\kern0pt}d{\isacharparenleft}{\kern0pt}s\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ a{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharasterisk}{\kern0pt}v\ v{\isacharparenright}{\kern0pt}\ {\isachardollar}{\kern0pt}\ s{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymlambda}a{\isachardot}{\kern0pt}\ a\ {\isasymin}\ A\ s{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}d\ s{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}P{\isacharunderscore}{\kern0pt}lower\ {\isacharparenleft}{\kern0pt}d{\isacharparenleft}{\kern0pt}s\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ a{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharasterisk}{\kern0pt}v\ GS{\isacharunderscore}{\kern0pt}rec\ v{\isacharparenright}{\kern0pt}\ {\isachardollar}{\kern0pt}\ s\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}P{\isacharunderscore}{\kern0pt}lower\ {\isacharparenleft}{\kern0pt}d{\isacharparenleft}{\kern0pt}s\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ a{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharasterisk}{\kern0pt}v\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}step\ d\ v{\isacharparenright}{\kern0pt}\ {\isachardollar}{\kern0pt}\ s{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{for}\ a\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}GS{\isacharunderscore}{\kern0pt}rec{\isacharprime}{\kern0pt}\ less{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}\ that\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ lower{\isacharunderscore}{\kern0pt}triangular{\isacharunderscore}{\kern0pt}mat{\isacharunderscore}{\kern0pt}mult{\isacharbrackleft}{\kern0pt}OF\ lt{\isacharunderscore}{\kern0pt}P{\isacharunderscore}{\kern0pt}lower{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}P{\isacharunderscore}{\kern0pt}lower\ {\isacharparenleft}{\kern0pt}d{\isacharparenleft}{\kern0pt}s\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ a{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharasterisk}{\kern0pt}v\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}step\ d\ v{\isacharparenright}{\kern0pt}\ {\isachardollar}{\kern0pt}\ s{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}P{\isacharunderscore}{\kern0pt}lower\ {\isacharparenleft}{\kern0pt}d{\isacharparenleft}{\kern0pt}s\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ a{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharasterisk}{\kern0pt}v\ GS{\isacharunderscore}{\kern0pt}inv\ d\ v{\isacharparenright}{\kern0pt}\ {\isachardollar}{\kern0pt}\ s{\isachardoublequoteclose}\ \isakeyword{for}\ a\ \ \ \ \ \ \isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ less{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}\ that\ GS{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}GS{\isacharunderscore}{\kern0pt}inv\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}fastforce\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ lower{\isacharunderscore}{\kern0pt}triangular{\isacharunderscore}{\kern0pt}mat{\isacharunderscore}{\kern0pt}mult{\isacharbrackleft}{\kern0pt}OF\ lt{\isacharunderscore}{\kern0pt}P{\isacharunderscore}{\kern0pt}lower{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}P{\isacharunderscore}{\kern0pt}lower\ {\isacharparenleft}{\kern0pt}d{\isacharparenleft}{\kern0pt}s\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ a{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharasterisk}{\kern0pt}v\ GS{\isacharunderscore}{\kern0pt}inv\ d\ v{\isacharparenright}{\kern0pt}\ {\isachardollar}{\kern0pt}\ s\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}P{\isacharunderscore}{\kern0pt}lower\ {\isacharparenleft}{\kern0pt}d{\isacharparenleft}{\kern0pt}s\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ a{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharasterisk}{\kern0pt}v\ GS{\isacharunderscore}{\kern0pt}inv\ {\isacharparenleft}{\kern0pt}d{\isacharparenleft}{\kern0pt}s\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ a{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ v{\isacharparenright}{\kern0pt}\ {\isachardollar}{\kern0pt}\ s{\isachardoublequoteclose}\ \isakeyword{for}\ a\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ GS{\isacharunderscore}{\kern0pt}indep{\isacharunderscore}{\kern0pt}high{\isacharunderscore}{\kern0pt}states{\isacharbrackleft}{\kern0pt}of\ {\isacharunderscore}{\kern0pt}\ d\ {\isachardoublequoteopen}d{\isacharparenleft}{\kern0pt}s\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ a{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}{\isacharbrackright}{\kern0pt}\ \isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}fastforce\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ strict{\isacharunderscore}{\kern0pt}lower{\isacharunderscore}{\kern0pt}triangular{\isacharunderscore}{\kern0pt}mat{\isacharunderscore}{\kern0pt}mult{\isacharbrackleft}{\kern0pt}OF\ slt{\isacharunderscore}{\kern0pt}P{\isacharunderscore}{\kern0pt}lower{\isacharbrackright}{\kern0pt}\ dest{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ leD{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isacharasterisk}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max\ {\isacharparenleft}{\kern0pt}{\isasymlambda}a{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}r{\isacharunderscore}{\kern0pt}vec\ {\isacharparenleft}{\kern0pt}d{\isacharparenleft}{\kern0pt}s\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ a{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharplus}{\kern0pt}\ l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}P{\isacharunderscore}{\kern0pt}lower\ {\isacharparenleft}{\kern0pt}d{\isacharparenleft}{\kern0pt}s\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ a{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharasterisk}{\kern0pt}v\ GS{\isacharunderscore}{\kern0pt}inv\ {\isacharparenleft}{\kern0pt}d{\isacharparenleft}{\kern0pt}s\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ a{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ v{\isacharparenright}{\kern0pt}\ {\isacharplus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}P{\isacharunderscore}{\kern0pt}upper\ {\isacharparenleft}{\kern0pt}d{\isacharparenleft}{\kern0pt}s\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ a{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharasterisk}{\kern0pt}v\ v{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isachardollar}{\kern0pt}\ s{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymlambda}a{\isachardot}{\kern0pt}\ a\ {\isasymin}\ A\ s{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}d\ s{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ {\isacharasterisk}{\kern0pt}{\isacharasterisk}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max\ {\isacharparenleft}{\kern0pt}{\isasymlambda}a{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}r{\isacharunderscore}{\kern0pt}vec\ {\isacharparenleft}{\kern0pt}d{\isacharparenleft}{\kern0pt}s\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ a{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharplus}{\kern0pt}\ R{\isacharunderscore}{\kern0pt}mat\ {\isacharparenleft}{\kern0pt}d{\isacharparenleft}{\kern0pt}s\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ a{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharasterisk}{\kern0pt}v\ v{\isacharparenright}{\kern0pt}\ {\isacharplus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ P{\isacharunderscore}{\kern0pt}lower\ {\isacharparenleft}{\kern0pt}d{\isacharparenleft}{\kern0pt}s\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ a{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharasterisk}{\kern0pt}v\ GS{\isacharunderscore}{\kern0pt}inv\ {\isacharparenleft}{\kern0pt}d{\isacharparenleft}{\kern0pt}s\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ a{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ v{\isacharparenright}{\kern0pt}\ {\isacharparenright}{\kern0pt}\ {\isachardollar}{\kern0pt}\ s{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymlambda}a{\isachardot}{\kern0pt}\ a\ {\isasymin}\ A\ s{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}d\ s{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ R{\isacharunderscore}{\kern0pt}mat{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ algebra{\isacharunderscore}{\kern0pt}simps\ \ scaleR{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}vector{\isacharunderscore}{\kern0pt}assoc{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{proof}\isamarkupfalse%
-{\isacharminus}{\kern0pt}\ \isanewline
-\ \ \ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}r{\isacharunderscore}{\kern0pt}vec\ d\ {\isacharplus}{\kern0pt}\ R{\isacharunderscore}{\kern0pt}mat\ d\ {\isacharasterisk}{\kern0pt}v\ v{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ Q{\isacharunderscore}{\kern0pt}mat\ d\ {\isacharasterisk}{\kern0pt}v\ {\isacharparenleft}{\kern0pt}GS{\isacharunderscore}{\kern0pt}inv\ d\ v{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{for}\ d\ v\isanewline
-\ \ \ \ \ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ GS{\isacharunderscore}{\kern0pt}inv{\isacharunderscore}{\kern0pt}def\ matrix{\isacharunderscore}{\kern0pt}vector{\isacharunderscore}{\kern0pt}mul{\isacharunderscore}{\kern0pt}assoc\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharcomma}{\kern0pt}\ lifting{\isacharparenright}{\kern0pt}\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}inverse{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}\ is{\isacharunderscore}{\kern0pt}splitting{\isacharunderscore}{\kern0pt}blin{\isacharunderscore}{\kern0pt}def{\isacharprime}{\kern0pt}\ matrix{\isacharunderscore}{\kern0pt}inv{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}\ matrix{\isacharunderscore}{\kern0pt}vector{\isacharunderscore}{\kern0pt}mul{\isacharunderscore}{\kern0pt}lid\ splitting{\isacharunderscore}{\kern0pt}gauss{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}r{\isacharunderscore}{\kern0pt}vec\ d\ {\isacharplus}{\kern0pt}\ R{\isacharunderscore}{\kern0pt}mat\ d\ {\isacharasterisk}{\kern0pt}v\ v{\isacharparenright}{\kern0pt}\ {\isacharplus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ P{\isacharunderscore}{\kern0pt}lower\ d{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharasterisk}{\kern0pt}v\ GS{\isacharunderscore}{\kern0pt}inv\ d\ v{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}inv\ d\ v{\isachardoublequoteclose}\ \isakeyword{for}\ d\isanewline
-\ \ \ \ \ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ Q{\isacharunderscore}{\kern0pt}mat{\isacharunderscore}{\kern0pt}def\ \isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ matrix{\isacharunderscore}{\kern0pt}vector{\isacharunderscore}{\kern0pt}mult{\isacharunderscore}{\kern0pt}diff{\isacharunderscore}{\kern0pt}rdistrib{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\ \isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isacharasterisk}{\kern0pt}{\isacharasterisk}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ presburger\isanewline
-\ \ \ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}case\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ less\ d\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}fastforce\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ is{\isacharunderscore}{\kern0pt}am{\isacharunderscore}{\kern0pt}GS{\isacharunderscore}{\kern0pt}inv{\isacharunderscore}{\kern0pt}extend{\isacharbrackleft}{\kern0pt}of\ x\ v\ d\ {\isachardoublequoteopen}d\ x{\isachardoublequoteclose}\ s{\isacharcomma}{\kern0pt}\ unfolded\ fun{\isacharunderscore}{\kern0pt}upd{\isacharunderscore}{\kern0pt}triv{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ is{\isacharunderscore}{\kern0pt}am{\isacharunderscore}{\kern0pt}GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}step{\isacharunderscore}{\kern0pt}indep{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}d\ s\ {\isacharequal}{\kern0pt}\ d{\isacharprime}{\kern0pt}\ s{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max\ {\isacharparenleft}{\kern0pt}{\isasymlambda}d{\isachardot}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}step\ d\ v\ {\isachardollar}{\kern0pt}\ s{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymlambda}d{\isachardot}{\kern0pt}\ d\ {\isasymin}\ D\isactrlsub D{\isacharparenright}{\kern0pt}\ d{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}GS{\isacharunderscore}{\kern0pt}rec\ v\ {\isachardollar}{\kern0pt}\ s\ {\isacharequal}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}step\ d{\isacharprime}{\kern0pt}\ v\ {\isachardollar}{\kern0pt}\ s{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}GS{\isacharunderscore}{\kern0pt}rec\ v\ {\isachardollar}{\kern0pt}\ s\ {\isacharequal}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}step\ d\ v\ {\isachardollar}{\kern0pt}\ s{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}GS{\isacharunderscore}{\kern0pt}rec{\isacharprime}{\kern0pt}\ assms{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}\ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}step\ d\ v\ {\isachardollar}{\kern0pt}\ s\ {\isacharequal}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}step\ d{\isacharprime}{\kern0pt}\ v\ {\isachardollar}{\kern0pt}\ s{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}step{\isacharunderscore}{\kern0pt}elem\ assms{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}\ \isacommand{by}\isamarkupfalse%
-\ fastforce\isanewline
-\ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ is{\isacharunderscore}{\kern0pt}am{\isacharunderscore}{\kern0pt}GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}step{\isacharunderscore}{\kern0pt}indep{\isacharprime}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}d\ s\ {\isacharequal}{\kern0pt}\ d{\isacharprime}{\kern0pt}\ s{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max\ {\isacharparenleft}{\kern0pt}{\isasymlambda}d{\isachardot}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}step\ d\ v\ {\isachardollar}{\kern0pt}\ s{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymlambda}d{\isachardot}{\kern0pt}\ d\ {\isasymin}\ D\isactrlsub D{\isacharparenright}{\kern0pt}\ d{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}GS{\isacharunderscore}{\kern0pt}rec\ v\ {\isachardollar}{\kern0pt}\ s\ {\isacharequal}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}step\ d{\isacharprime}{\kern0pt}\ v\ {\isachardollar}{\kern0pt}\ s{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}GS{\isacharunderscore}{\kern0pt}rec\ v\ {\isachardollar}{\kern0pt}\ s\ {\isacharequal}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}step\ d\ v\ {\isachardollar}{\kern0pt}\ s{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}GS{\isacharunderscore}{\kern0pt}rec{\isacharprime}{\kern0pt}\ assms{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}\ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}step\ d\ v\ {\isachardollar}{\kern0pt}\ s\ {\isacharequal}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}step\ d{\isacharprime}{\kern0pt}\ v\ {\isachardollar}{\kern0pt}\ s{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}step{\isacharunderscore}{\kern0pt}elem\ assms{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}\ \isacommand{by}\isamarkupfalse%
-\ fastforce\isanewline
-\ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}GS{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}splitting{\isacharprime}{\kern0pt}{\isacharprime}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}{\isasymAnd}s{\isachardot}{\kern0pt}\ s\ {\isasymle}\ k\ {\isasymLongrightarrow}\ is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max\ {\isacharparenleft}{\kern0pt}{\isasymlambda}d{\isachardot}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}inv\ d\ v\ {\isachardollar}{\kern0pt}\ s{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymlambda}d{\isachardot}{\kern0pt}\ d\ {\isasymin}\ D\isactrlsub D{\isacharparenright}{\kern0pt}\ d{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}s\ {\isasymle}\ k{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max\ {\isacharparenleft}{\kern0pt}{\isasymlambda}d{\isachardot}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}step\ d\ v\ {\isachardollar}{\kern0pt}\ s{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymlambda}d{\isachardot}{\kern0pt}\ d\ {\isasymin}\ D\isactrlsub D{\isacharparenright}{\kern0pt}\ d\ {\isasymand}\ GS{\isacharunderscore}{\kern0pt}inv\ d\ v\ {\isachardollar}{\kern0pt}\ s\ {\isacharequal}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}rec\ v\ {\isachardollar}{\kern0pt}\ s{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ assms\isanewline
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}induction\ k\ arbitrary{\isacharcolon}{\kern0pt}\ s\ rule{\isacharcolon}{\kern0pt}\ less{\isacharunderscore}{\kern0pt}induct{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{case}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}less\ x{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ d{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}d\ {\isasymin}\ D\isactrlsub D{\isachardoublequoteclose}\ \isacommand{using}\isamarkupfalse%
-\ assms\ \isacommand{unfolding}\isamarkupfalse%
-\ is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}linorder\ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max\ {\isacharparenleft}{\kern0pt}{\isasymlambda}a{\isachardot}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}step\ {\isacharparenleft}{\kern0pt}d{\isacharparenleft}{\kern0pt}s\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ a{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ v\ {\isachardollar}{\kern0pt}\ s{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymlambda}a{\isachardot}{\kern0pt}\ a\ {\isasymin}\ A\ s{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}d\ s{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{if}\ {\isachardoublequoteopen}s\ {\isasymle}\ x{\isachardoublequoteclose}\ \isakeyword{for}\ s\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max\ {\isacharparenleft}{\kern0pt}{\isasymlambda}a{\isachardot}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}inv\ {\isacharparenleft}{\kern0pt}d{\isacharparenleft}{\kern0pt}s\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ a{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ v\ {\isachardollar}{\kern0pt}\ s{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymlambda}a{\isachardot}{\kern0pt}\ a\ {\isasymin}\ A\ s{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}d\ s{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ less{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isacharbrackleft}{\kern0pt}OF\ that{\isacharbrackright}{\kern0pt}\ \isanewline
-\ \ \ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ is{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det{\isacharunderscore}{\kern0pt}def\ is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}linorder\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ {\isacharasterisk}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max\ {\isacharparenleft}{\kern0pt}{\isasymlambda}a{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}r{\isacharunderscore}{\kern0pt}vec\ {\isacharparenleft}{\kern0pt}d{\isacharparenleft}{\kern0pt}s\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ a{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharplus}{\kern0pt}\ l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isacharparenleft}{\kern0pt}P{\isacharunderscore}{\kern0pt}lower\ {\isacharparenleft}{\kern0pt}d{\isacharparenleft}{\kern0pt}s\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ a{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharasterisk}{\kern0pt}v\ {\isacharparenleft}{\kern0pt}GS{\isacharunderscore}{\kern0pt}inv\ {\isacharparenleft}{\kern0pt}d{\isacharparenleft}{\kern0pt}s\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ a{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ v{\isacharparenright}{\kern0pt}\ {\isacharplus}{\kern0pt}\ P{\isacharunderscore}{\kern0pt}upper\ {\isacharparenleft}{\kern0pt}d{\isacharparenleft}{\kern0pt}s\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ a{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharasterisk}{\kern0pt}v\ v{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isachardollar}{\kern0pt}\ s{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymlambda}a{\isachardot}{\kern0pt}\ a\ {\isasymin}\ A\ s{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}d\ s{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}subst\ {\isacharparenleft}{\kern0pt}asm{\isacharparenright}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}inv{\isacharunderscore}{\kern0pt}rec{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ add{\isachardot}{\kern0pt}commute{\isacharparenright}{\kern0pt}\isanewline
-\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ {\isacharasterisk}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max\ {\isacharparenleft}{\kern0pt}{\isasymlambda}a{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}r{\isacharunderscore}{\kern0pt}vec\ {\isacharparenleft}{\kern0pt}d{\isacharparenleft}{\kern0pt}s\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ a{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharplus}{\kern0pt}\ l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isacharparenleft}{\kern0pt}P{\isacharunderscore}{\kern0pt}lower\ {\isacharparenleft}{\kern0pt}d{\isacharparenleft}{\kern0pt}s\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ a{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharasterisk}{\kern0pt}v\ {\isacharparenleft}{\kern0pt}GS{\isacharunderscore}{\kern0pt}inv\ d\ v{\isacharparenright}{\kern0pt}\ {\isacharplus}{\kern0pt}\ P{\isacharunderscore}{\kern0pt}upper\ {\isacharparenleft}{\kern0pt}d{\isacharparenleft}{\kern0pt}s\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ a{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharasterisk}{\kern0pt}v\ v{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isachardollar}{\kern0pt}\ s{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymlambda}a{\isachardot}{\kern0pt}\ a\ {\isasymin}\ A\ s{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}d\ s{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}P{\isacharunderscore}{\kern0pt}lower\ {\isacharparenleft}{\kern0pt}d{\isacharparenleft}{\kern0pt}s\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ a{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharasterisk}{\kern0pt}v\ {\isacharparenleft}{\kern0pt}GS{\isacharunderscore}{\kern0pt}inv\ {\isacharparenleft}{\kern0pt}d{\isacharparenleft}{\kern0pt}s\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ a{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ v{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isachardollar}{\kern0pt}\ s\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}P{\isacharunderscore}{\kern0pt}lower\ {\isacharparenleft}{\kern0pt}d{\isacharparenleft}{\kern0pt}s\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ a{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharasterisk}{\kern0pt}v\ {\isacharparenleft}{\kern0pt}GS{\isacharunderscore}{\kern0pt}inv\ d\ v{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isachardollar}{\kern0pt}\ s{\isachardoublequoteclose}\ \isakeyword{for}\ a\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ GS{\isacharunderscore}{\kern0pt}indep{\isacharunderscore}{\kern0pt}high{\isacharunderscore}{\kern0pt}states{\isacharbrackleft}{\kern0pt}of\ {\isacharunderscore}{\kern0pt}\ {\isachardoublequoteopen}d{\isacharparenleft}{\kern0pt}s\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ a{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ d\ v{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}rule\ strict{\isacharunderscore}{\kern0pt}lower{\isacharunderscore}{\kern0pt}triangular{\isacharunderscore}{\kern0pt}mat{\isacharunderscore}{\kern0pt}mult{\isacharbrackleft}{\kern0pt}OF\ slt{\isacharunderscore}{\kern0pt}P{\isacharunderscore}{\kern0pt}lower{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}metis\ array{\isacharunderscore}{\kern0pt}rules{\isacharparenleft}{\kern0pt}{\isadigit{4}}{\isacharparenright}{\kern0pt}\ leD{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\ \isacommand{using}\isamarkupfalse%
-\ {\isacharasterisk}{\kern0pt}\ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max\ {\isacharparenleft}{\kern0pt}{\isasymlambda}a{\isachardot}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}step\ {\isacharparenleft}{\kern0pt}d{\isacharparenleft}{\kern0pt}s\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ a{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ v\ {\isachardollar}{\kern0pt}\ s{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymlambda}a{\isachardot}{\kern0pt}\ a\ {\isasymin}\ A\ s{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}d\ s{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}P{\isacharunderscore}{\kern0pt}lower\ {\isacharparenleft}{\kern0pt}d{\isacharparenleft}{\kern0pt}s\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ a{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharasterisk}{\kern0pt}v\ {\isacharparenleft}{\kern0pt}GS{\isacharunderscore}{\kern0pt}inv\ d\ v{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isachardollar}{\kern0pt}\ s\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}P{\isacharunderscore}{\kern0pt}lower\ {\isacharparenleft}{\kern0pt}d{\isacharparenleft}{\kern0pt}s\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ a{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharasterisk}{\kern0pt}v\ {\isacharparenleft}{\kern0pt}GS{\isacharunderscore}{\kern0pt}rec\ v{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isachardollar}{\kern0pt}\ s{\isachardoublequoteclose}\ \isakeyword{for}\ a\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ less{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}\ less{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}that\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}intro\ strict{\isacharunderscore}{\kern0pt}lower{\isacharunderscore}{\kern0pt}triangular{\isacharunderscore}{\kern0pt}mat{\isacharunderscore}{\kern0pt}mult{\isacharbrackleft}{\kern0pt}OF\ slt{\isacharunderscore}{\kern0pt}P{\isacharunderscore}{\kern0pt}lower{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\ force\isanewline
-\ \ \ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\ \isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isacharasterisk}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}step{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isacharasterisk}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymAnd}s{\isachardot}{\kern0pt}\ s\ {\isasymle}\ x\ {\isasymLongrightarrow}\ is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max\ {\isacharparenleft}{\kern0pt}{\isasymlambda}d{\isachardot}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}step\ d\ v\ {\isachardollar}{\kern0pt}\ s{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymlambda}d{\isachardot}{\kern0pt}\ d\ {\isasymin}\ D\isactrlsub D{\isacharparenright}{\kern0pt}\ d{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ d\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}intro\ is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}step{\isacharunderscore}{\kern0pt}act{\isacharprime}{\kern0pt}{\isacharbrackleft}{\kern0pt}of\ d\ d{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\ auto\isanewline
-\ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}GS{\isacharunderscore}{\kern0pt}inv\ d\ v\ {\isachardollar}{\kern0pt}\ s\ {\isacharequal}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}rec\ v\ {\isachardollar}{\kern0pt}\ s{\isachardoublequoteclose}\ \isakeyword{if}\ {\isachardoublequoteopen}s\ {\isasymle}\ x{\isachardoublequoteclose}\ \isakeyword{for}\ s\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}GS{\isacharunderscore}{\kern0pt}rec\ v\ {\isachardollar}{\kern0pt}\ s\ {\isacharequal}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}step\ d\ v\ {\isachardollar}{\kern0pt}\ s{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isacharasterisk}{\kern0pt}{\isacharbrackleft}{\kern0pt}OF\ that{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}GS{\isacharunderscore}{\kern0pt}rec{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isacharasterisk}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}GS{\isacharunderscore}{\kern0pt}inv\ that\ \isacommand{by}\isamarkupfalse%
-\ presburger\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}case\ \isacommand{using}\isamarkupfalse%
-\ less\ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}GS{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}splitting{\isacharprime}{\kern0pt}{\isacharprime}{\kern0pt}{\isacharprime}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}{\isasymAnd}s{\isachardot}{\kern0pt}\ s\ {\isasymle}\ k\ {\isasymLongrightarrow}\ is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max\ {\isacharparenleft}{\kern0pt}{\isasymlambda}d{\isachardot}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}inv\ d\ v\ {\isachardollar}{\kern0pt}\ s{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymlambda}d{\isachardot}{\kern0pt}\ d\ {\isasymin}\ D\isactrlsub D{\isacharparenright}{\kern0pt}\ d{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}s\ {\isasymle}\ k{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max\ {\isacharparenleft}{\kern0pt}{\isasymlambda}d{\isachardot}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}step\ d\ v\ {\isachardollar}{\kern0pt}\ s{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymlambda}d{\isachardot}{\kern0pt}\ d\ {\isasymin}\ D\isactrlsub D{\isacharparenright}{\kern0pt}\ d{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ assms\ \ is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}GS{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}splitting{\isacharprime}{\kern0pt}{\isacharprime}{\kern0pt}\ \isacommand{by}\isamarkupfalse%
-\ blast%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}GS{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}splitting{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}{\isasymAnd}s{\isachardot}{\kern0pt}\ is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max\ {\isacharparenleft}{\kern0pt}{\isasymlambda}d{\isachardot}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}step\ d\ v\ {\isachardollar}{\kern0pt}\ s{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymlambda}d{\isachardot}{\kern0pt}\ d\ {\isasymin}\ D\isactrlsub D{\isacharparenright}{\kern0pt}\ d{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max\ {\isacharparenleft}{\kern0pt}{\isasymlambda}d{\isachardot}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}inv\ d\ v\ {\isachardollar}{\kern0pt}\ k{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymlambda}d{\isachardot}{\kern0pt}\ d\ {\isasymin}\ D\isactrlsub D{\isacharparenright}{\kern0pt}\ d{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ assms\ is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}GS{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}splitting{\isacharprime}{\kern0pt}{\isacharbrackleft}{\kern0pt}of\ {\isachardoublequoteopen}Max\ UNIV{\isachardoublequoteclose}{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}linorder{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}gs{\isacharunderscore}{\kern0pt}iff{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}d\ {\isasymin}\ D\isactrlsub D{\isachardoublequoteclose}\ \isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}\ {\isacharparenleft}{\kern0pt}{\isasymforall}s\ {\isasymle}\ k{\isachardot}{\kern0pt}\ is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max\ {\isacharparenleft}{\kern0pt}{\isasymlambda}d{\isachardot}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}inv\ d\ v\ {\isachardollar}{\kern0pt}\ s{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymlambda}d{\isachardot}{\kern0pt}\ d\ {\isasymin}\ D\isactrlsub D{\isacharparenright}{\kern0pt}\ d{\isacharparenright}{\kern0pt}\ {\isasymlongleftrightarrow}\isanewline
-\ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}s\ {\isasymle}\ k{\isachardot}{\kern0pt}\ is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max\ {\isacharparenleft}{\kern0pt}{\isasymlambda}d{\isachardot}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}step\ d\ v\ {\isachardollar}{\kern0pt}\ s{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymlambda}d{\isachardot}{\kern0pt}\ d\ {\isasymin}\ D\isactrlsub D{\isacharparenright}{\kern0pt}\ d{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}GS{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}splitting{\isacharprime}{\kern0pt}\ is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}GS{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}splitting{\isacharprime}{\kern0pt}{\isacharprime}{\kern0pt}\ \ \isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ meson%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ GS{\isacharunderscore}{\kern0pt}opt{\isacharunderscore}{\kern0pt}indep{\isacharunderscore}{\kern0pt}high{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isasymAnd}s{\isacharprime}{\kern0pt}{\isachardot}{\kern0pt}\ s{\isacharprime}{\kern0pt}\ {\isacharless}{\kern0pt}\ s\ {\isasymLongrightarrow}\ is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max\ {\isacharparenleft}{\kern0pt}{\isasymlambda}d{\isachardot}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}step\ d\ v\ {\isachardollar}{\kern0pt}\ s{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}\ is{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\ d{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ {\isachardoublequoteopen}s{\isacharprime}{\kern0pt}\ {\isacharless}{\kern0pt}\ s{\isachardoublequoteclose}\ {\isachardoublequoteopen}a\ {\isasymin}\ A\ s{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max\ {\isacharparenleft}{\kern0pt}{\isasymlambda}d{\isachardot}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}step\ d\ v\ {\isachardollar}{\kern0pt}\ s{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}\ is{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\ {\isacharparenleft}{\kern0pt}d{\isacharparenleft}{\kern0pt}s\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ a{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}rule\ is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}linorderI{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\ y\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\ {\isachardoublequoteopen}is{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\ y{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}step\ y\ v\ {\isachardollar}{\kern0pt}\ s{\isacharprime}{\kern0pt}\ {\isasymle}\ r\ {\isacharparenleft}{\kern0pt}s{\isacharprime}{\kern0pt}{\isacharcomma}{\kern0pt}\ d\ s{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharplus}{\kern0pt}\ l\ {\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}P{\isacharunderscore}{\kern0pt}lower\ d\ {\isacharasterisk}{\kern0pt}v\ GS{\isacharunderscore}{\kern0pt}rec\ v{\isacharparenright}{\kern0pt}\ {\isachardollar}{\kern0pt}\ s{\isacharprime}{\kern0pt}\ {\isacharplus}{\kern0pt}\ l\ {\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}P{\isacharunderscore}{\kern0pt}upper\ d\ {\isacharasterisk}{\kern0pt}v\ v{\isacharparenright}{\kern0pt}\ {\isachardollar}{\kern0pt}\ s{\isacharprime}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}linorderD{\isacharbrackleft}{\kern0pt}OF\ assms{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}step{\isacharunderscore}{\kern0pt}def\ algebra{\isacharunderscore}{\kern0pt}simps\ assms{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isacharequal}{\kern0pt}\ r\ {\isacharparenleft}{\kern0pt}s{\isacharprime}{\kern0pt}{\isacharcomma}{\kern0pt}\ {\isacharparenleft}{\kern0pt}d{\isacharparenleft}{\kern0pt}s\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ a{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ s{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharplus}{\kern0pt}\ l\ {\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}P{\isacharunderscore}{\kern0pt}lower\ {\isacharparenleft}{\kern0pt}d{\isacharparenleft}{\kern0pt}s\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ a{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharasterisk}{\kern0pt}v\ GS{\isacharunderscore}{\kern0pt}rec\ v{\isacharparenright}{\kern0pt}\ {\isachardollar}{\kern0pt}\ s{\isacharprime}{\kern0pt}\ {\isacharplus}{\kern0pt}\ l\ {\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}P{\isacharunderscore}{\kern0pt}upper\ {\isacharparenleft}{\kern0pt}d{\isacharparenleft}{\kern0pt}s\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ a{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharasterisk}{\kern0pt}v\ v{\isacharparenright}{\kern0pt}\ {\isachardollar}{\kern0pt}\ s{\isacharprime}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}P{\isacharunderscore}{\kern0pt}lower\ d\ {\isacharasterisk}{\kern0pt}v\ GS{\isacharunderscore}{\kern0pt}rec\ v{\isacharparenright}{\kern0pt}\ {\isachardollar}{\kern0pt}\ s{\isacharprime}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}P{\isacharunderscore}{\kern0pt}lower\ {\isacharparenleft}{\kern0pt}d{\isacharparenleft}{\kern0pt}s\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ a{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharasterisk}{\kern0pt}v\ GS{\isacharunderscore}{\kern0pt}rec\ v{\isacharparenright}{\kern0pt}\ {\isachardollar}{\kern0pt}\ s{\isacharprime}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assms\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}fastforce\ simp{\isacharcolon}{\kern0pt}\ matrix{\isacharunderscore}{\kern0pt}vector{\isacharunderscore}{\kern0pt}mult{\isacharunderscore}{\kern0pt}def\ P{\isacharunderscore}{\kern0pt}lower{\isacharunderscore}{\kern0pt}def\ P{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}elem\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ sum{\isachardot}{\kern0pt}cong{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}P{\isacharunderscore}{\kern0pt}upper\ d\ {\isacharasterisk}{\kern0pt}v\ v{\isacharparenright}{\kern0pt}\ {\isachardollar}{\kern0pt}\ s{\isacharprime}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}P{\isacharunderscore}{\kern0pt}upper\ {\isacharparenleft}{\kern0pt}d{\isacharparenleft}{\kern0pt}s\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ a{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharasterisk}{\kern0pt}v\ v{\isacharparenright}{\kern0pt}\ {\isachardollar}{\kern0pt}\ s{\isacharprime}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assms\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}fastforce\ simp{\isacharcolon}{\kern0pt}\ matrix{\isacharunderscore}{\kern0pt}vector{\isacharunderscore}{\kern0pt}mult{\isacharunderscore}{\kern0pt}def\ P{\isacharunderscore}{\kern0pt}upper{\isacharunderscore}{\kern0pt}def\ P{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}elem\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ sum{\isachardot}{\kern0pt}cong{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assms{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}\ \isacommand{by}\isamarkupfalse%
-\ force\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isacharequal}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}step\ {\isacharparenleft}{\kern0pt}d{\isacharparenleft}{\kern0pt}s\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ a{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ v\ {\isachardollar}{\kern0pt}\ s{\isacharprime}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}step{\isacharunderscore}{\kern0pt}def\ algebra{\isacharunderscore}{\kern0pt}simps{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{finally}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}step\ y\ v\ {\isachardollar}{\kern0pt}\ s{\isacharprime}{\kern0pt}\ {\isasymle}\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}step\ {\isacharparenleft}{\kern0pt}d{\isacharparenleft}{\kern0pt}s\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ a{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ v\ {\isachardollar}{\kern0pt}\ s{\isacharprime}{\kern0pt}{\isachardoublequoteclose}\isacommand{{\isachardot}{\kern0pt}}\isamarkupfalse%
-\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}is{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\ {\isacharparenleft}{\kern0pt}d{\isacharparenleft}{\kern0pt}s\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ a{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}linorderD{\isacharbrackleft}{\kern0pt}OF\ assms{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}{\isacharbrackleft}{\kern0pt}OF\ assms{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isacharbrackright}{\kern0pt}{\isacharbrackright}{\kern0pt}\ assms{\isacharparenleft}{\kern0pt}{\isadigit{3}}{\isacharparenright}{\kern0pt}\ is{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det{\isacharunderscore}{\kern0pt}def\ \isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ fastforce\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ mult{\isacharunderscore}{\kern0pt}mat{\isacharunderscore}{\kern0pt}vec{\isacharunderscore}{\kern0pt}nth{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}X\ {\isacharasterisk}{\kern0pt}v\ x{\isacharparenright}{\kern0pt}\ {\isachardollar}{\kern0pt}\ i\ {\isacharequal}{\kern0pt}\ scalar{\isacharunderscore}{\kern0pt}product\ {\isacharparenleft}{\kern0pt}row\ i\ X{\isacharparenright}{\kern0pt}\ x{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ matrix{\isacharunderscore}{\kern0pt}vector{\isacharunderscore}{\kern0pt}mult{\isacharunderscore}{\kern0pt}def\ row{\isacharunderscore}{\kern0pt}def\ scalar{\isacharunderscore}{\kern0pt}product{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ ext{\isacharunderscore}{\kern0pt}GS{\isacharunderscore}{\kern0pt}opt{\isacharunderscore}{\kern0pt}le{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isasymAnd}s{\isacharprime}{\kern0pt}{\isachardot}{\kern0pt}\ s{\isacharprime}{\kern0pt}\ {\isacharless}{\kern0pt}\ s\ {\isasymLongrightarrow}\ is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max\ {\isacharparenleft}{\kern0pt}{\isasymlambda}d{\isachardot}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}step\ d\ v\ {\isachardollar}{\kern0pt}\ s{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymlambda}d{\isachardot}{\kern0pt}\ d\ {\isasymin}\ D\isactrlsub D{\isacharparenright}{\kern0pt}\ d{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isakeyword{and}\ {\isachardoublequoteopen}is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max\ {\isacharparenleft}{\kern0pt}{\isasymlambda}a{\isachardot}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}step\ {\isacharparenleft}{\kern0pt}d{\isacharparenleft}{\kern0pt}s\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ a{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ v\ {\isachardollar}{\kern0pt}\ s{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymlambda}a{\isachardot}{\kern0pt}\ a\ {\isasymin}\ A\ s{\isacharparenright}{\kern0pt}\ a{\isachardoublequoteclose}\ {\isachardoublequoteopen}s{\isacharprime}{\kern0pt}\ {\isasymle}\ s{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isakeyword{and}\ {\isachardoublequoteopen}d\ {\isasymin}\ D\isactrlsub D{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max\ {\isacharparenleft}{\kern0pt}{\isasymlambda}d{\isachardot}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}step\ d\ v\ {\isachardollar}{\kern0pt}\ s{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymlambda}d{\isachardot}{\kern0pt}\ d\ {\isasymin}\ D\isactrlsub D{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}d{\isacharparenleft}{\kern0pt}s\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ a{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ assms\ is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}step{\isacharunderscore}{\kern0pt}act\ is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}linorderD{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}cases\ {\isachardoublequoteopen}s\ {\isacharequal}{\kern0pt}\ s{\isacharprime}{\kern0pt}{\isachardoublequoteclose}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}auto\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}opt{\isacharunderscore}{\kern0pt}indep{\isacharunderscore}{\kern0pt}high{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ ex{\isacharunderscore}{\kern0pt}GS{\isacharunderscore}{\kern0pt}opt{\isacharunderscore}{\kern0pt}le{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}{\isasymexists}d{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymforall}s{\isacharprime}{\kern0pt}\ {\isasymle}\ s{\isachardot}{\kern0pt}\ is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max\ {\isacharparenleft}{\kern0pt}{\isasymlambda}d{\isachardot}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}step\ d\ v\ {\isachardollar}{\kern0pt}\ s{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymlambda}d{\isachardot}{\kern0pt}\ d\ {\isasymin}\ D\isactrlsub D{\isacharparenright}{\kern0pt}\ d{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}induction\ s\ rule{\isacharcolon}{\kern0pt}\ less{\isacharunderscore}{\kern0pt}induct{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{case}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}less\ x{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}case\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}cases\ {\isachardoublequoteopen}{\isasymexists}y{\isachardot}{\kern0pt}\ y\ {\isacharless}{\kern0pt}\ x{\isachardoublequoteclose}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{case}\isamarkupfalse%
-\ True\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}y{\isachardot}{\kern0pt}\ y\ {\isacharless}{\kern0pt}\ x{\isacharbraceright}{\kern0pt}\ {\isasymnoteq}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\ \isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymAnd}y{\isachardot}{\kern0pt}\ y\ {\isasymle}\ Max\ {\isacharbraceleft}{\kern0pt}y{\isachardot}{\kern0pt}\ y\ {\isacharless}{\kern0pt}\ x{\isacharbraceright}{\kern0pt}\ {\isasymlongleftrightarrow}\ y\ {\isacharless}{\kern0pt}\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ True\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ Max{\isacharunderscore}{\kern0pt}ge{\isacharunderscore}{\kern0pt}iff{\isacharbrackleft}{\kern0pt}OF\ finite{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{obtain}\isamarkupfalse%
-\ d\ \isakeyword{where}\ d{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max\ {\isacharparenleft}{\kern0pt}{\isasymlambda}d{\isachardot}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}step\ d\ v\ {\isachardollar}{\kern0pt}\ s{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymlambda}d{\isachardot}{\kern0pt}\ d\ {\isasymin}\ D\isactrlsub D{\isacharparenright}{\kern0pt}\ d{\isachardoublequoteclose}\ \isakeyword{if}\ {\isachardoublequoteopen}s{\isacharprime}{\kern0pt}{\isacharless}{\kern0pt}\ x{\isachardoublequoteclose}\ \isakeyword{for}\ s{\isacharprime}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ less{\isacharbrackleft}{\kern0pt}of\ {\isachardoublequoteopen}Max\ {\isacharbraceleft}{\kern0pt}y{\isachardot}{\kern0pt}\ y\ {\isacharless}{\kern0pt}\ x{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}{\isacharbrackright}{\kern0pt}\ {\isadigit{1}}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \isacommand{obtain}\isamarkupfalse%
-\ a\ \isakeyword{where}\ a{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max\ {\isacharparenleft}{\kern0pt}{\isasymlambda}a{\isachardot}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}step\ {\isacharparenleft}{\kern0pt}d{\isacharparenleft}{\kern0pt}x\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ a{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ v\ {\isachardollar}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymlambda}a{\isachardot}{\kern0pt}\ a\ {\isasymin}\ A\ x{\isacharparenright}{\kern0pt}\ a{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ finite{\isacharunderscore}{\kern0pt}is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharbrackleft}{\kern0pt}OF\ finite\ A{\isacharunderscore}{\kern0pt}ne{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ d{\isacharprime}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max\ {\isacharparenleft}{\kern0pt}{\isasymlambda}d{\isachardot}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}step\ d\ v\ {\isachardollar}{\kern0pt}\ s{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymlambda}d{\isachardot}{\kern0pt}\ d\ {\isasymin}\ D\isactrlsub D{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}d{\isacharparenleft}{\kern0pt}x\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ a{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{if}\ {\isachardoublequoteopen}s{\isacharprime}{\kern0pt}\ {\isacharless}{\kern0pt}\ x{\isachardoublequoteclose}\ \isakeyword{for}\ s{\isacharprime}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ d\ GS{\isacharunderscore}{\kern0pt}opt{\isacharunderscore}{\kern0pt}indep{\isacharunderscore}{\kern0pt}high\ that\ is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}linorderD{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}{\isacharbrackleft}{\kern0pt}OF\ a{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ d{\isacharprime}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max\ {\isacharparenleft}{\kern0pt}{\isasymlambda}d{\isachardot}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}step\ d\ v\ {\isachardollar}{\kern0pt}\ s{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymlambda}d{\isachardot}{\kern0pt}\ d\ {\isasymin}\ D\isactrlsub D{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}d{\isacharparenleft}{\kern0pt}x\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ a{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{if}\ {\isachardoublequoteopen}s{\isacharprime}{\kern0pt}\ {\isasymle}\ x{\isachardoublequoteclose}\ \isakeyword{for}\ s{\isacharprime}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ \ that\ a\ is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}linorderD{\isacharbrackleft}{\kern0pt}OF\ d{\isacharbrackright}{\kern0pt}\ True\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}fastforce\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ ext{\isacharunderscore}{\kern0pt}GS{\isacharunderscore}{\kern0pt}opt{\isacharunderscore}{\kern0pt}le{\isacharbrackleft}{\kern0pt}OF\ d{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{case}\isamarkupfalse%
-\ False\isanewline
-\ \ \ \ \isacommand{define}\isamarkupfalse%
-\ d\ \isakeyword{where}\ {\isachardoublequoteopen}d\ y\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}SOME\ a{\isachardot}{\kern0pt}\ a\ {\isasymin}\ A\ y{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{for}\ y\isanewline
-\ \ \ \ \isacommand{obtain}\isamarkupfalse%
-\ a\ \isakeyword{where}\ a{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max\ {\isacharparenleft}{\kern0pt}{\isasymlambda}a{\isachardot}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}step\ {\isacharparenleft}{\kern0pt}d{\isacharparenleft}{\kern0pt}x\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ a{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ v\ {\isachardollar}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymlambda}a{\isachardot}{\kern0pt}\ a\ {\isasymin}\ A\ x{\isacharparenright}{\kern0pt}\ a{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ finite{\isacharunderscore}{\kern0pt}is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharbrackleft}{\kern0pt}OF\ finite\ A{\isacharunderscore}{\kern0pt}ne{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}y\ {\isasymle}\ x\ {\isasymLongrightarrow}\ y\ {\isacharequal}{\kern0pt}\ x{\isachardoublequoteclose}\ \isakeyword{for}\ y\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ False\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}meson\ le{\isacharunderscore}{\kern0pt}neq{\isacharunderscore}{\kern0pt}trans{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max\ {\isacharparenleft}{\kern0pt}{\isasymlambda}d{\isachardot}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}step\ d\ v\ {\isachardollar}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymlambda}d{\isachardot}{\kern0pt}\ d\ {\isasymin}\ D\isactrlsub D{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}d{\isacharparenleft}{\kern0pt}x\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ a{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ False\ a\ SOME{\isacharunderscore}{\kern0pt}is{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\ \isacommand{unfolding}\isamarkupfalse%
-\ d{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}fastforce\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}step{\isacharunderscore}{\kern0pt}act{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{then}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isadigit{1}}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ ex{\isacharunderscore}{\kern0pt}GS{\isacharunderscore}{\kern0pt}opt{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}{\isasymexists}d{\isachardot}{\kern0pt}\ {\isasymforall}s{\isachardot}{\kern0pt}\ is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max\ {\isacharparenleft}{\kern0pt}{\isasymlambda}d{\isachardot}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}step\ d\ v\ {\isachardollar}{\kern0pt}\ s{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymlambda}d{\isachardot}{\kern0pt}\ d\ {\isasymin}\ D\isactrlsub D{\isacharparenright}{\kern0pt}\ d{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ ex{\isacharunderscore}{\kern0pt}GS{\isacharunderscore}{\kern0pt}opt{\isacharunderscore}{\kern0pt}le{\isacharbrackleft}{\kern0pt}of\ {\isachardoublequoteopen}Max\ UNIV{\isachardoublequoteclose}{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}GS{\isacharunderscore}{\kern0pt}inv{\isacharprime}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}GS{\isacharunderscore}{\kern0pt}rec\ v\ {\isachardollar}{\kern0pt}\ s\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSqunion}d{\isasymin}D\isactrlsub D{\isachardot}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}inv\ d\ v\ {\isachardollar}{\kern0pt}\ s{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{obtain}\isamarkupfalse%
-\ d\ \isakeyword{where}\ d{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isasymAnd}s{\isachardot}{\kern0pt}\ is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max\ {\isacharparenleft}{\kern0pt}{\isasymlambda}d{\isachardot}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}step\ d\ v\ {\isachardollar}{\kern0pt}\ s{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymlambda}d{\isachardot}{\kern0pt}\ d\ {\isasymin}\ D\isactrlsub D{\isacharparenright}{\kern0pt}\ d{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ ex{\isacharunderscore}{\kern0pt}GS{\isacharunderscore}{\kern0pt}opt\ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isasymSqunion}d{\isasymin}D\isactrlsub D{\isachardot}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}step\ d\ v\ {\isachardollar}{\kern0pt}\ s{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}step\ d\ v\ {\isachardollar}{\kern0pt}\ s{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ d\ is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}GS{\isacharunderscore}{\kern0pt}rec\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}vec\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isasymSqunion}d{\isasymin}D\isactrlsub D{\isachardot}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}inv\ d\ v\ {\isachardollar}{\kern0pt}\ s{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}inv\ d\ v\ {\isachardollar}{\kern0pt}\ s{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}GS{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}splitting{\isacharbrackleft}{\kern0pt}OF\ d{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}subst\ arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}SUP{\isacharbrackleft}{\kern0pt}symmetric{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\ auto\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}GS{\isacharunderscore}{\kern0pt}inv\ d\ \isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ presburger\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}fun{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}GS{\isacharunderscore}{\kern0pt}inv{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}fun\ v\ s\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSqunion}d{\isasymin}D\isactrlsub D{\isachardot}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}inv\ d\ {\isacharparenleft}{\kern0pt}vec{\isacharunderscore}{\kern0pt}lambda\ v{\isacharparenright}{\kern0pt}\ {\isachardollar}{\kern0pt}\ s{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}GS{\isacharunderscore}{\kern0pt}inv{\isacharprime}{\kern0pt}{\isacharbrackleft}{\kern0pt}of\ {\isachardoublequoteopen}vec{\isacharunderscore}{\kern0pt}lambda\ v{\isachardoublequoteclose}{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \isacommand{unfolding}\isamarkupfalse%
-\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ vec{\isacharunderscore}{\kern0pt}lambda{\isacharunderscore}{\kern0pt}inverse{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ invertible{\isacharunderscore}{\kern0pt}Q{\isacharunderscore}{\kern0pt}GS{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}invertible\isactrlsub L\ {\isacharparenleft}{\kern0pt}Q{\isacharunderscore}{\kern0pt}GS\ d{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{for}\ d\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ Q{\isacharunderscore}{\kern0pt}mat{\isacharunderscore}{\kern0pt}invertible\ invertible{\isacharunderscore}{\kern0pt}invertible\isactrlsub L{\isacharunderscore}{\kern0pt}I{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ ex{\isacharunderscore}{\kern0pt}opt{\isacharunderscore}{\kern0pt}blinfun{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymexists}d{\isachardot}{\kern0pt}\ {\isasymforall}s{\isachardot}{\kern0pt}\ is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max\ {\isacharparenleft}{\kern0pt}{\isasymlambda}d{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}inv\isactrlsub L\ {\isacharparenleft}{\kern0pt}Q{\isacharunderscore}{\kern0pt}GS\ d{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}r{\isacharunderscore}{\kern0pt}det\isactrlsub b\ d\ {\isacharplus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}R{\isacharunderscore}{\kern0pt}GS\ d{\isacharparenright}{\kern0pt}\ v{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ s{\isacharparenright}{\kern0pt}\ is{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\ d{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}GS{\isacharunderscore}{\kern0pt}inv\ d\ {\isacharparenleft}{\kern0pt}vec{\isacharunderscore}{\kern0pt}lambda\ v{\isacharparenright}{\kern0pt}\ {\isachardollar}{\kern0pt}\ s\ {\isacharequal}{\kern0pt}\ inv\isactrlsub L\ {\isacharparenleft}{\kern0pt}Q{\isacharunderscore}{\kern0pt}GS\ d{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}r{\isacharunderscore}{\kern0pt}det\isactrlsub b\ d\ {\isacharplus}{\kern0pt}\ R{\isacharunderscore}{\kern0pt}GS\ d\ v{\isacharparenright}{\kern0pt}\ s{\isachardoublequoteclose}\ \isakeyword{for}\ d\ s\isanewline
-\ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ GS{\isacharunderscore}{\kern0pt}inv{\isacharunderscore}{\kern0pt}def\ plus{\isacharunderscore}{\kern0pt}bfun{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ invertible{\isacharunderscore}{\kern0pt}Q{\isacharunderscore}{\kern0pt}GS\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}mult{\isacharprime}{\kern0pt}\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}inverse{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isacharbrackleft}{\kern0pt}symmetric{\isacharbrackright}{\kern0pt}\ apply{\isacharunderscore}{\kern0pt}bfun{\isacharunderscore}{\kern0pt}inverse{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{obtain}\isamarkupfalse%
-\ d\ \isakeyword{where}\ {\isachardoublequoteopen}is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max\ {\isacharparenleft}{\kern0pt}{\isasymlambda}d{\isachardot}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}inv\ d\ {\isacharparenleft}{\kern0pt}vec{\isacharunderscore}{\kern0pt}lambda\ v{\isacharparenright}{\kern0pt}\ {\isachardollar}{\kern0pt}\ s{\isacharparenright}{\kern0pt}\ is{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\ d{\isachardoublequoteclose}\ \isakeyword{for}\ s\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ ex{\isacharunderscore}{\kern0pt}GS{\isacharunderscore}{\kern0pt}opt{\isacharbrackleft}{\kern0pt}of\ {\isachardoublequoteopen}vec{\isacharunderscore}{\kern0pt}lambda\ v{\isachardoublequoteclose}{\isacharbrackright}{\kern0pt}\ is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}GS{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}splitting\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\ \isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ GS{\isacharunderscore}{\kern0pt}inv{\isacharunderscore}{\kern0pt}blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}inv\isactrlsub L\ {\isacharparenleft}{\kern0pt}Q{\isacharunderscore}{\kern0pt}GS\ d{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}r{\isacharunderscore}{\kern0pt}det\isactrlsub b\ d\ {\isacharplus}{\kern0pt}\ R{\isacharunderscore}{\kern0pt}GS\ d\ v{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ Bfun\ {\isacharparenleft}{\kern0pt}vec{\isacharunderscore}{\kern0pt}nth\ {\isacharparenleft}{\kern0pt}GS{\isacharunderscore}{\kern0pt}inv\ d\ {\isacharparenleft}{\kern0pt}vec{\isacharunderscore}{\kern0pt}lambda\ v{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ GS{\isacharunderscore}{\kern0pt}inv{\isacharunderscore}{\kern0pt}def\ plus{\isacharunderscore}{\kern0pt}bfun{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ invertible{\isacharunderscore}{\kern0pt}Q{\isacharunderscore}{\kern0pt}GS\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}inverse{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isacharbrackleft}{\kern0pt}symmetric{\isacharbrackright}{\kern0pt}\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}mult{\isacharprime}{\kern0pt}{\isacharprime}{\kern0pt}\ apply{\isacharunderscore}{\kern0pt}bfun{\isacharunderscore}{\kern0pt}inverse\ {\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ norm{\isacharunderscore}{\kern0pt}GS{\isacharunderscore}{\kern0pt}QR{\isacharunderscore}{\kern0pt}le{\isacharunderscore}{\kern0pt}disc{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}norm\ {\isacharparenleft}{\kern0pt}inv\isactrlsub L\ {\isacharparenleft}{\kern0pt}Q{\isacharunderscore}{\kern0pt}GS\ d{\isacharparenright}{\kern0pt}\ o\isactrlsub L\ R{\isacharunderscore}{\kern0pt}GS\ d{\isacharparenright}{\kern0pt}\ {\isasymle}\ l{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}norm\ {\isacharparenleft}{\kern0pt}inv\isactrlsub L\ {\isacharparenleft}{\kern0pt}Q{\isacharunderscore}{\kern0pt}GS\ d{\isacharparenright}{\kern0pt}\ o\isactrlsub L\ R{\isacharunderscore}{\kern0pt}GS\ d{\isacharparenright}{\kern0pt}\ {\isasymle}\ norm\ {\isacharparenleft}{\kern0pt}inv\isactrlsub L\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}{\isasymlambda}{\isacharunderscore}{\kern0pt}{\isachardot}{\kern0pt}\ id{\isacharunderscore}{\kern0pt}blinfun{\isacharparenright}{\kern0pt}\ d{\isacharparenright}{\kern0pt}\ o\isactrlsub L\ {\isacharparenleft}{\kern0pt}l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isasymP}\isactrlsub {\isadigit{1}}\ {\isacharparenleft}{\kern0pt}mk{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\ d{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isachardoublequoteclose}\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}rule\ norm{\isacharunderscore}{\kern0pt}splitting{\isacharunderscore}{\kern0pt}le{\isacharbrackleft}{\kern0pt}of\ {\isachardoublequoteopen}mk{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\ d{\isachardoublequoteclose}{\isacharbrackright}{\kern0pt}{\isacharcomma}{\kern0pt}\ goal{\isacharunderscore}{\kern0pt}cases{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{case}\isamarkupfalse%
-\ {\isadigit{1}}\isanewline
-\ \ \ \ \isacommand{then}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}case\ \isanewline
-\ \ \ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ is{\isacharunderscore}{\kern0pt}splitting{\isacharunderscore}{\kern0pt}blin{\isacharunderscore}{\kern0pt}def{\isacharprime}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ nonneg{\isacharunderscore}{\kern0pt}id{\isacharunderscore}{\kern0pt}blinfun\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}scaleR\ nonneg{\isacharunderscore}{\kern0pt}{\isasymP}\isactrlsub {\isadigit{1}}\ scaleR{\isacharunderscore}{\kern0pt}nonneg{\isacharunderscore}{\kern0pt}nonneg{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{case}\isamarkupfalse%
-\ {\isadigit{3}}\isanewline
-\ \ \ \ \isacommand{then}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}case\ \isanewline
-\ \ \ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ R{\isacharunderscore}{\kern0pt}mat{\isacharunderscore}{\kern0pt}def\ P{\isacharunderscore}{\kern0pt}upper{\isacharunderscore}{\kern0pt}def\ Finite{\isacharunderscore}{\kern0pt}Cartesian{\isacharunderscore}{\kern0pt}Product{\isachardot}{\kern0pt}less{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}vec{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ nonneg{\isacharunderscore}{\kern0pt}P{\isacharunderscore}{\kern0pt}dec\ \isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ P{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}def\ nonneg{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}nonneg\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}scaleR{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ splitting{\isacharunderscore}{\kern0pt}gauss{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isacharequal}{\kern0pt}\ norm\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isasymP}\isactrlsub {\isadigit{1}}\ {\isacharparenleft}{\kern0pt}mk{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\ d{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isasymle}\ l{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{finally}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isacommand{{\isachardot}{\kern0pt}}\isamarkupfalse%
-\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{sublocale}\isamarkupfalse%
-\ GS{\isacharcolon}{\kern0pt}\ MDP{\isacharunderscore}{\kern0pt}QR\ A\ K\ r\ l\ Q{\isacharunderscore}{\kern0pt}GS\ R{\isacharunderscore}{\kern0pt}GS\isanewline
-\ \ \isakeyword{rewrites}\ {\isachardoublequoteopen}GS{\isachardot}{\kern0pt}{\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}split\ {\isacharequal}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}fun\isactrlsub b{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isasymSqunion}d{\isasymin}D\isactrlsub D{\isachardot}{\kern0pt}\ norm\ {\isacharparenleft}{\kern0pt}inv\isactrlsub L\ {\isacharparenleft}{\kern0pt}Q{\isacharunderscore}{\kern0pt}GS\ d{\isacharparenright}{\kern0pt}\ o\isactrlsub L\ R{\isacharunderscore}{\kern0pt}GS\ d{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharless}{\kern0pt}\ {\isadigit{1}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ norm{\isacharunderscore}{\kern0pt}GS{\isacharunderscore}{\kern0pt}QR{\isacharunderscore}{\kern0pt}le{\isacharunderscore}{\kern0pt}disc\ ex{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}fastforce\ intro{\isacharcolon}{\kern0pt}\ le{\isacharunderscore}{\kern0pt}less{\isacharunderscore}{\kern0pt}trans{\isacharbrackleft}{\kern0pt}of\ {\isacharunderscore}{\kern0pt}\ l\ {\isadigit{1}}{\isacharbrackright}{\kern0pt}\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ cSUP{\isacharunderscore}{\kern0pt}least{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}MDP{\isacharunderscore}{\kern0pt}QR\ A\ K\ r\ l\ Q{\isacharunderscore}{\kern0pt}GS\ R{\isacharunderscore}{\kern0pt}GS{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ unfold{\isacharunderscore}{\kern0pt}locales\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ splitting{\isacharunderscore}{\kern0pt}gauss\ ex{\isacharunderscore}{\kern0pt}opt{\isacharunderscore}{\kern0pt}blinfun{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}MDP{\isacharunderscore}{\kern0pt}QR{\isachardot}{\kern0pt}{\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}split\ A\ r\ Q{\isacharunderscore}{\kern0pt}GS\ R{\isacharunderscore}{\kern0pt}GS\ {\isacharequal}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}fun\isactrlsub b{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}fastforce\ simp{\isacharcolon}{\kern0pt}\ MDP{\isacharunderscore}{\kern0pt}QR{\isachardot}{\kern0pt}{\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}split{\isachardot}{\kern0pt}rep{\isacharunderscore}{\kern0pt}eq\ MDP{\isacharunderscore}{\kern0pt}QR{\isachardot}{\kern0pt}{\isasymL}{\isacharunderscore}{\kern0pt}split{\isacharunderscore}{\kern0pt}def\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}fun\isactrlsub b{\isachardot}{\kern0pt}rep{\isacharunderscore}{\kern0pt}eq\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}fun{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}GS{\isacharunderscore}{\kern0pt}inv\ GS{\isacharunderscore}{\kern0pt}inv{\isacharunderscore}{\kern0pt}blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{abbreviation}\isamarkupfalse%
-\ {\isachardoublequoteopen}gs{\isacharunderscore}{\kern0pt}measure\ {\isasymequiv}\ {\isacharparenleft}{\kern0pt}{\isasymlambda}{\isacharparenleft}{\kern0pt}eps{\isacharcomma}{\kern0pt}\ v{\isacharparenright}{\kern0pt}{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ if\ v\ {\isacharequal}{\kern0pt}\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt\ {\isasymor}\ l\ {\isacharequal}{\kern0pt}\ {\isadigit{0}}\isanewline
-\ \ \ \ then\ {\isadigit{0}}\isanewline
-\ \ \ \ else\ nat\ {\isacharparenleft}{\kern0pt}ceiling\ {\isacharparenleft}{\kern0pt}log\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharslash}{\kern0pt}l{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}dist\ v\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt{\isacharparenright}{\kern0pt}\ {\isacharminus}{\kern0pt}\ log\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharslash}{\kern0pt}l{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}eps\ {\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharminus}{\kern0pt}l{\isacharparenright}{\kern0pt}\ {\isacharslash}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isadigit{8}}\ {\isacharasterisk}{\kern0pt}\ l{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ dist{\isacharunderscore}{\kern0pt}{\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}split{\isacharunderscore}{\kern0pt}lt{\isacharunderscore}{\kern0pt}dist{\isacharunderscore}{\kern0pt}opt{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}dist\ v\ {\isacharparenleft}{\kern0pt}GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}fun\isactrlsub b\ v{\isacharparenright}{\kern0pt}\ {\isasymle}\ {\isadigit{2}}\ {\isacharasterisk}{\kern0pt}\ dist\ v\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ le{\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}dist\ v\ {\isacharparenleft}{\kern0pt}GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}fun\isactrlsub b\ v{\isacharparenright}{\kern0pt}\ {\isasymle}\ dist\ v\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt\ {\isacharplus}{\kern0pt}\ dist\ {\isacharparenleft}{\kern0pt}GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}fun\isactrlsub b\ v{\isacharparenright}{\kern0pt}\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ dist{\isacharunderscore}{\kern0pt}triangle\ dist{\isacharunderscore}{\kern0pt}commute{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ le{\isadigit{2}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}dist\ {\isacharparenleft}{\kern0pt}GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}fun\isactrlsub b\ v{\isacharparenright}{\kern0pt}\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt\ {\isasymle}\ GS{\isachardot}{\kern0pt}QR{\isacharunderscore}{\kern0pt}disc\ {\isacharasterisk}{\kern0pt}\ dist\ v\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ GS{\isachardot}{\kern0pt}{\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}split{\isacharunderscore}{\kern0pt}contraction\ GS{\isachardot}{\kern0pt}{\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}split{\isacharunderscore}{\kern0pt}fix\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharcomma}{\kern0pt}\ lifting{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ mult{\isacharunderscore}{\kern0pt}right{\isacharunderscore}{\kern0pt}mono{\isacharbrackleft}{\kern0pt}of\ GS{\isachardot}{\kern0pt}QR{\isacharunderscore}{\kern0pt}disc\ {\isadigit{1}}{\isacharbrackright}{\kern0pt}\ GS{\isachardot}{\kern0pt}QR{\isacharunderscore}{\kern0pt}contraction\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}fastforce\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ order{\isachardot}{\kern0pt}trans{\isacharbrackleft}{\kern0pt}OF\ le{\isadigit{2}}{\isacharbrackright}{\kern0pt}\ order{\isachardot}{\kern0pt}trans{\isacharbrackleft}{\kern0pt}OF\ le{\isadigit{1}}{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ GS{\isacharunderscore}{\kern0pt}QR{\isacharunderscore}{\kern0pt}disc{\isacharunderscore}{\kern0pt}le{\isacharunderscore}{\kern0pt}disc{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}GS{\isachardot}{\kern0pt}QR{\isacharunderscore}{\kern0pt}disc\ {\isasymle}\ l{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ norm{\isacharunderscore}{\kern0pt}GS{\isacharunderscore}{\kern0pt}QR{\isacharunderscore}{\kern0pt}le{\isacharunderscore}{\kern0pt}disc\ ex{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}fastforce\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ cSUP{\isacharunderscore}{\kern0pt}least{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ gs{\isacharunderscore}{\kern0pt}rel{\isacharunderscore}{\kern0pt}dec{\isacharcolon}{\kern0pt}\ \isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}l\ {\isasymnoteq}\ {\isadigit{0}}{\isachardoublequoteclose}\ {\isachardoublequoteopen}GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}fun\isactrlsub b\ v\ {\isasymnoteq}\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}{\isasymlceil}log\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}\ {\isacharslash}{\kern0pt}\ l{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}dist\ {\isacharparenleft}{\kern0pt}GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}fun\isactrlsub b\ v{\isacharparenright}{\kern0pt}\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt{\isacharparenright}{\kern0pt}\ {\isacharminus}{\kern0pt}\ c{\isasymrceil}\ {\isacharless}{\kern0pt}\ {\isasymlceil}log\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}\ {\isacharslash}{\kern0pt}\ l{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}dist\ v\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt{\isacharparenright}{\kern0pt}\ {\isacharminus}{\kern0pt}\ c{\isasymrceil}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}log\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}\ {\isacharslash}{\kern0pt}\ l{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}dist\ {\isacharparenleft}{\kern0pt}GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}fun\isactrlsub b\ v{\isacharparenright}{\kern0pt}\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt{\isacharparenright}{\kern0pt}\ {\isacharminus}{\kern0pt}\ c\ {\isasymle}\ log\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}\ {\isacharslash}{\kern0pt}\ l{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}l\ {\isacharasterisk}{\kern0pt}\ dist\ v\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt{\isacharparenright}{\kern0pt}\ {\isacharminus}{\kern0pt}\ c{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ GS{\isachardot}{\kern0pt}{\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}split{\isacharunderscore}{\kern0pt}contraction{\isacharbrackleft}{\kern0pt}of\ {\isacharunderscore}{\kern0pt}\ {\isachardoublequoteopen}{\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt{\isachardoublequoteclose}{\isacharbrackright}{\kern0pt}\ GS{\isachardot}{\kern0pt}QR{\isacharunderscore}{\kern0pt}contraction\ norm{\isacharunderscore}{\kern0pt}GS{\isacharunderscore}{\kern0pt}QR{\isacharunderscore}{\kern0pt}le{\isacharunderscore}{\kern0pt}disc\ disc{\isacharunderscore}{\kern0pt}lt{\isacharunderscore}{\kern0pt}one\ GS{\isacharunderscore}{\kern0pt}QR{\isacharunderscore}{\kern0pt}disc{\isacharunderscore}{\kern0pt}le{\isacharunderscore}{\kern0pt}disc\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}fastforce\ simp{\isacharcolon}{\kern0pt}\ assms\ less{\isacharunderscore}{\kern0pt}le\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ log{\isacharunderscore}{\kern0pt}le\ order{\isachardot}{\kern0pt}trans{\isacharbrackleft}{\kern0pt}OF\ GS{\isachardot}{\kern0pt}{\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}split{\isacharunderscore}{\kern0pt}contraction{\isacharbrackleft}{\kern0pt}of\ v\ {\isachardoublequoteopen}{\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt{\isachardoublequoteclose}{\isacharcomma}{\kern0pt}\ simplified{\isacharbrackright}{\kern0pt}{\isacharbrackright}{\kern0pt}\ mult{\isacharunderscore}{\kern0pt}right{\isacharunderscore}{\kern0pt}mono{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isacharequal}{\kern0pt}\ log\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}\ {\isacharslash}{\kern0pt}\ l{\isacharparenright}{\kern0pt}\ l\ {\isacharplus}{\kern0pt}\ log\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharslash}{\kern0pt}l{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}dist\ v\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt{\isacharparenright}{\kern0pt}\ {\isacharminus}{\kern0pt}\ c{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assms\ disc{\isacharunderscore}{\kern0pt}lt{\isacharunderscore}{\kern0pt}one\ \isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ less{\isacharunderscore}{\kern0pt}le\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ log{\isacharunderscore}{\kern0pt}mult{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isacharequal}{\kern0pt}\ {\isacharminus}{\kern0pt}{\isacharparenleft}{\kern0pt}log\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}\ {\isacharslash}{\kern0pt}\ l{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharslash}{\kern0pt}l{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharplus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}log\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharslash}{\kern0pt}l{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}dist\ v\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharminus}{\kern0pt}\ c{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assms\ disc{\isacharunderscore}{\kern0pt}lt{\isacharunderscore}{\kern0pt}one\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}subst\ log{\isacharunderscore}{\kern0pt}inverse{\isacharbrackleft}{\kern0pt}symmetric{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ less{\isacharunderscore}{\kern0pt}le\ right{\isacharunderscore}{\kern0pt}inverse{\isacharunderscore}{\kern0pt}eq{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}log\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharslash}{\kern0pt}l{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}dist\ v\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharminus}{\kern0pt}\ {\isadigit{1}}\ {\isacharminus}{\kern0pt}\ c{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assms\ order{\isachardot}{\kern0pt}strict{\isacharunderscore}{\kern0pt}implies{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}eq{\isacharbrackleft}{\kern0pt}OF\ disc{\isacharunderscore}{\kern0pt}lt{\isacharunderscore}{\kern0pt}one{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ log{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}one\ neq{\isacharunderscore}{\kern0pt}le{\isacharunderscore}{\kern0pt}trans{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{finally}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}log\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}\ {\isacharslash}{\kern0pt}\ l{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}dist\ {\isacharparenleft}{\kern0pt}GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}fun\isactrlsub b\ v{\isacharparenright}{\kern0pt}\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt{\isacharparenright}{\kern0pt}\ {\isacharminus}{\kern0pt}\ c\ {\isasymle}\ log\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}\ {\isacharslash}{\kern0pt}\ l{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}dist\ v\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt{\isacharparenright}{\kern0pt}\ {\isacharminus}{\kern0pt}\ {\isadigit{1}}\ {\isacharminus}{\kern0pt}\ c{\isachardoublequoteclose}\ \isacommand{{\isachardot}{\kern0pt}}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ linarith\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{function}\isamarkupfalse%
-\ gs{\isacharunderscore}{\kern0pt}iteration\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}real\ {\isasymRightarrow}\ {\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}s\ {\isasymRightarrow}\isactrlsub b\ real{\isacharparenright}{\kern0pt}\ {\isasymRightarrow}\ {\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}s\ {\isasymRightarrow}\isactrlsub b\ real{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}gs{\isacharunderscore}{\kern0pt}iteration\ eps\ v\ {\isacharequal}{\kern0pt}\isanewline
-\ \ {\isacharparenleft}{\kern0pt}if\ {\isadigit{2}}\ {\isacharasterisk}{\kern0pt}\ l\ {\isacharasterisk}{\kern0pt}\ dist\ v\ {\isacharparenleft}{\kern0pt}GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}fun\isactrlsub b\ v{\isacharparenright}{\kern0pt}\ {\isacharless}{\kern0pt}\ eps\ {\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharminus}{\kern0pt}l{\isacharparenright}{\kern0pt}\ {\isasymor}\ eps\ {\isasymle}\ {\isadigit{0}}\ then\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}fun\isactrlsub b\ v\ else\ gs{\isacharunderscore}{\kern0pt}iteration\ eps\ {\isacharparenleft}{\kern0pt}GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}fun\isactrlsub b\ v{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isacommand{termination}\isamarkupfalse%
-\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}relation\ {\isachardoublequoteopen}Wellfounded{\isachardot}{\kern0pt}measure\ gs{\isacharunderscore}{\kern0pt}measure{\isachardoublequoteclose}{\isacharcomma}{\kern0pt}\ {\isacharparenleft}{\kern0pt}simp{\isacharsemicolon}{\kern0pt}\ fail{\isacharparenright}{\kern0pt}{\isacharcomma}{\kern0pt}\ cases\ {\isachardoublequoteopen}l\ {\isacharequal}{\kern0pt}\ {\isadigit{0}}{\isachardoublequoteclose}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{case}\isamarkupfalse%
-\ False\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\ eps\ v\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\ h{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymnot}\ {\isacharparenleft}{\kern0pt}{\isadigit{2}}\ {\isacharasterisk}{\kern0pt}\ l\ {\isacharasterisk}{\kern0pt}\ dist\ v\ {\isacharparenleft}{\kern0pt}GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}fun\isactrlsub b\ v{\isacharparenright}{\kern0pt}\ {\isacharless}{\kern0pt}\ eps\ {\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}\ {\isacharminus}{\kern0pt}\ l{\isacharparenright}{\kern0pt}\ {\isasymor}\ eps\ {\isasymle}\ {\isadigit{0}}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}eps{\isacharcomma}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}fun\isactrlsub b\ v{\isacharparenright}{\kern0pt}{\isacharcomma}{\kern0pt}\ eps{\isacharcomma}{\kern0pt}\ v{\isacharparenright}{\kern0pt}\ {\isasymin}\ Wellfounded{\isachardot}{\kern0pt}measure\ gs{\isacharunderscore}{\kern0pt}measure{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ gt{\isacharunderscore}{\kern0pt}zero{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}l\ {\isasymnoteq}\ {\isadigit{0}}{\isachardoublequoteclose}\ {\isachardoublequoteopen}eps\ {\isachargreater}{\kern0pt}\ {\isadigit{0}}{\isachardoublequoteclose}\ \isakeyword{and}\ dist{\isacharunderscore}{\kern0pt}ge{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}eps\ {\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}\ {\isacharminus}{\kern0pt}\ l{\isacharparenright}{\kern0pt}\ {\isasymle}\ dist\ v\ {\isacharparenleft}{\kern0pt}GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}fun\isactrlsub b\ v{\isacharparenright}{\kern0pt}\ {\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isadigit{2}}\ {\isacharasterisk}{\kern0pt}\ l{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ h\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ algebra{\isacharunderscore}{\kern0pt}simps{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ v{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}opt{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}v\ {\isasymnoteq}\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ h\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}log\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}\ {\isacharslash}{\kern0pt}\ l{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}eps\ {\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}\ {\isacharminus}{\kern0pt}\ l{\isacharparenright}{\kern0pt}\ {\isacharslash}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isadigit{8}}\ {\isacharasterisk}{\kern0pt}\ l{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharless}{\kern0pt}\ log\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}\ {\isacharslash}{\kern0pt}\ l{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}dist\ v\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}intro\ log{\isacharunderscore}{\kern0pt}less{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isadigit{1}}\ {\isacharless}{\kern0pt}\ {\isadigit{1}}\ {\isacharslash}{\kern0pt}\ l{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ mult{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}less{\isacharunderscore}{\kern0pt}div{\isacharunderscore}{\kern0pt}pos\ intro{\isacharcolon}{\kern0pt}\ neq{\isacharunderscore}{\kern0pt}le{\isacharunderscore}{\kern0pt}trans{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isadigit{0}}\ {\isacharless}{\kern0pt}\ eps\ {\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}\ {\isacharminus}{\kern0pt}\ l{\isacharparenright}{\kern0pt}\ {\isacharslash}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isadigit{8}}\ {\isacharasterisk}{\kern0pt}\ l{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ mult{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}less{\isacharunderscore}{\kern0pt}div{\isacharunderscore}{\kern0pt}pos\ intro{\isacharcolon}{\kern0pt}\ neq{\isacharunderscore}{\kern0pt}le{\isacharunderscore}{\kern0pt}trans{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}eps\ {\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}\ {\isacharminus}{\kern0pt}\ l{\isacharparenright}{\kern0pt}\ {\isacharslash}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isadigit{8}}\ {\isacharasterisk}{\kern0pt}\ l{\isacharparenright}{\kern0pt}\ {\isacharless}{\kern0pt}\ dist\ v\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt{\isachardoublequoteclose}\ \isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ dist{\isacharunderscore}{\kern0pt}pos{\isacharunderscore}{\kern0pt}lt{\isacharbrackleft}{\kern0pt}OF\ v{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}opt{\isacharbrackright}{\kern0pt}\ dist{\isacharunderscore}{\kern0pt}{\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}split{\isacharunderscore}{\kern0pt}lt{\isacharunderscore}{\kern0pt}dist{\isacharunderscore}{\kern0pt}opt{\isacharbrackleft}{\kern0pt}of\ v{\isacharbrackright}{\kern0pt}\ gt{\isacharunderscore}{\kern0pt}zero\ zero{\isacharunderscore}{\kern0pt}le{\isacharunderscore}{\kern0pt}disc\ \isanewline
-\ \ \ \ \ \ \ \ \ \ mult{\isacharunderscore}{\kern0pt}strict{\isacharunderscore}{\kern0pt}left{\isacharunderscore}{\kern0pt}mono{\isacharbrackleft}{\kern0pt}of\ {\isachardoublequoteopen}dist\ v\ {\isacharparenleft}{\kern0pt}GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}fun\isactrlsub b\ v{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isadigit{4}}\ {\isacharasterisk}{\kern0pt}\ dist\ v\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ l{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}intro\ mult{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}div{\isacharunderscore}{\kern0pt}pos{\isacharunderscore}{\kern0pt}less\ le{\isacharunderscore}{\kern0pt}less{\isacharunderscore}{\kern0pt}trans{\isacharbrackleft}{\kern0pt}OF\ dist{\isacharunderscore}{\kern0pt}ge{\isacharbrackright}{\kern0pt}{\isacharcomma}{\kern0pt}\ argo{\isacharplus}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ gs{\isacharunderscore}{\kern0pt}rel{\isacharunderscore}{\kern0pt}dec\ h\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\isacommand{qed}\isamarkupfalse%
-\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ THE{\isacharunderscore}{\kern0pt}fix{\isacharunderscore}{\kern0pt}GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}fun\isactrlsub b{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}THE\ v{\isachardot}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}fun\isactrlsub b\ v\ {\isacharequal}{\kern0pt}\ v{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ GS{\isachardot}{\kern0pt}{\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}lim{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}\ GS{\isachardot}{\kern0pt}{\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}split{\isacharunderscore}{\kern0pt}fix\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ blast{\isacharplus}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\begin{isamarkuptext}%
-The distance between an estimate for the value and the optimal value can be bounded with respect to 
-the distance between the estimate and the result of applying it to \isa{{\isasymL}\isactrlsub b}%
-\end{isamarkuptext}\isamarkuptrue%
-\isacommand{lemma}\isamarkupfalse%
-\ contraction{\isacharunderscore}{\kern0pt}{\isasymL}{\isacharunderscore}{\kern0pt}split{\isacharunderscore}{\kern0pt}dist{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isadigit{1}}\ {\isacharminus}{\kern0pt}\ l{\isacharparenright}{\kern0pt}\ {\isacharasterisk}{\kern0pt}\ dist\ v\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt\ {\isasymle}\ dist\ v\ {\isacharparenleft}{\kern0pt}GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}fun\isactrlsub b\ v{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ GS{\isacharunderscore}{\kern0pt}QR{\isacharunderscore}{\kern0pt}disc{\isacharunderscore}{\kern0pt}le{\isacharunderscore}{\kern0pt}disc\ \isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}fastforce\ \isanewline
-\ \ \ \ \ \ simp{\isacharcolon}{\kern0pt}\ THE{\isacharunderscore}{\kern0pt}fix{\isacharunderscore}{\kern0pt}GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}fun\isactrlsub b\ \isanewline
-\ \ \ \ \ \ intro{\isacharcolon}{\kern0pt}\ order{\isachardot}{\kern0pt}trans{\isacharbrackleft}{\kern0pt}OF\ {\isacharunderscore}{\kern0pt}\ contraction{\isacharunderscore}{\kern0pt}dist{\isacharcomma}{\kern0pt}\ of\ {\isacharunderscore}{\kern0pt}\ l{\isacharbrackright}{\kern0pt}\ order{\isachardot}{\kern0pt}trans{\isacharbrackleft}{\kern0pt}OF\ GS{\isachardot}{\kern0pt}{\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}split{\isacharunderscore}{\kern0pt}contraction{\isacharbrackright}{\kern0pt}\ mult{\isacharunderscore}{\kern0pt}right{\isacharunderscore}{\kern0pt}mono{\isacharparenright}{\kern0pt}{\isacharplus}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ dist{\isacharunderscore}{\kern0pt}{\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}split{\isacharunderscore}{\kern0pt}opt{\isacharunderscore}{\kern0pt}eps{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}eps\ {\isachargreater}{\kern0pt}\ {\isadigit{0}}{\isachardoublequoteclose}\ {\isachardoublequoteopen}{\isadigit{2}}\ {\isacharasterisk}{\kern0pt}\ l\ {\isacharasterisk}{\kern0pt}\ dist\ v\ {\isacharparenleft}{\kern0pt}GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}fun\isactrlsub b\ v{\isacharparenright}{\kern0pt}\ {\isacharless}{\kern0pt}\ eps\ {\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharminus}{\kern0pt}l{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}dist\ {\isacharparenleft}{\kern0pt}GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}fun\isactrlsub b\ v{\isacharparenright}{\kern0pt}\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt\ {\isacharless}{\kern0pt}\ eps\ {\isacharslash}{\kern0pt}\ {\isadigit{2}}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}dist\ v\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt\ {\isasymle}\ dist\ v\ {\isacharparenleft}{\kern0pt}GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}fun\isactrlsub b\ v{\isacharparenright}{\kern0pt}\ {\isacharslash}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}\ {\isacharminus}{\kern0pt}\ l{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ contraction{\isacharunderscore}{\kern0pt}{\isasymL}{\isacharunderscore}{\kern0pt}split{\isacharunderscore}{\kern0pt}dist\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ mult{\isachardot}{\kern0pt}commute\ pos{\isacharunderscore}{\kern0pt}le{\isacharunderscore}{\kern0pt}divide{\isacharunderscore}{\kern0pt}eq{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isadigit{2}}\ {\isacharasterisk}{\kern0pt}\ l\ {\isacharasterisk}{\kern0pt}\ dist\ v\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt\ {\isasymle}\ {\isadigit{2}}\ {\isacharasterisk}{\kern0pt}\ l\ {\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}dist\ v\ {\isacharparenleft}{\kern0pt}GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}fun\isactrlsub b\ v{\isacharparenright}{\kern0pt}\ {\isacharslash}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}\ {\isacharminus}{\kern0pt}\ l{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ contraction{\isacharunderscore}{\kern0pt}{\isasymL}{\isacharunderscore}{\kern0pt}dist\ assms\ mult{\isacharunderscore}{\kern0pt}le{\isacharunderscore}{\kern0pt}cancel{\isacharunderscore}{\kern0pt}left{\isacharunderscore}{\kern0pt}pos{\isacharbrackleft}{\kern0pt}of\ {\isachardoublequoteopen}{\isadigit{2}}\ {\isacharasterisk}{\kern0pt}\ l{\isachardoublequoteclose}{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}fastforce\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ mult{\isacharunderscore}{\kern0pt}left{\isacharunderscore}{\kern0pt}mono{\isacharbrackleft}{\kern0pt}of\ {\isacharunderscore}{\kern0pt}\ {\isacharunderscore}{\kern0pt}\ {\isachardoublequoteopen}{\isadigit{2}}\ {\isacharasterisk}{\kern0pt}\ l{\isachardoublequoteclose}{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isadigit{2}}\ {\isacharasterisk}{\kern0pt}\ l\ {\isacharasterisk}{\kern0pt}\ dist\ v\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt\ {\isacharless}{\kern0pt}\ eps{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ assms{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}\ pos{\isacharunderscore}{\kern0pt}divide{\isacharunderscore}{\kern0pt}less{\isacharunderscore}{\kern0pt}eq\ intro{\isacharcolon}{\kern0pt}\ order{\isachardot}{\kern0pt}strict{\isacharunderscore}{\kern0pt}trans{\isadigit{1}}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}dist\ v\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt\ {\isacharasterisk}{\kern0pt}\ l\ {\isacharless}{\kern0pt}\ eps\ {\isacharslash}{\kern0pt}\ {\isadigit{2}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ argo\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isacharasterisk}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}l\ {\isacharasterisk}{\kern0pt}\ dist\ v\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt\ {\isacharless}{\kern0pt}\ eps\ {\isacharslash}{\kern0pt}\ {\isadigit{2}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ algebra{\isacharunderscore}{\kern0pt}simps{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}dist\ {\isacharparenleft}{\kern0pt}GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}fun\isactrlsub b\ v{\isacharparenright}{\kern0pt}\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt\ {\isacharless}{\kern0pt}\ eps\ {\isacharslash}{\kern0pt}\ {\isadigit{2}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ GS{\isachardot}{\kern0pt}{\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}split{\isacharunderscore}{\kern0pt}contraction{\isacharbrackleft}{\kern0pt}of\ v\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt{\isacharbrackright}{\kern0pt}\ order{\isachardot}{\kern0pt}trans\ mult{\isacharunderscore}{\kern0pt}right{\isacharunderscore}{\kern0pt}mono{\isacharbrackleft}{\kern0pt}OF\ GS{\isacharunderscore}{\kern0pt}QR{\isacharunderscore}{\kern0pt}disc{\isacharunderscore}{\kern0pt}le{\isacharunderscore}{\kern0pt}disc\ zero{\isacharunderscore}{\kern0pt}le{\isacharunderscore}{\kern0pt}dist{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}fastforce\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ le{\isacharunderscore}{\kern0pt}less{\isacharunderscore}{\kern0pt}trans{\isacharbrackleft}{\kern0pt}OF\ {\isacharunderscore}{\kern0pt}\ {\isacharasterisk}{\kern0pt}{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isacommand{end}\isamarkupfalse%
-\isanewline
-\isanewline
-\isacommand{context}\isamarkupfalse%
-\ MDP{\isacharunderscore}{\kern0pt}ord\ \isanewline
-\isakeyword{begin}\ \ \ \ \ \ \ \isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ is{\isacharunderscore}{\kern0pt}am{\isacharunderscore}{\kern0pt}GS{\isacharunderscore}{\kern0pt}inv{\isacharunderscore}{\kern0pt}extend{\isacharprime}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isasymAnd}s{\isachardot}{\kern0pt}\ s\ {\isacharless}{\kern0pt}\ x\ {\isasymLongrightarrow}\ is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max\ {\isacharparenleft}{\kern0pt}{\isasymlambda}d{\isachardot}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}inv\ d\ v\ {\isachardollar}{\kern0pt}\ s{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymlambda}d{\isachardot}{\kern0pt}\ d\ {\isasymin}\ D\isactrlsub D{\isacharparenright}{\kern0pt}\ d{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max\ {\isacharparenleft}{\kern0pt}{\isasymlambda}d{\isachardot}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}step\ d\ v\ {\isachardollar}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymlambda}d{\isachardot}{\kern0pt}\ d\ {\isasymin}\ D\isactrlsub D{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}d{\isacharparenleft}{\kern0pt}x\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ a{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}s\ {\isasymle}\ x{\isachardoublequoteclose}\ {\isachardoublequoteopen}d\ {\isasymin}\ D\isactrlsub D{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max\ {\isacharparenleft}{\kern0pt}{\isasymlambda}d{\isachardot}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}inv\ d\ v\ {\isachardollar}{\kern0pt}\ s{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymlambda}d{\isachardot}{\kern0pt}\ d\ {\isasymin}\ D\isactrlsub D{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}d{\isacharparenleft}{\kern0pt}x\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ a{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ a{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}a\ {\isasymin}\ A\ x{\isachardoublequoteclose}\ \isacommand{using}\isamarkupfalse%
-\ assms{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}\ \isacommand{unfolding}\isamarkupfalse%
-\ is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}linorder\ is{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det{\isacharunderscore}{\kern0pt}def\ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ split{\isacharcolon}{\kern0pt}\ if{\isacharunderscore}{\kern0pt}splits{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isacharasterisk}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymexists}y{\isachardot}{\kern0pt}\ y\ {\isacharless}{\kern0pt}\ x\ {\isasymLongrightarrow}\ s{\isasymle}Max\ {\isacharbraceleft}{\kern0pt}y{\isachardot}{\kern0pt}\ y\ {\isacharless}{\kern0pt}\ x{\isacharbraceright}{\kern0pt}\ {\isasymlongleftrightarrow}\ s\ {\isacharless}{\kern0pt}\ x{\isachardoublequoteclose}\ \isakeyword{for}\ x\ s\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isacharprime}{\kern0pt}s\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ linorder{\isacharunderscore}{\kern0pt}class{\isachardot}{\kern0pt}Max{\isacharunderscore}{\kern0pt}ge{\isacharunderscore}{\kern0pt}iff{\isacharbrackleft}{\kern0pt}OF\ finite{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isasymAnd}s{\isachardot}{\kern0pt}\ s\ {\isacharless}{\kern0pt}\ x\ {\isasymLongrightarrow}\ is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max\ {\isacharparenleft}{\kern0pt}{\isasymlambda}d{\isachardot}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}step\ d\ v\ {\isachardollar}{\kern0pt}\ s{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymlambda}d{\isachardot}{\kern0pt}\ d\ {\isasymin}\ D\isactrlsub D{\isacharparenright}{\kern0pt}\ d{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}gs{\isacharunderscore}{\kern0pt}iff{\isacharbrackleft}{\kern0pt}OF\ assms{\isacharparenleft}{\kern0pt}{\isadigit{4}}{\isacharparenright}{\kern0pt}{\isacharcomma}{\kern0pt}\ of\ {\isachardoublequoteopen}Max\ {\isacharbraceleft}{\kern0pt}y{\isachardot}{\kern0pt}\ y\ {\isacharless}{\kern0pt}\ x{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}{\isacharbrackright}{\kern0pt}\ assms{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}cases\ {\isachardoublequoteopen}{\isasymexists}y{\isachardot}{\kern0pt}\ y\ {\isacharless}{\kern0pt}\ x{\isachardoublequoteclose}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ {\isacharasterisk}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isasymAnd}s{\isachardot}{\kern0pt}\ s\ {\isacharless}{\kern0pt}\ x\ {\isasymLongrightarrow}\ is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max\ {\isacharparenleft}{\kern0pt}{\isasymlambda}d{\isachardot}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}step\ d\ v\ {\isachardollar}{\kern0pt}\ s{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymlambda}d{\isachardot}{\kern0pt}\ d\ {\isasymin}\ D\isactrlsub D{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}d{\isacharparenleft}{\kern0pt}x\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ a{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ GS{\isacharunderscore}{\kern0pt}opt{\isacharunderscore}{\kern0pt}indep{\isacharunderscore}{\kern0pt}high\ a\ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isasymAnd}s{\isachardot}{\kern0pt}\ s\ {\isasymle}\ x\ {\isasymLongrightarrow}\ is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max\ {\isacharparenleft}{\kern0pt}{\isasymlambda}d{\isachardot}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}step\ d\ v\ {\isachardollar}{\kern0pt}\ s{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymlambda}d{\isachardot}{\kern0pt}\ d\ {\isasymin}\ D\isactrlsub D{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}d{\isacharparenleft}{\kern0pt}x\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ a{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assms{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}\ antisym{\isacharunderscore}{\kern0pt}conv{\isadigit{1}}\ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}gs{\isacharunderscore}{\kern0pt}iff{\isacharbrackleft}{\kern0pt}of\ {\isachardoublequoteopen}d{\isacharparenleft}{\kern0pt}x\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ a{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ s{\isacharbrackright}{\kern0pt}\ assms{\isacharparenleft}{\kern0pt}{\isadigit{4}}{\isacharparenright}{\kern0pt}\ assms\ a\ \isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}intro\ is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}GS{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}splitting{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}\ auto\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ {\isachardoublequoteopen}opt{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}gs{\isacharprime}{\kern0pt}\ d\ v\ s\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}LEAST\ a{\isachardot}{\kern0pt}\ is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max\ {\isacharparenleft}{\kern0pt}{\isasymlambda}a{\isachardot}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}step\ {\isacharparenleft}{\kern0pt}d{\isacharparenleft}{\kern0pt}s\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ a{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ v\ {\isachardollar}{\kern0pt}\ s{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymlambda}a{\isachardot}{\kern0pt}\ a\ {\isasymin}\ A\ s{\isacharparenright}{\kern0pt}\ a{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ {\isachardoublequoteopen}GS{\isacharunderscore}{\kern0pt}iter\ a\ v\ s\ {\isacharequal}{\kern0pt}\ r\ {\isacharparenleft}{\kern0pt}s{\isacharcomma}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isacharplus}{\kern0pt}\ l\ {\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSum}s{\isacharprime}{\kern0pt}\ {\isasymin}\ UNIV{\isachardot}{\kern0pt}\ pmf\ {\isacharparenleft}{\kern0pt}K{\isacharparenleft}{\kern0pt}s{\isacharcomma}{\kern0pt}a{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ s{\isacharprime}{\kern0pt}\ {\isacharasterisk}{\kern0pt}\ v\ {\isachardollar}{\kern0pt}\ s{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ {\isachardoublequoteopen}GS{\isacharunderscore}{\kern0pt}iter{\isacharunderscore}{\kern0pt}max\ v\ s\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSqunion}a\ {\isasymin}\ A\ s{\isachardot}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}iter\ a\ v\ s{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}iter{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}{\isasymAnd}s{\isachardot}{\kern0pt}\ s\ {\isacharless}{\kern0pt}\ k\ {\isasymLongrightarrow}\ v{\isacharprime}{\kern0pt}\ {\isachardollar}{\kern0pt}\ s\ {\isacharequal}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}rec\ v\ {\isachardollar}{\kern0pt}\ s{\isachardoublequoteclose}\ {\isachardoublequoteopen}{\isasymAnd}s{\isachardot}{\kern0pt}\ k\ {\isasymle}\ s\ {\isasymLongrightarrow}\ v{\isacharprime}{\kern0pt}\ {\isachardollar}{\kern0pt}\ s\ {\isacharequal}{\kern0pt}\ v\ {\isachardollar}{\kern0pt}\ s{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}step\ {\isacharparenleft}{\kern0pt}d{\isacharparenleft}{\kern0pt}k\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ a{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ v\ {\isachardollar}{\kern0pt}\ k\ {\isacharequal}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}iter\ a\ v{\isacharprime}{\kern0pt}\ k{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}P{\isacharunderscore}{\kern0pt}lower\ d\ {\isacharasterisk}{\kern0pt}v\ GS{\isacharunderscore}{\kern0pt}rec\ v{\isacharparenright}{\kern0pt}\ {\isachardollar}{\kern0pt}\ k\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}P{\isacharunderscore}{\kern0pt}lower\ d\ {\isacharasterisk}{\kern0pt}v\ v{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isachardollar}{\kern0pt}\ k{\isachardoublequoteclose}\ \isakeyword{for}\ d\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ slt{\isacharunderscore}{\kern0pt}P{\isacharunderscore}{\kern0pt}lower\ assms\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ strict{\isacharunderscore}{\kern0pt}lower{\isacharunderscore}{\kern0pt}triangular{\isacharunderscore}{\kern0pt}mat{\isacharunderscore}{\kern0pt}mult{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}P{\isacharunderscore}{\kern0pt}upper\ d\ {\isacharasterisk}{\kern0pt}v\ v{\isacharparenright}{\kern0pt}\ {\isachardollar}{\kern0pt}\ k\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}P{\isacharunderscore}{\kern0pt}upper\ d\ {\isacharasterisk}{\kern0pt}v\ v{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isachardollar}{\kern0pt}\ k{\isachardoublequoteclose}\ \isakeyword{for}\ d\isanewline
-\ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ P{\isacharunderscore}{\kern0pt}upper{\isacharunderscore}{\kern0pt}def\ \isacommand{using}\isamarkupfalse%
-\ assms\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ matrix{\isacharunderscore}{\kern0pt}vector{\isacharunderscore}{\kern0pt}mult{\isacharunderscore}{\kern0pt}def\ if{\isacharunderscore}{\kern0pt}distrib{\isacharbrackleft}{\kern0pt}of\ {\isachardoublequoteopen}{\isasymlambda}x{\isachardot}{\kern0pt}\ x\ {\isacharasterisk}{\kern0pt}\ {\isacharunderscore}{\kern0pt}\ {\isachardollar}{\kern0pt}\ {\isacharunderscore}{\kern0pt}{\isachardoublequoteclose}{\isacharbrackright}{\kern0pt}\ cong{\isacharcolon}{\kern0pt}\ if{\isacharunderscore}{\kern0pt}cong{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}P{\isacharunderscore}{\kern0pt}lower\ d\ {\isacharplus}{\kern0pt}\ P{\isacharunderscore}{\kern0pt}upper\ d\ {\isacharequal}{\kern0pt}\ P{\isacharunderscore}{\kern0pt}dec\ d{\isachardoublequoteclose}\ \isakeyword{for}\ d\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ P{\isacharunderscore}{\kern0pt}lower{\isacharunderscore}{\kern0pt}def\ P{\isacharunderscore}{\kern0pt}upper{\isacharunderscore}{\kern0pt}def\ Finite{\isacharunderscore}{\kern0pt}Cartesian{\isacharunderscore}{\kern0pt}Product{\isachardot}{\kern0pt}vec{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}iff{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ vector{\isacharunderscore}{\kern0pt}add{\isacharunderscore}{\kern0pt}component{\isacharbrackleft}{\kern0pt}symmetric{\isacharbrackright}{\kern0pt}\ matrix{\isacharunderscore}{\kern0pt}vector{\isacharunderscore}{\kern0pt}mult{\isacharunderscore}{\kern0pt}diff{\isacharunderscore}{\kern0pt}rdistrib{\isacharbrackleft}{\kern0pt}symmetric{\isacharbrackright}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}step{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ matrix{\isacharunderscore}{\kern0pt}vector{\isacharunderscore}{\kern0pt}mult{\isacharunderscore}{\kern0pt}def\ P{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}elem\ P{\isacharunderscore}{\kern0pt}lower{\isacharunderscore}{\kern0pt}def\ P{\isacharunderscore}{\kern0pt}upper{\isacharunderscore}{\kern0pt}def\ GS{\isacharunderscore}{\kern0pt}iter{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}fastforce\ simp{\isacharcolon}{\kern0pt}\ sum{\isachardot}{\kern0pt}distrib{\isacharbrackleft}{\kern0pt}symmetric{\isacharbrackright}{\kern0pt}\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ sum{\isachardot}{\kern0pt}cong{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}iter{\isacharunderscore}{\kern0pt}max{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}{\isasymAnd}s{\isachardot}{\kern0pt}\ s\ {\isacharless}{\kern0pt}\ k\ {\isasymLongrightarrow}\ v{\isacharprime}{\kern0pt}\ {\isachardollar}{\kern0pt}\ s\ {\isacharequal}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}rec\ v\ {\isachardollar}{\kern0pt}\ s{\isachardoublequoteclose}\ {\isachardoublequoteopen}{\isasymAnd}s{\isachardot}{\kern0pt}\ k\ {\isasymle}\ s\ {\isasymLongrightarrow}\ v{\isacharprime}{\kern0pt}\ {\isachardollar}{\kern0pt}\ s\ {\isacharequal}{\kern0pt}\ v\ {\isachardollar}{\kern0pt}\ s{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}GS{\isacharunderscore}{\kern0pt}rec\ v\ {\isachardollar}{\kern0pt}\ k\ {\isacharequal}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}iter{\isacharunderscore}{\kern0pt}max\ v{\isacharprime}{\kern0pt}\ k{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}iter{\isacharbrackleft}{\kern0pt}OF\ assms{\isacharbrackright}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}eq{\isacharprime}{\kern0pt}{\isacharbrackleft}{\kern0pt}of\ {\isacharunderscore}{\kern0pt}\ {\isacharunderscore}{\kern0pt}\ undefined{\isacharbrackright}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}iter{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}def\ \isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ {\isachardoublequoteopen}GS{\isacharunderscore}{\kern0pt}iter{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max\ v\ s\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}LEAST\ a{\isachardot}{\kern0pt}\ is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max\ {\isacharparenleft}{\kern0pt}{\isasymlambda}a{\isachardot}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}iter\ a\ v\ s{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymlambda}a{\isachardot}{\kern0pt}\ a\ {\isasymin}\ A\ s{\isacharparenright}{\kern0pt}\ a{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ {\isachardoublequoteopen}GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}am{\isacharunderscore}{\kern0pt}code\ v\ d\ s\ {\isacharequal}{\kern0pt}\ foldl\ {\isacharparenleft}{\kern0pt}{\isasymlambda}vd\ s{\isachardot}{\kern0pt}\ vd{\isacharparenleft}{\kern0pt}s\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}GS{\isacharunderscore}{\kern0pt}iter{\isacharunderscore}{\kern0pt}max\ {\isacharparenleft}{\kern0pt}{\isasymchi}\ s{\isachardot}{\kern0pt}\ fst\ {\isacharparenleft}{\kern0pt}vd\ s{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ s{\isacharcomma}{\kern0pt}\ \ GS{\isacharunderscore}{\kern0pt}iter{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max\ {\isacharparenleft}{\kern0pt}{\isasymchi}\ s{\isachardot}{\kern0pt}\ fst\ {\isacharparenleft}{\kern0pt}vd\ s{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ s{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymlambda}s{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}v\ {\isachardollar}{\kern0pt}\ s{\isacharcomma}{\kern0pt}\ d\ s{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}sorted{\isacharunderscore}{\kern0pt}list{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}set\ {\isacharbraceleft}{\kern0pt}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}s{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ s{\isachardoublequoteclose}\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ {\isachardoublequoteopen}GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}am{\isacharunderscore}{\kern0pt}code{\isacharprime}{\kern0pt}\ v\ d\ s\ {\isacharequal}{\kern0pt}\ foldl\ {\isacharparenleft}{\kern0pt}{\isasymlambda}vd\ s{\isachardot}{\kern0pt}\ vd{\isacharparenleft}{\kern0pt}s\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}GS{\isacharunderscore}{\kern0pt}iter{\isacharunderscore}{\kern0pt}max\ {\isacharparenleft}{\kern0pt}{\isasymchi}\ s{\isachardot}{\kern0pt}\ fst\ {\isacharparenleft}{\kern0pt}vd\ s{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ s{\isacharcomma}{\kern0pt}\ \ GS{\isacharunderscore}{\kern0pt}iter{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max\ {\isacharparenleft}{\kern0pt}{\isasymchi}\ s{\isachardot}{\kern0pt}\ fst\ {\isacharparenleft}{\kern0pt}vd\ s{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ s{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymlambda}s{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}v\ {\isachardollar}{\kern0pt}\ s{\isacharcomma}{\kern0pt}\ d\ s{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}sorted{\isacharunderscore}{\kern0pt}list{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}set\ UNIV{\isacharparenright}{\kern0pt}\ s{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}am{\isacharunderscore}{\kern0pt}code{\isacharprime}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}am{\isacharunderscore}{\kern0pt}code\ {\isacharequal}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}am{\isacharunderscore}{\kern0pt}code{\isacharprime}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isacharasterisk}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}sorted{\isacharunderscore}{\kern0pt}list{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}set\ UNIV\ {\isacharequal}{\kern0pt}\ sorted{\isacharunderscore}{\kern0pt}list{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}set{\isacharbraceleft}{\kern0pt}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}s{\isacharbraceright}{\kern0pt}\ {\isacharat}{\kern0pt}\ sorted{\isacharunderscore}{\kern0pt}list{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}set{\isacharbraceleft}{\kern0pt}s{\isacharless}{\kern0pt}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{for}\ s\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isacharprime}{\kern0pt}s\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ sorted{\isacharunderscore}{\kern0pt}list{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}set{\isacharunderscore}{\kern0pt}split{\isacharprime}{\kern0pt}{\isacharbrackleft}{\kern0pt}OF\ finite{\isacharcomma}{\kern0pt}\ of\ UNIV\ s{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ atMost{\isacharunderscore}{\kern0pt}def\ greaterThan{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}am{\isacharunderscore}{\kern0pt}code\ v\ d\ s\ {\isacharequal}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}am{\isacharunderscore}{\kern0pt}code{\isacharprime}{\kern0pt}\ v\ d\ s{\isachardoublequoteclose}\ \isakeyword{for}\ v\ d\ s\isanewline
-\ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}am{\isacharunderscore}{\kern0pt}code{\isacharunderscore}{\kern0pt}def\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}am{\isacharunderscore}{\kern0pt}code{\isacharprime}{\kern0pt}{\isacharunderscore}{\kern0pt}def\ {\isacharasterisk}{\kern0pt}{\isacharbrackleft}{\kern0pt}of\ s{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}fastforce\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ foldl{\isacharunderscore}{\kern0pt}upd{\isacharunderscore}{\kern0pt}notin{\isacharprime}{\kern0pt}{\isacharbrackleft}{\kern0pt}symmetric{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\ \isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ opt{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}gs{\isacharprime}{\kern0pt}{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}GS{\isacharunderscore}{\kern0pt}iter{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}{\isasymAnd}s{\isachardot}{\kern0pt}\ s\ {\isacharless}{\kern0pt}\ k\ {\isasymLongrightarrow}\ v{\isacharprime}{\kern0pt}\ {\isachardollar}{\kern0pt}\ s\ {\isacharequal}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}rec\ v\ {\isachardollar}{\kern0pt}\ s{\isachardoublequoteclose}\ {\isachardoublequoteopen}{\isasymAnd}s{\isachardot}{\kern0pt}\ k\ {\isasymle}\ s\ {\isasymLongrightarrow}\ v{\isacharprime}{\kern0pt}\ {\isachardollar}{\kern0pt}\ s\ {\isacharequal}{\kern0pt}\ v\ {\isachardollar}{\kern0pt}\ s{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}opt{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}gs{\isacharprime}{\kern0pt}\ d\ v\ k\ {\isacharequal}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}iter{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max\ v{\isacharprime}{\kern0pt}\ k{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ opt{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}gs{\isacharprime}{\kern0pt}{\isacharunderscore}{\kern0pt}def\ GS{\isacharunderscore}{\kern0pt}iter{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}subst\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}iter{\isacharbrackleft}{\kern0pt}OF\ assms{\isacharcomma}{\kern0pt}\ of\ k\ d{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ opt{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}gs{\isacharprime}{\kern0pt}{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}GS{\isacharunderscore}{\kern0pt}iter{\isacharprime}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ {\isachardoublequoteopen}opt{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}gs{\isacharprime}{\kern0pt}\ d\ v\ k\ {\isacharequal}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}iter{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max\ {\isacharparenleft}{\kern0pt}{\isasymchi}\ s{\isachardot}{\kern0pt}\ if\ s\ {\isacharless}{\kern0pt}\ k\ then\ GS{\isacharunderscore}{\kern0pt}rec\ v\ {\isachardollar}{\kern0pt}\ s\ else\ v\ {\isachardollar}{\kern0pt}\ s{\isacharparenright}{\kern0pt}\ k{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ opt{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}gs{\isacharprime}{\kern0pt}{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}GS{\isacharunderscore}{\kern0pt}iter\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ leD{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ opt{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}gs{\isacharprime}{\kern0pt}{\isacharunderscore}{\kern0pt}is{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}opt{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}gs{\isacharprime}{\kern0pt}\ d\ v\ {\isasymin}\ D\isactrlsub D{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ opt{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}gs{\isacharprime}{\kern0pt}{\isacharunderscore}{\kern0pt}def\ is{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \isacommand{using}\isamarkupfalse%
-\ finite{\isacharunderscore}{\kern0pt}is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharbrackleft}{\kern0pt}OF\ finite\ A{\isacharunderscore}{\kern0pt}ne{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ intro{\isacharcolon}{\kern0pt}\ LeastI{\isadigit{2}}{\isacharunderscore}{\kern0pt}ex{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ opt{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}gs{\isacharprime}{\kern0pt}{\isacharunderscore}{\kern0pt}is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max\ {\isacharparenleft}{\kern0pt}{\isasymlambda}d{\isachardot}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}inv\ d\ v\ {\isachardollar}{\kern0pt}\ s{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymlambda}d{\isachardot}{\kern0pt}\ d\ {\isasymin}\ D\isactrlsub D{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}opt{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}gs{\isacharprime}{\kern0pt}\ d\ v{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}induction\ arbitrary{\isacharcolon}{\kern0pt}\ d\ rule{\isacharcolon}{\kern0pt}\ less{\isacharunderscore}{\kern0pt}induct{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{case}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}less\ x{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}s\ {\isacharless}{\kern0pt}\ x\ {\isasymLongrightarrow}\ is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max\ {\isacharparenleft}{\kern0pt}{\isasymlambda}d{\isachardot}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}inv\ d\ v\ {\isachardollar}{\kern0pt}\ s{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymlambda}d{\isachardot}{\kern0pt}\ d\ {\isasymin}\ D\isactrlsub D{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}opt{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}gs{\isacharprime}{\kern0pt}\ d\ v{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{for}\ d\ s\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ less\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isacharasterisk}{\kern0pt}{\isacharcolon}{\kern0pt}{\isachardoublequoteopen}s\ {\isacharless}{\kern0pt}\ x\ {\isasymLongrightarrow}\ is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max\ {\isacharparenleft}{\kern0pt}{\isasymlambda}d{\isachardot}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}step\ d\ v\ {\isachardollar}{\kern0pt}\ s{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymlambda}d{\isachardot}{\kern0pt}\ d\ {\isasymin}\ D\isactrlsub D{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}opt{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}gs{\isacharprime}{\kern0pt}\ d\ v{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{for}\ d\ s\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}intro\ is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}GS{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}splitting{\isacharprime}{\kern0pt}{\isacharprime}{\kern0pt}{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}\ auto\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max\ {\isacharparenleft}{\kern0pt}{\isasymlambda}a{\isachardot}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}step\ {\isacharparenleft}{\kern0pt}d{\isacharparenleft}{\kern0pt}x\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ a{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ v\ {\isachardollar}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymlambda}a{\isachardot}{\kern0pt}\ a\ {\isasymin}\ A\ x{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}opt{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}gs{\isacharprime}{\kern0pt}\ d\ v\ x{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{for}\ d\isanewline
-\ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ opt{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}gs{\isacharprime}{\kern0pt}{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ finite{\isacharunderscore}{\kern0pt}is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharbrackleft}{\kern0pt}OF\ {\isacharunderscore}{\kern0pt}\ A{\isacharunderscore}{\kern0pt}ne{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ intro{\isacharcolon}{\kern0pt}\ LeastI{\isacharunderscore}{\kern0pt}ex{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max\ {\isacharparenleft}{\kern0pt}{\isasymlambda}d{\isachardot}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}step\ d\ v\ {\isachardollar}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymlambda}d{\isachardot}{\kern0pt}\ d\ {\isasymin}\ D\isactrlsub D{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}opt{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}gs{\isacharprime}{\kern0pt}\ d\ v{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{for}\ d\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ opt{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}gs{\isacharprime}{\kern0pt}{\isacharunderscore}{\kern0pt}is{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}intro\ is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}step{\isacharunderscore}{\kern0pt}act{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}\ auto\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}s\ {\isasymle}\ x\ {\isasymLongrightarrow}\ is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max\ {\isacharparenleft}{\kern0pt}{\isasymlambda}d{\isachardot}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}step\ d\ v\ {\isachardollar}{\kern0pt}\ s{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymlambda}d{\isachardot}{\kern0pt}\ d\ {\isasymin}\ D\isactrlsub D{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}opt{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}gs{\isacharprime}{\kern0pt}\ d\ v{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{for}\ d\ s\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isacharasterisk}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ order{\isachardot}{\kern0pt}order{\isacharunderscore}{\kern0pt}iff{\isacharunderscore}{\kern0pt}strict{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}s\ {\isasymle}\ x\ {\isasymLongrightarrow}\ is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max\ {\isacharparenleft}{\kern0pt}{\isasymlambda}d{\isachardot}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}inv\ d\ v\ {\isachardollar}{\kern0pt}\ s{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymlambda}d{\isachardot}{\kern0pt}\ d\ {\isasymin}\ D\isactrlsub D{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}opt{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}gs{\isacharprime}{\kern0pt}\ d\ v{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{for}\ d\ s\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}GS{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}splitting{\isacharprime}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}case\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ {\isachardoublequoteopen}GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}am{\isacharunderscore}{\kern0pt}code\ v\ d\ s\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}GS{\isacharunderscore}{\kern0pt}rec\ v\ {\isachardollar}{\kern0pt}\ s{\isacharcomma}{\kern0pt}\ opt{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}gs{\isacharprime}{\kern0pt}\ d\ v\ s{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}induction\ s\ arbitrary{\isacharcolon}{\kern0pt}\ d\ rule{\isacharcolon}{\kern0pt}\ less{\isacharunderscore}{\kern0pt}induct{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{case}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}less\ x{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}case\ \isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}cases\ {\isachardoublequoteopen}{\isasymexists}x{\isacharprime}{\kern0pt}{\isachardot}{\kern0pt}\ x{\isacharprime}{\kern0pt}\ {\isacharless}{\kern0pt}\ x{\isachardoublequoteclose}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{case}\isamarkupfalse%
-\ True\isanewline
-\ \ \ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}f\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isasymlambda}vd\ s{\isachardot}{\kern0pt}\ vd{\isacharparenleft}{\kern0pt}s\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}GS{\isacharunderscore}{\kern0pt}iter{\isacharunderscore}{\kern0pt}max\ {\isacharparenleft}{\kern0pt}{\isasymchi}\ s{\isachardot}{\kern0pt}\ fst\ {\isacharparenleft}{\kern0pt}vd\ s{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ s{\isacharcomma}{\kern0pt}\ \ GS{\isacharunderscore}{\kern0pt}iter{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max\ {\isacharparenleft}{\kern0pt}{\isasymchi}\ s{\isachardot}{\kern0pt}\ fst\ {\isacharparenleft}{\kern0pt}vd\ s{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ s{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{define}\isamarkupfalse%
-\ x{\isacharprime}{\kern0pt}\ \isakeyword{where}\ {\isachardoublequoteopen}x{\isacharprime}{\kern0pt}\ {\isacharequal}{\kern0pt}\ Max\ {\isacharbraceleft}{\kern0pt}x{\isacharprime}{\kern0pt}{\isachardot}{\kern0pt}\ x{\isacharprime}{\kern0pt}\ {\isacharless}{\kern0pt}\ x{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}old\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}foldl\ {\isacharquery}{\kern0pt}f\ {\isacharparenleft}{\kern0pt}{\isasymlambda}s{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}v\ {\isachardollar}{\kern0pt}\ s{\isacharcomma}{\kern0pt}\ d\ s{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}sorted{\isacharunderscore}{\kern0pt}list{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}set\ {\isacharbraceleft}{\kern0pt}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}x{\isacharprime}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}s\ {\isacharless}{\kern0pt}\ x\ {\isasymLongrightarrow}\ {\isacharparenleft}{\kern0pt}s\ {\isasymnotin}\ set\ {\isacharparenleft}{\kern0pt}sorted{\isacharunderscore}{\kern0pt}list{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}set\ {\isacharbraceleft}{\kern0pt}s{\isacharprime}{\kern0pt}\ {\isasymin}\ {\isacharbraceleft}{\kern0pt}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}x{\isacharprime}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ s\ {\isacharless}{\kern0pt}\ s{\isacharprime}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{for}\ s\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isacharprime}{\kern0pt}s\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}s\ {\isacharless}{\kern0pt}\ x\ {\isasymLongrightarrow}\ foldl\ {\isacharquery}{\kern0pt}f\ {\isacharparenleft}{\kern0pt}{\isasymlambda}s{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}v\ {\isachardollar}{\kern0pt}\ s{\isacharcomma}{\kern0pt}\ d\ s{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}sorted{\isacharunderscore}{\kern0pt}list{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}set\ {\isacharbraceleft}{\kern0pt}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}x{\isacharprime}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ s\ {\isacharequal}{\kern0pt}\ foldl\ {\isacharquery}{\kern0pt}f\ {\isacharparenleft}{\kern0pt}{\isasymlambda}s{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}v\ {\isachardollar}{\kern0pt}\ s{\isacharcomma}{\kern0pt}\ d\ s{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}sorted{\isacharunderscore}{\kern0pt}list{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}set\ {\isacharbraceleft}{\kern0pt}s{\isacharprime}{\kern0pt}\ {\isasymin}\ {\isacharbraceleft}{\kern0pt}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}x{\isacharprime}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ s{\isacharprime}{\kern0pt}\ {\isasymle}\ s{\isacharbraceright}{\kern0pt}\ {\isacharat}{\kern0pt}\ sorted{\isacharunderscore}{\kern0pt}list{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}set\ {\isacharbraceleft}{\kern0pt}s{\isacharprime}{\kern0pt}\ {\isasymin}\ {\isacharbraceleft}{\kern0pt}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}x{\isacharprime}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ s\ {\isacharless}{\kern0pt}\ s{\isacharprime}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ s{\isachardoublequoteclose}\ \isakeyword{for}\ s\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}subst\ sorted{\isacharunderscore}{\kern0pt}list{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}set{\isacharunderscore}{\kern0pt}split{\isacharprime}{\kern0pt}{\isacharbrackleft}{\kern0pt}symmetric{\isacharcomma}{\kern0pt}\ OF\ finite{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\ blast\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}s\ {\isacharless}{\kern0pt}\ x\ {\isasymLongrightarrow}\ foldl\ {\isacharquery}{\kern0pt}f\ {\isacharparenleft}{\kern0pt}{\isasymlambda}s{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}v\ {\isachardollar}{\kern0pt}\ s{\isacharcomma}{\kern0pt}\ d\ s{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}sorted{\isacharunderscore}{\kern0pt}list{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}set\ {\isacharbraceleft}{\kern0pt}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}x{\isacharprime}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ s\ {\isacharequal}{\kern0pt}\ foldl\ {\isacharquery}{\kern0pt}f\ {\isacharparenleft}{\kern0pt}{\isasymlambda}s{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}v\ {\isachardollar}{\kern0pt}\ s{\isacharcomma}{\kern0pt}\ d\ s{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}sorted{\isacharunderscore}{\kern0pt}list{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}set\ {\isacharbraceleft}{\kern0pt}s{\isacharprime}{\kern0pt}\ {\isasymin}\ {\isacharbraceleft}{\kern0pt}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}x{\isacharprime}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ s{\isacharprime}{\kern0pt}\ {\isasymle}\ s{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ s{\isachardoublequoteclose}\ \isakeyword{for}\ s\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ foldl{\isacharunderscore}{\kern0pt}upd{\isacharunderscore}{\kern0pt}notin{\isacharprime}{\kern0pt}{\isacharbrackleft}{\kern0pt}OF\ {\isadigit{1}}{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ fastforce\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ {\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}s\ {\isacharless}{\kern0pt}\ x\ {\isasymLongrightarrow}\ foldl\ {\isacharquery}{\kern0pt}f\ {\isacharparenleft}{\kern0pt}{\isasymlambda}s{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}v\ {\isachardollar}{\kern0pt}\ s{\isacharcomma}{\kern0pt}\ d\ s{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}sorted{\isacharunderscore}{\kern0pt}list{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}set\ {\isacharbraceleft}{\kern0pt}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}x{\isacharprime}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ s\ {\isacharequal}{\kern0pt}\ foldl\ {\isacharquery}{\kern0pt}f\ {\isacharparenleft}{\kern0pt}{\isasymlambda}s{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}v\ {\isachardollar}{\kern0pt}\ s{\isacharcomma}{\kern0pt}\ d\ s{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}sorted{\isacharunderscore}{\kern0pt}list{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}set\ {\isacharbraceleft}{\kern0pt}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}s{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ s{\isachardoublequoteclose}\ \isakeyword{for}\ s\isanewline
-\ \ \ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ x{\isacharprime}{\kern0pt}{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ True\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ atMost{\isacharunderscore}{\kern0pt}def\ Max{\isacharunderscore}{\kern0pt}ge{\isacharunderscore}{\kern0pt}iff{\isacharbrackleft}{\kern0pt}OF\ finite{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\ meson\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ fst{\isacharunderscore}{\kern0pt}IH{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}fst\ {\isacharparenleft}{\kern0pt}{\isacharquery}{\kern0pt}old\ s{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}rec\ v\ {\isachardollar}{\kern0pt}\ s{\isachardoublequoteclose}\ \isakeyword{if}\ {\isachardoublequoteopen}s\ {\isacharless}{\kern0pt}\ x{\isachardoublequoteclose}\ \isakeyword{for}\ s\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isadigit{1}}{\isacharbrackleft}{\kern0pt}OF\ that{\isacharbrackright}{\kern0pt}\ less{\isacharbrackleft}{\kern0pt}unfolded\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}am{\isacharunderscore}{\kern0pt}code{\isacharunderscore}{\kern0pt}def{\isacharbrackright}{\kern0pt}\ that\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ fst{\isacharunderscore}{\kern0pt}IH{\isacharprime}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}fst\ {\isacharparenleft}{\kern0pt}{\isacharquery}{\kern0pt}old\ s{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ v\ {\isachardollar}{\kern0pt}\ s{\isachardoublequoteclose}\ \isakeyword{if}\ {\isachardoublequoteopen}x\ {\isasymle}\ s{\isachardoublequoteclose}\ \isakeyword{for}\ s\ \isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ True\ that\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}subst\ foldl{\isacharunderscore}{\kern0pt}upd{\isacharunderscore}{\kern0pt}notin{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ x{\isacharprime}{\kern0pt}{\isacharunderscore}{\kern0pt}def\ \ Max{\isacharunderscore}{\kern0pt}ge{\isacharunderscore}{\kern0pt}iff{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ fst{\isacharunderscore}{\kern0pt}IH{\isacharprime}{\kern0pt}{\isacharprime}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}fst\ {\isacharparenleft}{\kern0pt}{\isacharquery}{\kern0pt}old\ s{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}if\ s\ {\isacharless}{\kern0pt}\ x\ then\ GS{\isacharunderscore}{\kern0pt}rec\ v\ {\isachardollar}{\kern0pt}\ s\ else\ v\ {\isachardollar}{\kern0pt}\ s{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{for}\ s\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ fst{\isacharunderscore}{\kern0pt}IH\ fst{\isacharunderscore}{\kern0pt}IH{\isacharprime}{\kern0pt}\ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}foldl\ {\isacharquery}{\kern0pt}f\ {\isacharparenleft}{\kern0pt}{\isasymlambda}s{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}v\ {\isachardollar}{\kern0pt}\ s{\isacharcomma}{\kern0pt}\ d\ s{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}sorted{\isacharunderscore}{\kern0pt}list{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}set\ {\isacharbraceleft}{\kern0pt}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}x{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ foldl\ {\isacharquery}{\kern0pt}f\ {\isacharparenleft}{\kern0pt}{\isasymlambda}s{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}v\ {\isachardollar}{\kern0pt}\ s{\isacharcomma}{\kern0pt}\ d\ s{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}sorted{\isacharunderscore}{\kern0pt}list{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}set\ {\isacharbraceleft}{\kern0pt}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}x{\isacharprime}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharat}{\kern0pt}\ sorted{\isacharunderscore}{\kern0pt}list{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}set\ {\isacharbraceleft}{\kern0pt}x{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isacharasterisk}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}x{\isacharprime}{\kern0pt}{\isachardot}{\kern0pt}\ x{\isacharprime}{\kern0pt}\ {\isacharless}{\kern0pt}\ x{\isacharbraceright}{\kern0pt}\ {\isasymnoteq}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\ \isacommand{using}\isamarkupfalse%
-\ True\ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ {\isacharasterisk}{\kern0pt}{\isacharasterisk}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}x{\isacharprime}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}y\ {\isasymin}\ {\isacharbraceleft}{\kern0pt}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}x{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ y\ {\isacharless}{\kern0pt}\ x{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}x{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}y\ {\isasymin}\ {\isacharbraceleft}{\kern0pt}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}x{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ x\ {\isasymle}\ y{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ x{\isacharprime}{\kern0pt}{\isacharunderscore}{\kern0pt}def\ Max{\isacharunderscore}{\kern0pt}ge{\isacharunderscore}{\kern0pt}iff{\isacharbrackleft}{\kern0pt}OF\ finite\ {\isacharasterisk}{\kern0pt}{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ {\isacharasterisk}{\kern0pt}{\isacharasterisk}{\kern0pt}\ sorted{\isacharunderscore}{\kern0pt}list{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}set{\isacharunderscore}{\kern0pt}split{\isacharbrackleft}{\kern0pt}symmetric{\isacharcomma}{\kern0pt}\ OF\ finite{\isacharbrackright}{\kern0pt}\ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ \ {\isacharequal}{\kern0pt}\ {\isacharquery}{\kern0pt}f\ {\isacharparenleft}{\kern0pt}foldl\ {\isacharquery}{\kern0pt}f\ {\isacharparenleft}{\kern0pt}{\isasymlambda}s{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}v\ {\isachardollar}{\kern0pt}\ s{\isacharcomma}{\kern0pt}\ d\ s{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}sorted{\isacharunderscore}{\kern0pt}list{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}set\ {\isacharbraceleft}{\kern0pt}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}x{\isacharprime}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharquery}{\kern0pt}old\ {\isacharparenleft}{\kern0pt}x\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}GS{\isacharunderscore}{\kern0pt}rec\ v\ {\isachardollar}{\kern0pt}\ x{\isacharcomma}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}iter{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max\ {\isacharparenleft}{\kern0pt}{\isasymchi}\ s{\isachardot}{\kern0pt}\ fst\ {\isacharparenleft}{\kern0pt}{\isacharquery}{\kern0pt}old\ s{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ x{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}subst\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}iter{\isacharunderscore}{\kern0pt}max{\isacharbrackleft}{\kern0pt}of\ {\isacharunderscore}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isasymchi}\ s{\isachardot}{\kern0pt}\ fst\ {\isacharparenleft}{\kern0pt}{\isacharquery}{\kern0pt}old\ s{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}{\isacharbrackright}{\kern0pt}{\isacharcomma}{\kern0pt}\ goal{\isacharunderscore}{\kern0pt}cases{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{case}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}\ s{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{then}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}case\ \isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ fst{\isacharunderscore}{\kern0pt}IH\ \isacommand{by}\isamarkupfalse%
-\ auto\ \isanewline
-\ \ \ \ \isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \isacommand{case}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}{\isadigit{2}}\ s{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{then}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}case\isanewline
-\ \ \ \ \ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ vec{\isacharunderscore}{\kern0pt}lambda{\isacharunderscore}{\kern0pt}inverse{\isacharbrackleft}{\kern0pt}OF\ UNIV{\isacharunderscore}{\kern0pt}I{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ True\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}subst\ foldl{\isacharunderscore}{\kern0pt}upd{\isacharunderscore}{\kern0pt}notin{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ x{\isacharprime}{\kern0pt}{\isacharunderscore}{\kern0pt}def\ Max{\isacharunderscore}{\kern0pt}ge{\isacharunderscore}{\kern0pt}iff{\isacharbrackleft}{\kern0pt}OF\ finite{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{qed}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharquery}{\kern0pt}old\ {\isacharparenleft}{\kern0pt}x\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}GS{\isacharunderscore}{\kern0pt}rec\ v\ {\isachardollar}{\kern0pt}\ x{\isacharcomma}{\kern0pt}\ opt{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}gs{\isacharprime}{\kern0pt}\ d\ v\ x{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ fst{\isacharunderscore}{\kern0pt}IH{\isacharprime}{\kern0pt}{\isacharprime}{\kern0pt}\ opt{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}gs{\isacharprime}{\kern0pt}{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}GS{\isacharunderscore}{\kern0pt}iter{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{finally}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}foldl\ {\isacharquery}{\kern0pt}f\ {\isacharparenleft}{\kern0pt}{\isasymlambda}s{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}v\ {\isachardollar}{\kern0pt}\ s{\isacharcomma}{\kern0pt}\ d\ s{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}sorted{\isacharunderscore}{\kern0pt}list{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}set\ {\isacharbraceleft}{\kern0pt}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}x{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharquery}{\kern0pt}old\ {\isacharparenleft}{\kern0pt}x\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}GS{\isacharunderscore}{\kern0pt}rec\ v\ {\isachardollar}{\kern0pt}\ x{\isacharcomma}{\kern0pt}\ opt{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}gs{\isacharprime}{\kern0pt}\ d\ v\ x{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isacommand{{\isachardot}{\kern0pt}}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\ \ \ \ \isanewline
-\ \ \ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}am{\isacharunderscore}{\kern0pt}code{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{case}\isamarkupfalse%
-\ False\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}x{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}x{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ not{\isacharunderscore}{\kern0pt}less\ antisym{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}am{\isacharunderscore}{\kern0pt}code{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ opt{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}gs{\isacharprime}{\kern0pt}{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}GS{\isacharunderscore}{\kern0pt}iter{\isacharbrackleft}{\kern0pt}of\ x\ v{\isacharbrackright}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}iter{\isacharunderscore}{\kern0pt}max{\isacharbrackleft}{\kern0pt}of\ x\ v{\isacharbrackright}{\kern0pt}\ False\ \isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ fastforce\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}am{\isacharunderscore}{\kern0pt}code{\isacharunderscore}{\kern0pt}eq{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}am{\isacharunderscore}{\kern0pt}code\ v\ d\ s\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}GS{\isacharunderscore}{\kern0pt}rec\ v\ {\isachardollar}{\kern0pt}\ s{\isacharcomma}{\kern0pt}\ opt{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}gs{\isacharprime}{\kern0pt}\ d\ v\ s{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}induction\ s\ arbitrary{\isacharcolon}{\kern0pt}\ d\ rule{\isacharcolon}{\kern0pt}\ less{\isacharunderscore}{\kern0pt}induct{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{case}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}less\ x{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}case\ \isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}cases\ {\isachardoublequoteopen}{\isasymexists}x{\isacharprime}{\kern0pt}{\isachardot}{\kern0pt}\ x{\isacharprime}{\kern0pt}\ {\isacharless}{\kern0pt}\ x{\isachardoublequoteclose}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{case}\isamarkupfalse%
-\ True\isanewline
-\ \ \ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}f\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isasymlambda}vd\ s{\isachardot}{\kern0pt}\ vd{\isacharparenleft}{\kern0pt}s\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}GS{\isacharunderscore}{\kern0pt}iter{\isacharunderscore}{\kern0pt}max\ {\isacharparenleft}{\kern0pt}{\isasymchi}\ s{\isachardot}{\kern0pt}\ fst\ {\isacharparenleft}{\kern0pt}vd\ s{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ s{\isacharcomma}{\kern0pt}\ \ GS{\isacharunderscore}{\kern0pt}iter{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max\ {\isacharparenleft}{\kern0pt}{\isasymchi}\ s{\isachardot}{\kern0pt}\ fst\ {\isacharparenleft}{\kern0pt}vd\ s{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ s{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{define}\isamarkupfalse%
-\ x{\isacharprime}{\kern0pt}\ \isakeyword{where}\ {\isachardoublequoteopen}x{\isacharprime}{\kern0pt}\ {\isacharequal}{\kern0pt}\ Max\ {\isacharbraceleft}{\kern0pt}x{\isacharprime}{\kern0pt}{\isachardot}{\kern0pt}\ x{\isacharprime}{\kern0pt}\ {\isacharless}{\kern0pt}\ x{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}old\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}foldl\ {\isacharquery}{\kern0pt}f\ {\isacharparenleft}{\kern0pt}{\isasymlambda}s{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}v\ {\isachardollar}{\kern0pt}\ s{\isacharcomma}{\kern0pt}\ d\ s{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}sorted{\isacharunderscore}{\kern0pt}list{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}set\ {\isacharbraceleft}{\kern0pt}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}x{\isacharprime}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}s\ {\isacharless}{\kern0pt}\ x\ {\isasymLongrightarrow}\ {\isacharparenleft}{\kern0pt}s\ {\isasymnotin}\ set\ {\isacharparenleft}{\kern0pt}sorted{\isacharunderscore}{\kern0pt}list{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}set\ {\isacharbraceleft}{\kern0pt}s{\isacharprime}{\kern0pt}\ {\isasymin}\ {\isacharbraceleft}{\kern0pt}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}x{\isacharprime}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ s\ {\isacharless}{\kern0pt}\ s{\isacharprime}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{for}\ s\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isacharprime}{\kern0pt}s\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}s\ {\isacharless}{\kern0pt}\ x\ {\isasymLongrightarrow}\ foldl\ {\isacharquery}{\kern0pt}f\ {\isacharparenleft}{\kern0pt}{\isasymlambda}s{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}v\ {\isachardollar}{\kern0pt}\ s{\isacharcomma}{\kern0pt}\ d\ s{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}sorted{\isacharunderscore}{\kern0pt}list{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}set\ {\isacharbraceleft}{\kern0pt}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}x{\isacharprime}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ s\ {\isacharequal}{\kern0pt}\ \ foldl\ {\isacharquery}{\kern0pt}f\ {\isacharparenleft}{\kern0pt}{\isasymlambda}s{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}v\ {\isachardollar}{\kern0pt}\ s{\isacharcomma}{\kern0pt}\ d\ s{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}sorted{\isacharunderscore}{\kern0pt}list{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}set\ {\isacharbraceleft}{\kern0pt}s{\isacharprime}{\kern0pt}\ {\isasymin}\ {\isacharbraceleft}{\kern0pt}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}x{\isacharprime}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ s{\isacharprime}{\kern0pt}\ {\isasymle}\ s{\isacharbraceright}{\kern0pt}\ {\isacharat}{\kern0pt}\ sorted{\isacharunderscore}{\kern0pt}list{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}set\ {\isacharbraceleft}{\kern0pt}s{\isacharprime}{\kern0pt}\ {\isasymin}\ {\isacharbraceleft}{\kern0pt}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}x{\isacharprime}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ s\ {\isacharless}{\kern0pt}\ s{\isacharprime}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ s{\isachardoublequoteclose}\ \isakeyword{for}\ s\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}subst\ sorted{\isacharunderscore}{\kern0pt}list{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}set{\isacharunderscore}{\kern0pt}split{\isacharprime}{\kern0pt}{\isacharbrackleft}{\kern0pt}symmetric{\isacharcomma}{\kern0pt}\ OF\ finite{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\ blast\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}s\ {\isacharless}{\kern0pt}\ x\ {\isasymLongrightarrow}\ foldl\ {\isacharquery}{\kern0pt}f\ {\isacharparenleft}{\kern0pt}{\isasymlambda}s{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}v\ {\isachardollar}{\kern0pt}\ s{\isacharcomma}{\kern0pt}\ d\ s{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}sorted{\isacharunderscore}{\kern0pt}list{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}set\ {\isacharbraceleft}{\kern0pt}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}x{\isacharprime}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ s\ {\isacharequal}{\kern0pt}\ \ foldl\ {\isacharquery}{\kern0pt}f\ {\isacharparenleft}{\kern0pt}{\isasymlambda}s{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}v\ {\isachardollar}{\kern0pt}\ s{\isacharcomma}{\kern0pt}\ d\ s{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}sorted{\isacharunderscore}{\kern0pt}list{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}set\ {\isacharbraceleft}{\kern0pt}s{\isacharprime}{\kern0pt}\ {\isasymin}\ {\isacharbraceleft}{\kern0pt}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}x{\isacharprime}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ s{\isacharprime}{\kern0pt}\ {\isasymle}\ s{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ s{\isachardoublequoteclose}\ \isakeyword{for}\ s\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ foldl{\isacharunderscore}{\kern0pt}upd{\isacharunderscore}{\kern0pt}notin{\isacharprime}{\kern0pt}{\isacharbrackleft}{\kern0pt}OF\ {\isadigit{1}}{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ fastforce\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ {\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}s\ {\isacharless}{\kern0pt}\ x\ {\isasymLongrightarrow}\ foldl\ {\isacharquery}{\kern0pt}f\ {\isacharparenleft}{\kern0pt}{\isasymlambda}s{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}v\ {\isachardollar}{\kern0pt}\ s{\isacharcomma}{\kern0pt}\ d\ s{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}sorted{\isacharunderscore}{\kern0pt}list{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}set\ {\isacharbraceleft}{\kern0pt}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}x{\isacharprime}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ s\ {\isacharequal}{\kern0pt}\ \ foldl\ {\isacharquery}{\kern0pt}f\ {\isacharparenleft}{\kern0pt}{\isasymlambda}s{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}v\ {\isachardollar}{\kern0pt}\ s{\isacharcomma}{\kern0pt}\ d\ s{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}sorted{\isacharunderscore}{\kern0pt}list{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}set\ {\isacharbraceleft}{\kern0pt}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}s{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ s{\isachardoublequoteclose}\ \isakeyword{for}\ s\isanewline
-\ \ \ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ x{\isacharprime}{\kern0pt}{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ True\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ atMost{\isacharunderscore}{\kern0pt}def\ Max{\isacharunderscore}{\kern0pt}ge{\isacharunderscore}{\kern0pt}iff{\isacharbrackleft}{\kern0pt}OF\ finite{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\ meson\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ fst{\isacharunderscore}{\kern0pt}IH{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}fst\ {\isacharparenleft}{\kern0pt}{\isacharquery}{\kern0pt}old\ s{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}rec\ v\ {\isachardollar}{\kern0pt}\ s{\isachardoublequoteclose}\ \isakeyword{if}\ {\isachardoublequoteopen}s\ {\isacharless}{\kern0pt}\ x{\isachardoublequoteclose}\ \isakeyword{for}\ s\isanewline
-\ \ \ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ {\isadigit{1}}{\isacharbrackleft}{\kern0pt}OF\ that{\isacharbrackright}{\kern0pt}\ less{\isacharbrackleft}{\kern0pt}unfolded\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}am{\isacharunderscore}{\kern0pt}code{\isacharunderscore}{\kern0pt}def{\isacharcomma}{\kern0pt}\ OF\ that{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ fst{\isacharunderscore}{\kern0pt}IH{\isacharprime}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}fst\ {\isacharparenleft}{\kern0pt}{\isacharquery}{\kern0pt}old\ s{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ v\ {\isachardollar}{\kern0pt}\ s{\isachardoublequoteclose}\ \isakeyword{if}\ {\isachardoublequoteopen}x\ {\isasymle}\ s{\isachardoublequoteclose}\ \isakeyword{for}\ s\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ True\ that\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}subst\ foldl{\isacharunderscore}{\kern0pt}upd{\isacharunderscore}{\kern0pt}notin{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ x{\isacharprime}{\kern0pt}{\isacharunderscore}{\kern0pt}def\ atMost{\isacharunderscore}{\kern0pt}def\ Max{\isacharunderscore}{\kern0pt}ge{\isacharunderscore}{\kern0pt}iff{\isacharbrackleft}{\kern0pt}OF\ finite{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ fst{\isacharunderscore}{\kern0pt}IH{\isacharprime}{\kern0pt}{\isacharprime}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}fst\ {\isacharparenleft}{\kern0pt}{\isacharquery}{\kern0pt}old\ s{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}if\ s\ {\isacharless}{\kern0pt}\ x\ then\ GS{\isacharunderscore}{\kern0pt}rec\ v\ {\isachardollar}{\kern0pt}\ s\ else\ v\ {\isachardollar}{\kern0pt}\ s{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{for}\ s\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ fst{\isacharunderscore}{\kern0pt}IH\ fst{\isacharunderscore}{\kern0pt}IH{\isacharprime}{\kern0pt}\ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}foldl\ {\isacharquery}{\kern0pt}f\ {\isacharparenleft}{\kern0pt}{\isasymlambda}s{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}v\ {\isachardollar}{\kern0pt}\ s{\isacharcomma}{\kern0pt}\ d\ s{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}sorted{\isacharunderscore}{\kern0pt}list{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}set\ {\isacharbraceleft}{\kern0pt}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}x{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ foldl\ {\isacharquery}{\kern0pt}f\ {\isacharparenleft}{\kern0pt}{\isasymlambda}s{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}v\ {\isachardollar}{\kern0pt}\ s{\isacharcomma}{\kern0pt}\ d\ s{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}sorted{\isacharunderscore}{\kern0pt}list{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}set\ {\isacharbraceleft}{\kern0pt}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}x{\isacharprime}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharat}{\kern0pt}\ sorted{\isacharunderscore}{\kern0pt}list{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}set\ {\isacharbraceleft}{\kern0pt}x{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isacharasterisk}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}x{\isacharprime}{\kern0pt}{\isachardot}{\kern0pt}\ x{\isacharprime}{\kern0pt}\ {\isacharless}{\kern0pt}\ x{\isacharbraceright}{\kern0pt}\ {\isasymnoteq}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\ \isacommand{using}\isamarkupfalse%
-\ True\ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ {\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}x{\isacharprime}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}y\ {\isasymin}\ {\isacharbraceleft}{\kern0pt}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}x{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ y\ {\isacharless}{\kern0pt}\ x{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ x{\isacharprime}{\kern0pt}{\isacharunderscore}{\kern0pt}def\ Max{\isacharunderscore}{\kern0pt}ge{\isacharunderscore}{\kern0pt}iff{\isacharbrackleft}{\kern0pt}OF\ finite\ {\isacharasterisk}{\kern0pt}{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{2}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}x{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}y\ {\isasymin}\ {\isacharbraceleft}{\kern0pt}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}x{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ x\ {\isasymle}\ y{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ {\isadigit{1}}\ {\isadigit{2}}\ sorted{\isacharunderscore}{\kern0pt}list{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}set{\isacharunderscore}{\kern0pt}split{\isacharbrackleft}{\kern0pt}symmetric{\isacharcomma}{\kern0pt}\ OF\ finite{\isacharbrackright}{\kern0pt}\ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ \ {\isacharequal}{\kern0pt}\ {\isacharquery}{\kern0pt}f\ {\isacharparenleft}{\kern0pt}foldl\ {\isacharquery}{\kern0pt}f\ {\isacharparenleft}{\kern0pt}{\isasymlambda}s{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}v\ {\isachardollar}{\kern0pt}\ s{\isacharcomma}{\kern0pt}\ d\ s{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}sorted{\isacharunderscore}{\kern0pt}list{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}set\ {\isacharbraceleft}{\kern0pt}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}x{\isacharprime}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharquery}{\kern0pt}old\ {\isacharparenleft}{\kern0pt}x\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}GS{\isacharunderscore}{\kern0pt}rec\ v\ {\isachardollar}{\kern0pt}\ x{\isacharcomma}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}iter{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max\ {\isacharparenleft}{\kern0pt}{\isasymchi}\ s{\isachardot}{\kern0pt}\ fst\ {\isacharparenleft}{\kern0pt}{\isacharquery}{\kern0pt}old\ s{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ x{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}subst\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}iter{\isacharunderscore}{\kern0pt}max{\isacharbrackleft}{\kern0pt}of\ {\isacharunderscore}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isasymchi}\ s{\isachardot}{\kern0pt}\ fst\ {\isacharparenleft}{\kern0pt}{\isacharquery}{\kern0pt}old\ s{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}{\isacharbrackright}{\kern0pt}{\isacharcomma}{\kern0pt}\ goal{\isacharunderscore}{\kern0pt}cases{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{case}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}{\isadigit{2}}\ s{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{then}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}case\isanewline
-\ \ \ \ \ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ vec{\isacharunderscore}{\kern0pt}lambda{\isacharunderscore}{\kern0pt}inverse{\isacharbrackleft}{\kern0pt}OF\ UNIV{\isacharunderscore}{\kern0pt}I{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ True\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}subst\ foldl{\isacharunderscore}{\kern0pt}upd{\isacharunderscore}{\kern0pt}notin{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ x{\isacharprime}{\kern0pt}{\isacharunderscore}{\kern0pt}def\ Max{\isacharunderscore}{\kern0pt}ge{\isacharunderscore}{\kern0pt}iff{\isacharbrackleft}{\kern0pt}OF\ finite{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{qed}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ fst{\isacharunderscore}{\kern0pt}IH{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharquery}{\kern0pt}old\ {\isacharparenleft}{\kern0pt}x\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}GS{\isacharunderscore}{\kern0pt}rec\ v\ {\isachardollar}{\kern0pt}\ x{\isacharcomma}{\kern0pt}\ opt{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}gs{\isacharprime}{\kern0pt}\ d\ v\ x{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ fst{\isacharunderscore}{\kern0pt}IH{\isacharprime}{\kern0pt}{\isacharprime}{\kern0pt}\ opt{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}gs{\isacharprime}{\kern0pt}{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}GS{\isacharunderscore}{\kern0pt}iter{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{finally}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}foldl\ {\isacharquery}{\kern0pt}f\ {\isacharparenleft}{\kern0pt}{\isasymlambda}s{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}v\ {\isachardollar}{\kern0pt}\ s{\isacharcomma}{\kern0pt}\ d\ s{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}sorted{\isacharunderscore}{\kern0pt}list{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}set\ {\isacharbraceleft}{\kern0pt}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}x{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharquery}{\kern0pt}old\ {\isacharparenleft}{\kern0pt}x\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}GS{\isacharunderscore}{\kern0pt}rec\ v\ {\isachardollar}{\kern0pt}\ x{\isacharcomma}{\kern0pt}\ opt{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}gs{\isacharprime}{\kern0pt}\ d\ v\ x{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isacommand{{\isachardot}{\kern0pt}}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\ \ \ \ \isanewline
-\ \ \ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}am{\isacharunderscore}{\kern0pt}code{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{case}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}False{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}x{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}x{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ not{\isacharunderscore}{\kern0pt}less\ antisym{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ {\isacharasterisk}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}sorted{\isacharunderscore}{\kern0pt}list{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}set\ {\isacharbraceleft}{\kern0pt}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}x{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharbrackleft}{\kern0pt}x{\isacharbrackright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}am{\isacharunderscore}{\kern0pt}code{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ opt{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}gs{\isacharprime}{\kern0pt}{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}GS{\isacharunderscore}{\kern0pt}iter{\isacharbrackleft}{\kern0pt}of\ x\ v{\isacharbrackright}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}iter{\isacharunderscore}{\kern0pt}max{\isacharbrackleft}{\kern0pt}of\ x\ v{\isacharbrackright}{\kern0pt}\ False\ \isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}fastforce\ simp{\isacharcolon}{\kern0pt}\ {\isacharasterisk}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}iter{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}iter{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max\ v\ s\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}LEAST\ a{\isachardot}{\kern0pt}\ is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max\ {\isacharparenleft}{\kern0pt}{\isasymlambda}a{\isachardot}{\kern0pt}\ r\ {\isacharparenleft}{\kern0pt}s{\isacharcomma}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isacharplus}{\kern0pt}\ l\ {\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSum}s{\isacharprime}{\kern0pt}\ {\isasymin}\ UNIV{\isachardot}{\kern0pt}\ pmf\ {\isacharparenleft}{\kern0pt}K\ {\isacharparenleft}{\kern0pt}s{\isacharcomma}{\kern0pt}a{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ s{\isacharprime}{\kern0pt}\ {\isacharasterisk}{\kern0pt}\ v\ s{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymlambda}a{\isachardot}{\kern0pt}\ a\ {\isasymin}\ A\ s{\isacharparenright}{\kern0pt}\ a{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ {\isachardoublequoteopen}opt{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}gs\ v\ s\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}LEAST\ a{\isachardot}{\kern0pt}\ is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max\ {\isacharparenleft}{\kern0pt}{\isasymlambda}a{\isachardot}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}fun{\isacharunderscore}{\kern0pt}inner\ v\ s\ a{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymlambda}a{\isachardot}{\kern0pt}\ a\ {\isasymin}\ A\ s{\isacharparenright}{\kern0pt}\ a{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ opt{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}gs{\isacharunderscore}{\kern0pt}eq{\isacharprime}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}opt{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}gs\ v\ {\isacharequal}{\kern0pt}\ opt{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}gs{\isacharprime}{\kern0pt}\ d\ {\isacharparenleft}{\kern0pt}vec{\isacharunderscore}{\kern0pt}lambda\ v{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ opt{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}gs{\isacharunderscore}{\kern0pt}def\ opt{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}gs{\isacharprime}{\kern0pt}{\isacharunderscore}{\kern0pt}def\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}fun{\isacharunderscore}{\kern0pt}inner{\isacharunderscore}{\kern0pt}def\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}step{\isacharunderscore}{\kern0pt}elem\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}fun\isactrlsub b{\isachardot}{\kern0pt}rep{\isacharunderscore}{\kern0pt}eq\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}def\ vec{\isacharunderscore}{\kern0pt}lambda{\isacharunderscore}{\kern0pt}inverse{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{declare}\isamarkupfalse%
-\ gs{\isacharunderscore}{\kern0pt}iteration{\isachardot}{\kern0pt}simps{\isacharbrackleft}{\kern0pt}simp\ del{\isacharbrackright}{\kern0pt}\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ gs{\isacharunderscore}{\kern0pt}iteration{\isacharunderscore}{\kern0pt}error{\isacharcolon}{\kern0pt}\ \isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}eps\ {\isachargreater}{\kern0pt}\ {\isadigit{0}}{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}dist\ {\isacharparenleft}{\kern0pt}gs{\isacharunderscore}{\kern0pt}iteration\ eps\ v{\isacharparenright}{\kern0pt}\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt\ {\isacharless}{\kern0pt}\ eps\ {\isacharslash}{\kern0pt}\ {\isadigit{2}}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ assms\ dist{\isacharunderscore}{\kern0pt}{\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}split{\isacharunderscore}{\kern0pt}opt{\isacharunderscore}{\kern0pt}eps\ gs{\isacharunderscore}{\kern0pt}iteration{\isachardot}{\kern0pt}simps\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}induction\ eps\ v\ rule{\isacharcolon}{\kern0pt}\ gs{\isacharunderscore}{\kern0pt}iteration{\isachardot}{\kern0pt}induct{\isacharparenright}{\kern0pt}\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}fun{\isacharunderscore}{\kern0pt}inner{\isacharunderscore}{\kern0pt}vec{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}fun{\isacharunderscore}{\kern0pt}inner\ v\ s\ a\ {\isacharequal}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}step\ {\isacharparenleft}{\kern0pt}d{\isacharparenleft}{\kern0pt}s\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ a{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}vec{\isacharunderscore}{\kern0pt}lambda\ v{\isacharparenright}{\kern0pt}\ {\isachardollar}{\kern0pt}\ s{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}step{\isacharunderscore}{\kern0pt}elem\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}fun{\isacharunderscore}{\kern0pt}inner{\isacharunderscore}{\kern0pt}def\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}def\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}fun\isactrlsub b{\isachardot}{\kern0pt}rep{\isacharunderscore}{\kern0pt}eq\ vec{\isacharunderscore}{\kern0pt}lambda{\isacharunderscore}{\kern0pt}inverse{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ find{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}error{\isacharunderscore}{\kern0pt}bound{\isacharunderscore}{\kern0pt}gs{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}eps\ {\isachargreater}{\kern0pt}\ {\isadigit{0}}{\isachardoublequoteclose}\ {\isachardoublequoteopen}{\isadigit{2}}\ {\isacharasterisk}{\kern0pt}\ l\ {\isacharasterisk}{\kern0pt}\ dist\ v\ {\isacharparenleft}{\kern0pt}GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}fun\isactrlsub b\ v{\isacharparenright}{\kern0pt}\ {\isacharless}{\kern0pt}\ eps\ {\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharminus}{\kern0pt}l{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}dist\ {\isacharparenleft}{\kern0pt}{\isasymnu}\isactrlsub b\ {\isacharparenleft}{\kern0pt}mk{\isacharunderscore}{\kern0pt}stationary{\isacharunderscore}{\kern0pt}det\ {\isacharparenleft}{\kern0pt}opt{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}gs\ {\isacharparenleft}{\kern0pt}GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}fun\isactrlsub b\ v{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt\ {\isacharless}{\kern0pt}\ eps{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}rule\ GS{\isachardot}{\kern0pt}find{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}QR{\isacharunderscore}{\kern0pt}error{\isacharunderscore}{\kern0pt}bound{\isacharbrackleft}{\kern0pt}OF\ assms{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isadigit{2}}\ {\isacharasterisk}{\kern0pt}\ GS{\isachardot}{\kern0pt}QR{\isacharunderscore}{\kern0pt}disc\ {\isacharasterisk}{\kern0pt}\ dist\ v\ {\isacharparenleft}{\kern0pt}GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}fun\isactrlsub b\ v{\isacharparenright}{\kern0pt}\ {\isasymle}\ {\isadigit{2}}\ {\isacharasterisk}{\kern0pt}\ l\ {\isacharasterisk}{\kern0pt}\ dist\ v\ {\isacharparenleft}{\kern0pt}GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}fun\isactrlsub b\ v{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ GS{\isacharunderscore}{\kern0pt}QR{\isacharunderscore}{\kern0pt}disc{\isacharunderscore}{\kern0pt}le{\isacharunderscore}{\kern0pt}disc\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ mult{\isacharunderscore}{\kern0pt}right{\isacharunderscore}{\kern0pt}mono{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isacharless}{\kern0pt}\ \ eps\ {\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharminus}{\kern0pt}l{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isacommand{using}\isamarkupfalse%
-\ assms\ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isasymle}\ eps\ {\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}\ {\isacharminus}{\kern0pt}\ GS{\isachardot}{\kern0pt}QR{\isacharunderscore}{\kern0pt}disc{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assms\ GS{\isacharunderscore}{\kern0pt}QR{\isacharunderscore}{\kern0pt}disc{\isacharunderscore}{\kern0pt}le{\isacharunderscore}{\kern0pt}disc\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ mult{\isacharunderscore}{\kern0pt}left{\isacharunderscore}{\kern0pt}mono{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{finally}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isadigit{2}}\ {\isacharasterisk}{\kern0pt}\ GS{\isachardot}{\kern0pt}QR{\isacharunderscore}{\kern0pt}disc\ {\isacharasterisk}{\kern0pt}\ dist\ v\ {\isacharparenleft}{\kern0pt}GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}fun\isactrlsub b\ v{\isacharparenright}{\kern0pt}\ {\isacharless}{\kern0pt}\ eps\ {\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}\ {\isacharminus}{\kern0pt}\ GS{\isachardot}{\kern0pt}QR{\isacharunderscore}{\kern0pt}disc{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isacommand{{\isachardot}{\kern0pt}}\isamarkupfalse%
-\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{obtain}\isamarkupfalse%
-\ d\ \isakeyword{where}\ d{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}is{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\ d{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ ex{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max\ {\isacharparenleft}{\kern0pt}{\isasymlambda}d{\isachardot}{\kern0pt}\ apply{\isacharunderscore}{\kern0pt}bfun\ {\isacharparenleft}{\kern0pt}GS{\isachardot}{\kern0pt}L{\isacharunderscore}{\kern0pt}split\ d\ {\isacharparenleft}{\kern0pt}GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}fun\isactrlsub b\ v{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ s{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymlambda}d{\isachardot}{\kern0pt}\ d\ {\isasymin}\ D\isactrlsub D{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}opt{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}gs\ {\isacharparenleft}{\kern0pt}GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}fun\isactrlsub b\ v{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{for}\ s\isanewline
-\ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ opt{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}gs{\isacharunderscore}{\kern0pt}eq{\isacharprime}{\kern0pt}{\isacharbrackleft}{\kern0pt}of\ {\isacharunderscore}{\kern0pt}\ d{\isacharbrackright}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}inv{\isacharunderscore}{\kern0pt}blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ \isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ opt{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}gs{\isacharprime}{\kern0pt}{\isacharunderscore}{\kern0pt}is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ {\isachardoublequoteopen}vi{\isacharunderscore}{\kern0pt}gs{\isacharunderscore}{\kern0pt}policy\ eps\ v\ {\isacharequal}{\kern0pt}\ opt{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}gs\ {\isacharparenleft}{\kern0pt}gs{\isacharunderscore}{\kern0pt}iteration\ eps\ v{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ vi{\isacharunderscore}{\kern0pt}gs{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}opt{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}{\isadigit{0}}\ {\isacharless}{\kern0pt}\ eps{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}dist\ {\isacharparenleft}{\kern0pt}{\isasymnu}\isactrlsub b\ {\isacharparenleft}{\kern0pt}mk{\isacharunderscore}{\kern0pt}stationary{\isacharunderscore}{\kern0pt}det\ {\isacharparenleft}{\kern0pt}vi{\isacharunderscore}{\kern0pt}gs{\isacharunderscore}{\kern0pt}policy\ eps\ v{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt\ {\isacharless}{\kern0pt}\ eps{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ vi{\isacharunderscore}{\kern0pt}gs{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}def\ \isanewline
-\ \ \isacommand{using}\isamarkupfalse%
-\ assms\isanewline
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}induction\ eps\ v\ rule{\isacharcolon}{\kern0pt}\ gs{\isacharunderscore}{\kern0pt}iteration{\isachardot}{\kern0pt}induct{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{case}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}\ v{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{then}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}case\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ find{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}error{\isacharunderscore}{\kern0pt}bound{\isacharunderscore}{\kern0pt}gs\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}subst\ gs{\isacharunderscore}{\kern0pt}iteration{\isachardot}{\kern0pt}simps{\isacharparenright}{\kern0pt}\ auto\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}iter{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}iter{\isacharunderscore}{\kern0pt}max{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}iter\ v\ {\isacharequal}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}iter{\isacharunderscore}{\kern0pt}max\ {\isacharparenleft}{\kern0pt}vec{\isacharunderscore}{\kern0pt}lambda\ v{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}iter{\isacharunderscore}{\kern0pt}def\ GS{\isacharunderscore}{\kern0pt}iter{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}def\ GS{\isacharunderscore}{\kern0pt}iter{\isacharunderscore}{\kern0pt}def\ \isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isacommand{end}\isamarkupfalse%
-\isanewline
-%
-\isadelimtheory
-\isanewline
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{end}\isamarkupfalse%
-%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\end{isabellebody}%
-\endinput
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diff --git a/thys/MDP-Algorithms/output/document/Value_Iteration.tex b/thys/MDP-Algorithms/output/document/Value_Iteration.tex
deleted file mode 100644
--- a/thys/MDP-Algorithms/output/document/Value_Iteration.tex
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@@ -1,1507 +0,0 @@
-%
-\begin{isabellebody}%
-\setisabellecontext{Value{\isacharunderscore}{\kern0pt}Iteration}%
-%
-\isadelimtheory
-\isanewline
-\isanewline
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{theory}\isamarkupfalse%
-\ Value{\isacharunderscore}{\kern0pt}Iteration\isanewline
-\ \ \isakeyword{imports}\ {\isachardoublequoteopen}MDP{\isacharminus}{\kern0pt}Rewards{\isachardot}{\kern0pt}MDP{\isacharunderscore}{\kern0pt}reward{\isachardoublequoteclose}\isanewline
-\isakeyword{begin}%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-\isanewline
-%
-\endisadelimtheory
-\isanewline
-\isacommand{context}\isamarkupfalse%
-\ MDP{\isacharunderscore}{\kern0pt}att{\isacharunderscore}{\kern0pt}{\isasymL}\isanewline
-\isakeyword{begin}%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsection{Value Iteration%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\begin{isamarkuptext}%
-In the previous sections we derived that repeated application of \isa{{\isasymL}\isactrlsub b} to any bounded 
-function from states to the reals converges to the optimal value of the MDP \isa{{\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt}.
-
-We can turn this procedure into an algorithm that computes not only an approximation of 
-\isa{{\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt} but also a policy that is arbitrarily close to optimal.
-
-Most of the proofs rely on the assumption that the supremum in \isa{{\isasymL}\isactrlsub b} can always be attained.%
-\end{isamarkuptext}\isamarkuptrue%
-%
-\begin{isamarkuptext}%
-The following lemma shows that the relation we use to prove termination of the value iteration 
-algorithm decreases in each step.
-In essence, the distance of the estimate to the optimal value decreases by a factor of at 
-least \isa{l} per iteration.%
-\end{isamarkuptext}\isamarkuptrue%
-\isacommand{lemma}\isamarkupfalse%
-\ vi{\isacharunderscore}{\kern0pt}rel{\isacharunderscore}{\kern0pt}dec{\isacharcolon}{\kern0pt}\ \isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}l\ {\isasymnoteq}\ {\isadigit{0}}{\isachardoublequoteclose}\ {\isachardoublequoteopen}{\isasymL}\isactrlsub b\ v\ {\isasymnoteq}\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}{\isasymlceil}log\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}\ {\isacharslash}{\kern0pt}\ l{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}dist\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ v{\isacharparenright}{\kern0pt}\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt{\isacharparenright}{\kern0pt}\ {\isacharminus}{\kern0pt}\ c{\isasymrceil}\ {\isacharless}{\kern0pt}\ {\isasymlceil}log\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}\ {\isacharslash}{\kern0pt}\ l{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}dist\ v\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt{\isacharparenright}{\kern0pt}\ {\isacharminus}{\kern0pt}\ c{\isasymrceil}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}log\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}\ {\isacharslash}{\kern0pt}\ l{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}dist\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ v{\isacharparenright}{\kern0pt}\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt{\isacharparenright}{\kern0pt}\ {\isacharminus}{\kern0pt}\ c\ {\isasymle}\ log\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}\ {\isacharslash}{\kern0pt}\ l{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}l\ {\isacharasterisk}{\kern0pt}\ dist\ v\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt{\isacharparenright}{\kern0pt}\ {\isacharminus}{\kern0pt}\ c{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ contraction{\isacharunderscore}{\kern0pt}{\isasymL}{\isacharbrackleft}{\kern0pt}of\ {\isacharunderscore}{\kern0pt}\ {\isachardoublequoteopen}{\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt{\isachardoublequoteclose}{\isacharbrackright}{\kern0pt}\ disc{\isacharunderscore}{\kern0pt}lt{\isacharunderscore}{\kern0pt}one\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ assms\ less{\isacharunderscore}{\kern0pt}le\ intro{\isacharcolon}{\kern0pt}\ log{\isacharunderscore}{\kern0pt}le{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isacharequal}{\kern0pt}\ log\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}\ {\isacharslash}{\kern0pt}\ l{\isacharparenright}{\kern0pt}\ l\ {\isacharplus}{\kern0pt}\ log\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharslash}{\kern0pt}l{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}dist\ v\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt{\isacharparenright}{\kern0pt}\ {\isacharminus}{\kern0pt}\ c{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assms\ disc{\isacharunderscore}{\kern0pt}lt{\isacharunderscore}{\kern0pt}one\ \isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ less{\isacharunderscore}{\kern0pt}le\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ log{\isacharunderscore}{\kern0pt}mult{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isacharequal}{\kern0pt}\ {\isacharminus}{\kern0pt}{\isacharparenleft}{\kern0pt}log\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}\ {\isacharslash}{\kern0pt}\ l{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharslash}{\kern0pt}l{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharplus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}log\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharslash}{\kern0pt}l{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}dist\ v\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharminus}{\kern0pt}\ c{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assms\ disc{\isacharunderscore}{\kern0pt}lt{\isacharunderscore}{\kern0pt}one\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}subst\ log{\isacharunderscore}{\kern0pt}inverse{\isacharbrackleft}{\kern0pt}symmetric{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ less{\isacharunderscore}{\kern0pt}le\ right{\isacharunderscore}{\kern0pt}inverse{\isacharunderscore}{\kern0pt}eq{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}log\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharslash}{\kern0pt}l{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}dist\ v\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharminus}{\kern0pt}\ {\isadigit{1}}\ {\isacharminus}{\kern0pt}\ c{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assms\ order{\isachardot}{\kern0pt}strict{\isacharunderscore}{\kern0pt}implies{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}eq{\isacharbrackleft}{\kern0pt}OF\ disc{\isacharunderscore}{\kern0pt}lt{\isacharunderscore}{\kern0pt}one{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ log{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}one\ neq{\isacharunderscore}{\kern0pt}le{\isacharunderscore}{\kern0pt}trans{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{finally}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}log\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}\ {\isacharslash}{\kern0pt}\ l{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}dist\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ v{\isacharparenright}{\kern0pt}\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt{\isacharparenright}{\kern0pt}\ {\isacharminus}{\kern0pt}\ c\ {\isasymle}\ log\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}\ {\isacharslash}{\kern0pt}\ l{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}dist\ v\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt{\isacharparenright}{\kern0pt}\ {\isacharminus}{\kern0pt}\ {\isadigit{1}}\ {\isacharminus}{\kern0pt}\ c{\isachardoublequoteclose}\ \isacommand{{\isachardot}{\kern0pt}}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ linarith\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ dist{\isacharunderscore}{\kern0pt}{\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}lt{\isacharunderscore}{\kern0pt}dist{\isacharunderscore}{\kern0pt}opt{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}dist\ v\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ v{\isacharparenright}{\kern0pt}\ {\isasymle}\ {\isadigit{2}}\ {\isacharasterisk}{\kern0pt}\ dist\ v\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ le{\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}dist\ v\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ v{\isacharparenright}{\kern0pt}\ {\isasymle}\ dist\ v\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt\ {\isacharplus}{\kern0pt}\ dist\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ v{\isacharparenright}{\kern0pt}\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ dist{\isacharunderscore}{\kern0pt}triangle\ dist{\isacharunderscore}{\kern0pt}commute{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ le{\isadigit{2}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}dist\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ v{\isacharparenright}{\kern0pt}\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt\ {\isasymle}\ l\ {\isacharasterisk}{\kern0pt}\ dist\ v\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}opt\ contraction{\isacharunderscore}{\kern0pt}{\isasymL}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ mult{\isacharunderscore}{\kern0pt}right{\isacharunderscore}{\kern0pt}mono{\isacharbrackleft}{\kern0pt}of\ l\ {\isadigit{1}}{\isacharbrackright}{\kern0pt}\ disc{\isacharunderscore}{\kern0pt}lt{\isacharunderscore}{\kern0pt}one\ \isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}fastforce\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ order{\isachardot}{\kern0pt}trans{\isacharbrackleft}{\kern0pt}OF\ le{\isadigit{2}}{\isacharbrackright}{\kern0pt}\ order{\isachardot}{\kern0pt}trans{\isacharbrackleft}{\kern0pt}OF\ le{\isadigit{1}}{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{abbreviation}\isamarkupfalse%
-\ {\isachardoublequoteopen}term{\isacharunderscore}{\kern0pt}measure\ {\isasymequiv}\ {\isacharparenleft}{\kern0pt}{\isasymlambda}{\isacharparenleft}{\kern0pt}eps{\isacharcomma}{\kern0pt}\ v{\isacharparenright}{\kern0pt}{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ if\ v\ {\isacharequal}{\kern0pt}\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt\ {\isasymor}\ l\ {\isacharequal}{\kern0pt}\ {\isadigit{0}}\isanewline
-\ \ \ \ then\ {\isadigit{0}}\isanewline
-\ \ \ \ else\ nat\ {\isacharparenleft}{\kern0pt}ceiling\ {\isacharparenleft}{\kern0pt}log\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharslash}{\kern0pt}l{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}dist\ v\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt{\isacharparenright}{\kern0pt}\ {\isacharminus}{\kern0pt}\ log\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharslash}{\kern0pt}l{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}eps\ {\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharminus}{\kern0pt}l{\isacharparenright}{\kern0pt}\ {\isacharslash}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isadigit{8}}\ {\isacharasterisk}{\kern0pt}\ l{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{function}\isamarkupfalse%
-\ value{\isacharunderscore}{\kern0pt}iteration\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}real\ {\isasymRightarrow}\ {\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}s\ {\isasymRightarrow}\isactrlsub b\ real{\isacharparenright}{\kern0pt}\ {\isasymRightarrow}\ {\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}s\ {\isasymRightarrow}\isactrlsub b\ real{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}value{\isacharunderscore}{\kern0pt}iteration\ eps\ v\ {\isacharequal}{\kern0pt}\isanewline
-\ \ {\isacharparenleft}{\kern0pt}if\ {\isadigit{2}}\ {\isacharasterisk}{\kern0pt}\ l\ {\isacharasterisk}{\kern0pt}\ dist\ v\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ v{\isacharparenright}{\kern0pt}\ {\isacharless}{\kern0pt}\ eps\ {\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharminus}{\kern0pt}l{\isacharparenright}{\kern0pt}\ {\isasymor}\ eps\ {\isasymle}\ {\isadigit{0}}\ then\ {\isasymL}\isactrlsub b\ v\ else\ value{\isacharunderscore}{\kern0pt}iteration\ eps\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ v{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{termination}\isamarkupfalse%
-\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}relation\ {\isachardoublequoteopen}Wellfounded{\isachardot}{\kern0pt}measure\ term{\isacharunderscore}{\kern0pt}measure{\isachardoublequoteclose}{\isacharcomma}{\kern0pt}\ {\isacharparenleft}{\kern0pt}simp{\isacharsemicolon}{\kern0pt}\ fail{\isacharparenright}{\kern0pt}{\isacharcomma}{\kern0pt}\ cases\ {\isachardoublequoteopen}l\ {\isacharequal}{\kern0pt}\ {\isadigit{0}}{\isachardoublequoteclose}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{case}\isamarkupfalse%
-\ False\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\ eps\ v\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\ h{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymnot}\ {\isacharparenleft}{\kern0pt}{\isadigit{2}}\ {\isacharasterisk}{\kern0pt}\ l\ {\isacharasterisk}{\kern0pt}\ dist\ v\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ v{\isacharparenright}{\kern0pt}\ {\isacharless}{\kern0pt}\ eps\ {\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}\ {\isacharminus}{\kern0pt}\ l{\isacharparenright}{\kern0pt}\ {\isasymor}\ eps\ {\isasymle}\ {\isadigit{0}}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}eps{\isacharcomma}{\kern0pt}\ {\isasymL}\isactrlsub b\ v{\isacharparenright}{\kern0pt}{\isacharcomma}{\kern0pt}\ eps{\isacharcomma}{\kern0pt}\ v{\isacharparenright}{\kern0pt}\ {\isasymin}\ Wellfounded{\isachardot}{\kern0pt}measure\ term{\isacharunderscore}{\kern0pt}measure{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ gt{\isacharunderscore}{\kern0pt}zero{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}l\ {\isasymnoteq}\ {\isadigit{0}}{\isachardoublequoteclose}\ {\isachardoublequoteopen}eps\ {\isachargreater}{\kern0pt}\ {\isadigit{0}}{\isachardoublequoteclose}\ \isakeyword{and}\ dist{\isacharunderscore}{\kern0pt}ge{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}eps\ {\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}\ {\isacharminus}{\kern0pt}\ l{\isacharparenright}{\kern0pt}\ {\isasymle}\ dist\ v\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ v{\isacharparenright}{\kern0pt}\ {\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isadigit{2}}\ {\isacharasterisk}{\kern0pt}\ l{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ h\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ algebra{\isacharunderscore}{\kern0pt}simps{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ v{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}opt{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}v\ {\isasymnoteq}\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ h\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ force\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}log\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}\ {\isacharslash}{\kern0pt}\ l{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}eps\ {\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}\ {\isacharminus}{\kern0pt}\ l{\isacharparenright}{\kern0pt}\ {\isacharslash}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isadigit{8}}\ {\isacharasterisk}{\kern0pt}\ l{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharless}{\kern0pt}\ log\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}\ {\isacharslash}{\kern0pt}\ l{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}dist\ v\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}intro\ log{\isacharunderscore}{\kern0pt}less{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isadigit{1}}\ {\isacharless}{\kern0pt}\ {\isadigit{1}}\ {\isacharslash}{\kern0pt}\ l{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ mult{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}less{\isacharunderscore}{\kern0pt}div{\isacharunderscore}{\kern0pt}pos\ intro{\isacharcolon}{\kern0pt}\ neq{\isacharunderscore}{\kern0pt}le{\isacharunderscore}{\kern0pt}trans{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isadigit{0}}\ {\isacharless}{\kern0pt}\ eps\ {\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}\ {\isacharminus}{\kern0pt}\ l{\isacharparenright}{\kern0pt}\ {\isacharslash}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isadigit{8}}\ {\isacharasterisk}{\kern0pt}\ l{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ mult{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}less{\isacharunderscore}{\kern0pt}div{\isacharunderscore}{\kern0pt}pos\ intro{\isacharcolon}{\kern0pt}\ neq{\isacharunderscore}{\kern0pt}le{\isacharunderscore}{\kern0pt}trans{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}eps\ {\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}\ {\isacharminus}{\kern0pt}\ l{\isacharparenright}{\kern0pt}\ {\isacharslash}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isadigit{8}}\ {\isacharasterisk}{\kern0pt}\ l{\isacharparenright}{\kern0pt}\ {\isacharless}{\kern0pt}\ dist\ v\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt{\isachardoublequoteclose}\ \isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ dist{\isacharunderscore}{\kern0pt}pos{\isacharunderscore}{\kern0pt}lt{\isacharbrackleft}{\kern0pt}OF\ v{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}opt{\isacharbrackright}{\kern0pt}\ dist{\isacharunderscore}{\kern0pt}{\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}lt{\isacharunderscore}{\kern0pt}dist{\isacharunderscore}{\kern0pt}opt{\isacharbrackleft}{\kern0pt}of\ v{\isacharbrackright}{\kern0pt}\ gt{\isacharunderscore}{\kern0pt}zero\ zero{\isacharunderscore}{\kern0pt}le{\isacharunderscore}{\kern0pt}disc\ \isanewline
-\ \ \ \ \ \ \ \ \ \ mult{\isacharunderscore}{\kern0pt}strict{\isacharunderscore}{\kern0pt}left{\isacharunderscore}{\kern0pt}mono{\isacharbrackleft}{\kern0pt}of\ {\isachardoublequoteopen}dist\ v\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ v{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isadigit{4}}\ {\isacharasterisk}{\kern0pt}\ dist\ v\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ l{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}intro\ mult{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}div{\isacharunderscore}{\kern0pt}pos{\isacharunderscore}{\kern0pt}less\ le{\isacharunderscore}{\kern0pt}less{\isacharunderscore}{\kern0pt}trans{\isacharbrackleft}{\kern0pt}OF\ dist{\isacharunderscore}{\kern0pt}ge{\isacharbrackright}{\kern0pt}{\isacharcomma}{\kern0pt}\ argo{\isacharplus}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ vi{\isacharunderscore}{\kern0pt}rel{\isacharunderscore}{\kern0pt}dec\ h\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\isacommand{qed}\isamarkupfalse%
-\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\begin{isamarkuptext}%
-The distance between an estimate for the value and the optimal value can be bounded with respect to 
-the distance between the estimate and the result of applying it to \isa{{\isasymL}\isactrlsub b}%
-\end{isamarkuptext}\isamarkuptrue%
-\isacommand{lemma}\isamarkupfalse%
-\ contraction{\isacharunderscore}{\kern0pt}{\isasymL}{\isacharunderscore}{\kern0pt}dist{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isadigit{1}}\ {\isacharminus}{\kern0pt}\ l{\isacharparenright}{\kern0pt}\ {\isacharasterisk}{\kern0pt}\ dist\ v\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt\ {\isasymle}\ dist\ v\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ v{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ contraction{\isacharunderscore}{\kern0pt}dist\ contraction{\isacharunderscore}{\kern0pt}{\isasymL}\ disc{\isacharunderscore}{\kern0pt}lt{\isacharunderscore}{\kern0pt}one\ zero{\isacharunderscore}{\kern0pt}le{\isacharunderscore}{\kern0pt}disc\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ fastforce%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ dist{\isacharunderscore}{\kern0pt}{\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}opt{\isacharunderscore}{\kern0pt}eps{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}eps\ {\isachargreater}{\kern0pt}\ {\isadigit{0}}{\isachardoublequoteclose}\ {\isachardoublequoteopen}{\isadigit{2}}\ {\isacharasterisk}{\kern0pt}\ l\ {\isacharasterisk}{\kern0pt}\ dist\ v\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ v{\isacharparenright}{\kern0pt}\ {\isacharless}{\kern0pt}\ eps\ {\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharminus}{\kern0pt}l{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}dist\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ v{\isacharparenright}{\kern0pt}\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt\ {\isacharless}{\kern0pt}\ eps\ {\isacharslash}{\kern0pt}\ {\isadigit{2}}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}dist\ v\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt\ {\isasymle}\ dist\ v\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ v{\isacharparenright}{\kern0pt}\ {\isacharslash}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}\ {\isacharminus}{\kern0pt}\ l{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ contraction{\isacharunderscore}{\kern0pt}{\isasymL}{\isacharunderscore}{\kern0pt}dist\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ mult{\isachardot}{\kern0pt}commute\ pos{\isacharunderscore}{\kern0pt}le{\isacharunderscore}{\kern0pt}divide{\isacharunderscore}{\kern0pt}eq{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isadigit{2}}\ {\isacharasterisk}{\kern0pt}\ l\ {\isacharasterisk}{\kern0pt}\ dist\ v\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt\ {\isasymle}\ {\isadigit{2}}\ {\isacharasterisk}{\kern0pt}\ l\ {\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}dist\ v\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ v{\isacharparenright}{\kern0pt}\ {\isacharslash}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}\ {\isacharminus}{\kern0pt}\ l{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ contraction{\isacharunderscore}{\kern0pt}{\isasymL}{\isacharunderscore}{\kern0pt}dist\ assms\ mult{\isacharunderscore}{\kern0pt}le{\isacharunderscore}{\kern0pt}cancel{\isacharunderscore}{\kern0pt}left{\isacharunderscore}{\kern0pt}pos{\isacharbrackleft}{\kern0pt}of\ {\isachardoublequoteopen}{\isadigit{2}}\ {\isacharasterisk}{\kern0pt}\ l{\isachardoublequoteclose}{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}fastforce\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ mult{\isacharunderscore}{\kern0pt}left{\isacharunderscore}{\kern0pt}mono{\isacharbrackleft}{\kern0pt}of\ {\isacharunderscore}{\kern0pt}\ {\isacharunderscore}{\kern0pt}\ {\isachardoublequoteopen}{\isadigit{2}}\ {\isacharasterisk}{\kern0pt}\ l{\isachardoublequoteclose}{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isadigit{2}}\ {\isacharasterisk}{\kern0pt}\ l\ {\isacharasterisk}{\kern0pt}\ dist\ v\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt\ {\isacharless}{\kern0pt}\ eps{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ assms{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}\ pos{\isacharunderscore}{\kern0pt}divide{\isacharunderscore}{\kern0pt}less{\isacharunderscore}{\kern0pt}eq\ intro{\isacharcolon}{\kern0pt}\ order{\isachardot}{\kern0pt}strict{\isacharunderscore}{\kern0pt}trans{\isadigit{1}}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}dist\ v\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt\ {\isacharasterisk}{\kern0pt}\ l\ {\isacharless}{\kern0pt}\ eps\ {\isacharslash}{\kern0pt}\ {\isadigit{2}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ argo\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}l\ {\isacharasterisk}{\kern0pt}\ dist\ v\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt\ {\isacharless}{\kern0pt}\ eps\ {\isacharslash}{\kern0pt}\ {\isadigit{2}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ algebra{\isacharunderscore}{\kern0pt}simps{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}dist\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ v{\isacharparenright}{\kern0pt}\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt\ {\isacharless}{\kern0pt}\ eps\ {\isacharslash}{\kern0pt}\ {\isadigit{2}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ contraction{\isacharunderscore}{\kern0pt}{\isasymL}{\isacharbrackleft}{\kern0pt}of\ v\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt{\isacharbrackright}{\kern0pt}\ \isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\begin{isamarkuptext}%
-The estimates above allow to give a bound on the error of \isa{value{\isacharunderscore}{\kern0pt}iteration}.%
-\end{isamarkuptext}\isamarkuptrue%
-\isacommand{declare}\isamarkupfalse%
-\ value{\isacharunderscore}{\kern0pt}iteration{\isachardot}{\kern0pt}simps{\isacharbrackleft}{\kern0pt}simp\ del{\isacharbrackright}{\kern0pt}\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ value{\isacharunderscore}{\kern0pt}iteration{\isacharunderscore}{\kern0pt}error{\isacharcolon}{\kern0pt}\ \isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}eps\ {\isachargreater}{\kern0pt}\ {\isadigit{0}}{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}dist\ {\isacharparenleft}{\kern0pt}value{\isacharunderscore}{\kern0pt}iteration\ eps\ v{\isacharparenright}{\kern0pt}\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt\ {\isacharless}{\kern0pt}\ eps\ {\isacharslash}{\kern0pt}\ {\isadigit{2}}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ assms\ dist{\isacharunderscore}{\kern0pt}{\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}opt{\isacharunderscore}{\kern0pt}eps\ value{\isacharunderscore}{\kern0pt}iteration{\isachardot}{\kern0pt}simps\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}induction\ eps\ v\ rule{\isacharcolon}{\kern0pt}\ value{\isacharunderscore}{\kern0pt}iteration{\isachardot}{\kern0pt}induct{\isacharparenright}{\kern0pt}\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\begin{isamarkuptext}%
-After the value iteration terminates, one can easily obtain a stationary deterministic 
-epsilon-optimal policy.
-
-Such a policy does not exist in general, attainment of the supremum in \isa{{\isasymL}\isactrlsub b} is required.%
-\end{isamarkuptext}\isamarkuptrue%
-\isacommand{definition}\isamarkupfalse%
-\ {\isachardoublequoteopen}find{\isacharunderscore}{\kern0pt}policy\ {\isacharparenleft}{\kern0pt}v\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isacharprime}{\kern0pt}s\ {\isasymRightarrow}\isactrlsub b\ real{\isacharparenright}{\kern0pt}\ s\ {\isacharequal}{\kern0pt}\ arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}on\ {\isacharparenleft}{\kern0pt}{\isasymlambda}a{\isachardot}{\kern0pt}\ L\isactrlsub a\ a\ v\ s{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}A\ s{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ {\isachardoublequoteopen}vi{\isacharunderscore}{\kern0pt}policy\ eps\ v\ {\isacharequal}{\kern0pt}\ find{\isacharunderscore}{\kern0pt}policy\ {\isacharparenleft}{\kern0pt}value{\isacharunderscore}{\kern0pt}iteration\ eps\ v{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}%
-\begin{isamarkuptext}%
-We formalize the attainment of the supremum using a predicate \isa{has{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max}.%
-\end{isamarkuptext}\isamarkuptrue%
-\isacommand{abbreviation}\isamarkupfalse%
-\ {\isachardoublequoteopen}vi\ u\ n\ {\isasymequiv}\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ {\isacharcircum}{\kern0pt}{\isacharcircum}{\kern0pt}n{\isacharparenright}{\kern0pt}\ u{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ {\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}iter{\isacharunderscore}{\kern0pt}mono{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}u\ {\isasymle}\ v{\isachardoublequoteclose}\ \isakeyword{shows}\ {\isachardoublequoteopen}vi\ u\ n\ {\isasymle}\ vi\ v\ n{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ assms\ {\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}mono\ \isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}induction\ n{\isacharparenright}{\kern0pt}\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ \isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}vi\ v\ {\isacharparenleft}{\kern0pt}Suc\ n{\isacharparenright}{\kern0pt}\ {\isasymle}\ vi\ v\ n{\isachardoublequoteclose}\ \isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}vi\ v\ {\isacharparenleft}{\kern0pt}Suc\ n\ {\isacharplus}{\kern0pt}\ m{\isacharparenright}{\kern0pt}\ {\isasymle}\ vi\ v\ {\isacharparenleft}{\kern0pt}n\ {\isacharplus}{\kern0pt}\ m{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}vi\ v\ {\isacharparenleft}{\kern0pt}Suc\ n\ {\isacharplus}{\kern0pt}\ m{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ vi\ {\isacharparenleft}{\kern0pt}vi\ v\ {\isacharparenleft}{\kern0pt}Suc\ n{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ m{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ Groups{\isachardot}{\kern0pt}add{\isacharunderscore}{\kern0pt}ac{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}\ funpow{\isacharunderscore}{\kern0pt}add\ funpow{\isacharunderscore}{\kern0pt}swap{\isadigit{1}}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}\ {\isasymle}\ vi\ {\isacharparenleft}{\kern0pt}vi\ v\ n{\isacharparenright}{\kern0pt}\ m{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}iter{\isacharunderscore}{\kern0pt}mono{\isacharbrackleft}{\kern0pt}OF\ assms{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}\ {\isacharequal}{\kern0pt}\ vi\ v\ {\isacharparenleft}{\kern0pt}n\ {\isacharplus}{\kern0pt}\ m{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ add{\isachardot}{\kern0pt}commute\ funpow{\isacharunderscore}{\kern0pt}add{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{finally}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\ \isacommand{{\isachardot}{\kern0pt}}\isamarkupfalse%
-\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ \isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}vi\ v\ n\ {\isasymle}\ vi\ v\ {\isacharparenleft}{\kern0pt}Suc\ n{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}vi\ v\ {\isacharparenleft}{\kern0pt}n\ {\isacharplus}{\kern0pt}\ m{\isacharparenright}{\kern0pt}\ {\isasymle}\ vi\ v\ {\isacharparenleft}{\kern0pt}Suc\ n\ {\isacharplus}{\kern0pt}\ m{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}vi\ v\ {\isacharparenleft}{\kern0pt}n\ {\isacharplus}{\kern0pt}\ m{\isacharparenright}{\kern0pt}\ {\isasymle}\ vi\ {\isacharparenleft}{\kern0pt}vi\ v\ n{\isacharparenright}{\kern0pt}\ m{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ Groups{\isachardot}{\kern0pt}add{\isacharunderscore}{\kern0pt}ac{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}\ funpow{\isacharunderscore}{\kern0pt}add\ funpow{\isacharunderscore}{\kern0pt}swap{\isadigit{1}}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isasymle}\ vi\ v\ {\isacharparenleft}{\kern0pt}Suc\ n\ {\isacharplus}{\kern0pt}\ m{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}iter{\isacharunderscore}{\kern0pt}mono{\isacharbrackleft}{\kern0pt}OF\ assms{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp\ only{\isacharcolon}{\kern0pt}\ add{\isachardot}{\kern0pt}commute\ funpow{\isacharunderscore}{\kern0pt}add\ o{\isacharunderscore}{\kern0pt}apply{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{finally}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\ \isacommand{{\isachardot}{\kern0pt}}\isamarkupfalse%
-\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ {\isachardoublequoteopen}vi\ v\ {\isasymlonglonglongrightarrow}\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ {\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}lim\isacommand{{\isachardot}{\kern0pt}}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isasymlambda}n{\isachardot}{\kern0pt}\ dist\ {\isacharparenleft}{\kern0pt}vi\ v\ {\isacharparenleft}{\kern0pt}Suc\ n{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}vi\ v\ n{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymlonglonglongrightarrow}\ {\isadigit{0}}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ thm{\isacharunderscore}{\kern0pt}{\isadigit{6}}{\isacharunderscore}{\kern0pt}{\isadigit{3}}{\isacharunderscore}{\kern0pt}{\isadigit{1}}{\isacharunderscore}{\kern0pt}b{\isacharunderscore}{\kern0pt}aux{\isacharbrackleft}{\kern0pt}of\ v{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp\ only{\isacharcolon}{\kern0pt}\ dist{\isacharunderscore}{\kern0pt}commute{\isacharbrackleft}{\kern0pt}of\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ {\isacharcircum}{\kern0pt}{\isacharcircum}{\kern0pt}\ Suc\ {\isacharunderscore}{\kern0pt}{\isacharparenright}{\kern0pt}\ v{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isanewline
-\isacommand{end}\isamarkupfalse%
-\isanewline
-\isanewline
-\isacommand{context}\isamarkupfalse%
-\ MDP{\isacharunderscore}{\kern0pt}att{\isacharunderscore}{\kern0pt}{\isasymL}\ \isanewline
-\isakeyword{begin}%
-\begin{isamarkuptext}%
-The error of the resulting policy is bounded by the distance from its value to the value computed 
-by the value iteration plus the error in the value iteration itself.
-We show that both are less than \isa{eps\ {\isacharslash}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}{\isacharprime}{\kern0pt}b{\isacharparenright}{\kern0pt}} when the algorithm terminates.%
-\end{isamarkuptext}\isamarkuptrue%
-\isacommand{lemma}\isamarkupfalse%
-\ find{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}error{\isacharunderscore}{\kern0pt}bound{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}eps\ {\isachargreater}{\kern0pt}\ {\isadigit{0}}{\isachardoublequoteclose}\ {\isachardoublequoteopen}{\isadigit{2}}\ {\isacharasterisk}{\kern0pt}\ l\ {\isacharasterisk}{\kern0pt}\ dist\ v\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ v{\isacharparenright}{\kern0pt}\ {\isacharless}{\kern0pt}\ eps\ {\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharminus}{\kern0pt}l{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}dist\ {\isacharparenleft}{\kern0pt}{\isasymnu}\isactrlsub b\ {\isacharparenleft}{\kern0pt}mk{\isacharunderscore}{\kern0pt}stationary{\isacharunderscore}{\kern0pt}det\ {\isacharparenleft}{\kern0pt}find{\isacharunderscore}{\kern0pt}policy\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ v{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt\ {\isacharless}{\kern0pt}\ eps{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}d\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}mk{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\ {\isacharparenleft}{\kern0pt}find{\isacharunderscore}{\kern0pt}policy\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ v{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}p\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}mk{\isacharunderscore}{\kern0pt}stationary\ {\isacharquery}{\kern0pt}d{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ L{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}{\isasymL}\isactrlsub b{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}L\ {\isacharparenleft}{\kern0pt}mk{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\ {\isacharparenleft}{\kern0pt}find{\isacharunderscore}{\kern0pt}policy\ v{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ v\ {\isacharequal}{\kern0pt}\ {\isasymL}\isactrlsub b\ v{\isachardoublequoteclose}\ \isakeyword{for}\ v\isanewline
-\ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ find{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}intro\ antisym{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}L\ {\isacharparenleft}{\kern0pt}mk{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\ {\isacharparenleft}{\kern0pt}{\isasymlambda}s{\isachardot}{\kern0pt}\ arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}on\ {\isacharparenleft}{\kern0pt}{\isasymlambda}a{\isachardot}{\kern0pt}\ L\isactrlsub a\ a\ v\ s{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}A\ s{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ v\ {\isasymle}\ {\isasymL}\isactrlsub b\ v{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ Sup{\isacharunderscore}{\kern0pt}att\ has{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max\ abs{\isacharunderscore}{\kern0pt}L{\isacharunderscore}{\kern0pt}le\isanewline
-\ \ \ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ {\isasymL}\isactrlsub b{\isachardot}{\kern0pt}rep{\isacharunderscore}{\kern0pt}eq\ {\isasymL}{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}SUP{\isacharunderscore}{\kern0pt}det\ less{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}bfun{\isacharunderscore}{\kern0pt}def\ arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}on{\isacharunderscore}{\kern0pt}def\ is{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det{\isacharunderscore}{\kern0pt}def\ max{\isacharunderscore}{\kern0pt}L{\isacharunderscore}{\kern0pt}ex{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ cSUP{\isacharunderscore}{\kern0pt}upper\ bounded{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}bdd{\isacharunderscore}{\kern0pt}above\ boundedI{\isacharbrackleft}{\kern0pt}of\ {\isacharunderscore}{\kern0pt}\ {\isachardoublequoteopen}r\isactrlsub M\ {\isacharplus}{\kern0pt}\ l\ {\isacharasterisk}{\kern0pt}\ norm\ v{\isachardoublequoteclose}{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymL}\isactrlsub b\ v\ {\isasymle}\ L\ {\isacharparenleft}{\kern0pt}mk{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\ {\isacharparenleft}{\kern0pt}{\isasymlambda}s{\isachardot}{\kern0pt}\ arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}on\ {\isacharparenleft}{\kern0pt}{\isasymlambda}a{\isachardot}{\kern0pt}\ L\isactrlsub a\ a\ v\ s{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}A\ s{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ v{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ less{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}bfun{\isacharunderscore}{\kern0pt}def\ {\isasymL}\isactrlsub b{\isachardot}{\kern0pt}rep{\isacharunderscore}{\kern0pt}eq\ {\isasymL}{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}SUP{\isacharunderscore}{\kern0pt}det\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ Sup{\isacharunderscore}{\kern0pt}att\ ex{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ cSUP{\isacharunderscore}{\kern0pt}least\ app{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}ge\ simp{\isacharcolon}{\kern0pt}\ L{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}L\isactrlsub a{\isacharunderscore}{\kern0pt}det\ max{\isacharunderscore}{\kern0pt}L{\isacharunderscore}{\kern0pt}ex{\isacharunderscore}{\kern0pt}def\ is{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}dist\ {\isacharparenleft}{\kern0pt}{\isasymnu}\isactrlsub b\ {\isacharquery}{\kern0pt}p{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ v{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ dist\ {\isacharparenleft}{\kern0pt}L\ {\isacharquery}{\kern0pt}d\ {\isacharparenleft}{\kern0pt}{\isasymnu}\isactrlsub b\ {\isacharquery}{\kern0pt}p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ v{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ L{\isacharunderscore}{\kern0pt}{\isasymnu}{\isacharunderscore}{\kern0pt}fix\ \isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ force\isanewline
-\ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isasymle}\ dist\ {\isacharparenleft}{\kern0pt}L\ {\isacharquery}{\kern0pt}d\ {\isacharparenleft}{\kern0pt}{\isasymnu}\isactrlsub b\ {\isacharquery}{\kern0pt}p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ v{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharplus}{\kern0pt}\ dist\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ v{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ v{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ dist{\isacharunderscore}{\kern0pt}triangle\ \isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isacharequal}{\kern0pt}\ dist\ {\isacharparenleft}{\kern0pt}L\ {\isacharquery}{\kern0pt}d\ {\isacharparenleft}{\kern0pt}{\isasymnu}\isactrlsub b\ {\isacharquery}{\kern0pt}p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}L\ {\isacharquery}{\kern0pt}d\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ v{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharplus}{\kern0pt}\ dist\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ v{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ v{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ L{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}{\isasymL}\isactrlsub b{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isasymle}\ l\ {\isacharasterisk}{\kern0pt}\ \ dist\ {\isacharparenleft}{\kern0pt}{\isasymnu}\isactrlsub b\ {\isacharquery}{\kern0pt}p{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ v{\isacharparenright}{\kern0pt}\ {\isacharplus}{\kern0pt}\ l\ {\isacharasterisk}{\kern0pt}\ dist\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ v{\isacharparenright}{\kern0pt}\ v{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ contraction{\isacharunderscore}{\kern0pt}{\isasymL}\ contraction{\isacharunderscore}{\kern0pt}L\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}fastforce\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ add{\isacharunderscore}{\kern0pt}mono{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{finally}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ aux{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}dist\ {\isacharparenleft}{\kern0pt}{\isasymnu}\isactrlsub b\ {\isacharquery}{\kern0pt}p{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ v{\isacharparenright}{\kern0pt}\ {\isasymle}\ l\ {\isacharasterisk}{\kern0pt}\ dist\ {\isacharparenleft}{\kern0pt}{\isasymnu}\isactrlsub b\ {\isacharquery}{\kern0pt}p{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ v{\isacharparenright}{\kern0pt}\ {\isacharplus}{\kern0pt}\ l\ {\isacharasterisk}{\kern0pt}\ dist\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ v{\isacharparenright}{\kern0pt}\ v{\isachardoublequoteclose}\ \isacommand{{\isachardot}{\kern0pt}}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}dist\ {\isacharparenleft}{\kern0pt}{\isasymnu}\isactrlsub b\ {\isacharquery}{\kern0pt}p{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ v{\isacharparenright}{\kern0pt}\ {\isacharminus}{\kern0pt}\ l\ {\isacharasterisk}{\kern0pt}\ dist\ {\isacharparenleft}{\kern0pt}{\isasymnu}\isactrlsub b\ {\isacharquery}{\kern0pt}p{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ v{\isacharparenright}{\kern0pt}\ {\isasymle}\ l\ {\isacharasterisk}{\kern0pt}\ dist\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ v{\isacharparenright}{\kern0pt}\ v{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}dist\ {\isacharparenleft}{\kern0pt}{\isasymnu}\isactrlsub b\ {\isacharquery}{\kern0pt}p{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ v{\isacharparenright}{\kern0pt}\ {\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}\ {\isacharminus}{\kern0pt}\ l{\isacharparenright}{\kern0pt}\ {\isasymle}\ l\ {\isacharasterisk}{\kern0pt}\ dist\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ v{\isacharparenright}{\kern0pt}\ v{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ argo\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ \ {\isachardoublequoteopen}{\isadigit{2}}\ {\isacharasterisk}{\kern0pt}\ dist\ {\isacharparenleft}{\kern0pt}{\isasymnu}\isactrlsub b\ {\isacharquery}{\kern0pt}p{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ v{\isacharparenright}{\kern0pt}\ {\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharminus}{\kern0pt}l{\isacharparenright}{\kern0pt}\ {\isasymle}\ {\isadigit{2}}\ {\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}l\ {\isacharasterisk}{\kern0pt}\ dist\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ v{\isacharparenright}{\kern0pt}\ v{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ zero{\isacharunderscore}{\kern0pt}le{\isacharunderscore}{\kern0pt}disc\ mult{\isacharunderscore}{\kern0pt}left{\isacharunderscore}{\kern0pt}mono\ \isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isasymle}\ eps\ {\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharminus}{\kern0pt}l{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assms\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ mult{\isacharunderscore}{\kern0pt}left{\isacharunderscore}{\kern0pt}mono\ simp{\isacharcolon}{\kern0pt}\ dist{\isacharunderscore}{\kern0pt}commute\ pos{\isacharunderscore}{\kern0pt}divide{\isacharunderscore}{\kern0pt}le{\isacharunderscore}{\kern0pt}eq{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{finally}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isadigit{2}}\ {\isacharasterisk}{\kern0pt}\ dist\ {\isacharparenleft}{\kern0pt}{\isasymnu}\isactrlsub b\ {\isacharquery}{\kern0pt}p{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ v{\isacharparenright}{\kern0pt}\ {\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}\ {\isacharminus}{\kern0pt}\ l{\isacharparenright}{\kern0pt}\ {\isasymle}\ eps\ {\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}\ {\isacharminus}{\kern0pt}\ l{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isacommand{{\isachardot}{\kern0pt}}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isadigit{2}}\ {\isacharasterisk}{\kern0pt}\ dist\ {\isacharparenleft}{\kern0pt}{\isasymnu}\isactrlsub b\ {\isacharquery}{\kern0pt}p{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ v{\isacharparenright}{\kern0pt}\ {\isasymle}\ eps{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ disc{\isacharunderscore}{\kern0pt}lt{\isacharunderscore}{\kern0pt}one\ mult{\isacharunderscore}{\kern0pt}right{\isacharunderscore}{\kern0pt}le{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}le\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isadigit{2}}\ {\isacharasterisk}{\kern0pt}\ dist\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ v{\isacharparenright}{\kern0pt}\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt\ {\isacharless}{\kern0pt}\ eps{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ dist{\isacharunderscore}{\kern0pt}{\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}opt{\isacharunderscore}{\kern0pt}eps\ assms\ \isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ fastforce\isanewline
-\ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}dist\ {\isacharparenleft}{\kern0pt}{\isasymnu}\isactrlsub b\ {\isacharquery}{\kern0pt}p{\isacharparenright}{\kern0pt}\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt\ {\isasymle}\ dist\ {\isacharparenleft}{\kern0pt}{\isasymnu}\isactrlsub b\ {\isacharquery}{\kern0pt}p{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ v{\isacharparenright}{\kern0pt}\ {\isacharplus}{\kern0pt}\ dist\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ v{\isacharparenright}{\kern0pt}\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ dist{\isacharunderscore}{\kern0pt}triangle\ \isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\ \ \isanewline
-\ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\ \isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ vi{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}opt{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}{\isadigit{0}}\ {\isacharless}{\kern0pt}\ eps{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}dist\ {\isacharparenleft}{\kern0pt}{\isasymnu}\isactrlsub b\ {\isacharparenleft}{\kern0pt}mk{\isacharunderscore}{\kern0pt}stationary{\isacharunderscore}{\kern0pt}det\ {\isacharparenleft}{\kern0pt}vi{\isacharunderscore}{\kern0pt}policy\ eps\ v{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt\ {\isacharless}{\kern0pt}\ eps{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ vi{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}def\ \isanewline
-\ \ \isacommand{using}\isamarkupfalse%
-\ assms\isanewline
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}induction\ eps\ v\ rule{\isacharcolon}{\kern0pt}\ value{\isacharunderscore}{\kern0pt}iteration{\isachardot}{\kern0pt}induct{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{case}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}\ v{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{then}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}case\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ find{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}error{\isacharunderscore}{\kern0pt}bound\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}subst\ value{\isacharunderscore}{\kern0pt}iteration{\isachardot}{\kern0pt}simps{\isacharparenright}{\kern0pt}\ auto\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ lemma{\isacharunderscore}{\kern0pt}{\isadigit{6}}{\isacharunderscore}{\kern0pt}{\isadigit{3}}{\isacharunderscore}{\kern0pt}{\isadigit{1}}{\isacharunderscore}{\kern0pt}d{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}eps\ {\isachargreater}{\kern0pt}\ {\isadigit{0}}{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}{\isadigit{2}}\ {\isacharasterisk}{\kern0pt}\ l\ {\isacharasterisk}{\kern0pt}\ dist\ {\isacharparenleft}{\kern0pt}vi\ v\ {\isacharparenleft}{\kern0pt}Suc\ n{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}vi\ v\ n{\isacharparenright}{\kern0pt}\ {\isacharless}{\kern0pt}\ eps\ {\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharminus}{\kern0pt}l{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}dist\ {\isacharparenleft}{\kern0pt}vi\ v\ {\isacharparenleft}{\kern0pt}Suc\ n{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt\ {\isacharless}{\kern0pt}\ eps\ {\isacharslash}{\kern0pt}\ {\isadigit{2}}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ dist{\isacharunderscore}{\kern0pt}{\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}opt{\isacharunderscore}{\kern0pt}eps\ assms\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ dist{\isacharunderscore}{\kern0pt}commute{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{end}\isamarkupfalse%
-\isanewline
-\isanewline
-\isacommand{context}\isamarkupfalse%
-\ MDP{\isacharunderscore}{\kern0pt}act\ \isakeyword{begin}\isanewline
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ {\isachardoublequoteopen}find{\isacharunderscore}{\kern0pt}policy{\isacharprime}{\kern0pt}\ {\isacharparenleft}{\kern0pt}v\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isacharprime}{\kern0pt}s\ {\isasymRightarrow}\isactrlsub b\ real{\isacharparenright}{\kern0pt}\ s\ {\isacharequal}{\kern0pt}\ arb{\isacharunderscore}{\kern0pt}act\ {\isacharparenleft}{\kern0pt}opt{\isacharunderscore}{\kern0pt}acts\ v\ s{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ {\isachardoublequoteopen}vi{\isacharunderscore}{\kern0pt}policy{\isacharprime}{\kern0pt}\ eps\ v\ {\isacharequal}{\kern0pt}\ find{\isacharunderscore}{\kern0pt}policy{\isacharprime}{\kern0pt}\ {\isacharparenleft}{\kern0pt}value{\isacharunderscore}{\kern0pt}iteration\ eps\ v{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ find{\isacharunderscore}{\kern0pt}policy{\isacharprime}{\kern0pt}{\isacharunderscore}{\kern0pt}error{\isacharunderscore}{\kern0pt}bound{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}eps\ {\isachargreater}{\kern0pt}\ {\isadigit{0}}{\isachardoublequoteclose}\ {\isachardoublequoteopen}{\isadigit{2}}\ {\isacharasterisk}{\kern0pt}\ l\ {\isacharasterisk}{\kern0pt}\ dist\ v\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ v{\isacharparenright}{\kern0pt}\ {\isacharless}{\kern0pt}\ eps\ {\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharminus}{\kern0pt}l{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}dist\ {\isacharparenleft}{\kern0pt}{\isasymnu}\isactrlsub b\ {\isacharparenleft}{\kern0pt}mk{\isacharunderscore}{\kern0pt}stationary{\isacharunderscore}{\kern0pt}det\ {\isacharparenleft}{\kern0pt}find{\isacharunderscore}{\kern0pt}policy{\isacharprime}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ v{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt\ {\isacharless}{\kern0pt}\ eps{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}d\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}mk{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\ {\isacharparenleft}{\kern0pt}find{\isacharunderscore}{\kern0pt}policy{\isacharprime}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ v{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}p\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}mk{\isacharunderscore}{\kern0pt}stationary\ {\isacharquery}{\kern0pt}d{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ L{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}{\isasymL}\isactrlsub b{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}L\ {\isacharparenleft}{\kern0pt}mk{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\ {\isacharparenleft}{\kern0pt}find{\isacharunderscore}{\kern0pt}policy{\isacharprime}{\kern0pt}\ v{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ v\ {\isacharequal}{\kern0pt}\ {\isasymL}\isactrlsub b\ v{\isachardoublequoteclose}\ \isakeyword{for}\ v\isanewline
-\ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ find{\isacharunderscore}{\kern0pt}policy{\isacharprime}{\kern0pt}{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isasymnu}{\isacharunderscore}{\kern0pt}improving{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}{\isasymL}\isactrlsub b\ {\isasymnu}{\isacharunderscore}{\kern0pt}improving{\isacharunderscore}{\kern0pt}opt{\isacharunderscore}{\kern0pt}acts{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}dist\ {\isacharparenleft}{\kern0pt}{\isasymnu}\isactrlsub b\ {\isacharquery}{\kern0pt}p{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ v{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ dist\ {\isacharparenleft}{\kern0pt}L\ {\isacharquery}{\kern0pt}d\ {\isacharparenleft}{\kern0pt}{\isasymnu}\isactrlsub b\ {\isacharquery}{\kern0pt}p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ v{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ L{\isacharunderscore}{\kern0pt}{\isasymnu}{\isacharunderscore}{\kern0pt}fix\ \isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ force\isanewline
-\ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isasymle}\ dist\ {\isacharparenleft}{\kern0pt}L\ {\isacharquery}{\kern0pt}d\ {\isacharparenleft}{\kern0pt}{\isasymnu}\isactrlsub b\ {\isacharquery}{\kern0pt}p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ v{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharplus}{\kern0pt}\ dist\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ v{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ v{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ dist{\isacharunderscore}{\kern0pt}triangle\ \isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isacharequal}{\kern0pt}\ dist\ {\isacharparenleft}{\kern0pt}L\ {\isacharquery}{\kern0pt}d\ {\isacharparenleft}{\kern0pt}{\isasymnu}\isactrlsub b\ {\isacharquery}{\kern0pt}p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}L\ {\isacharquery}{\kern0pt}d\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ v{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharplus}{\kern0pt}\ dist\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ v{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ v{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ L{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}{\isasymL}\isactrlsub b{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isasymle}\ l\ {\isacharasterisk}{\kern0pt}\ \ dist\ {\isacharparenleft}{\kern0pt}{\isasymnu}\isactrlsub b\ {\isacharquery}{\kern0pt}p{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ v{\isacharparenright}{\kern0pt}\ {\isacharplus}{\kern0pt}\ l\ {\isacharasterisk}{\kern0pt}\ dist\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ v{\isacharparenright}{\kern0pt}\ v{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ contraction{\isacharunderscore}{\kern0pt}{\isasymL}\ contraction{\isacharunderscore}{\kern0pt}L\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}fastforce\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ add{\isacharunderscore}{\kern0pt}mono{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{finally}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ aux{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}dist\ {\isacharparenleft}{\kern0pt}{\isasymnu}\isactrlsub b\ {\isacharquery}{\kern0pt}p{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ v{\isacharparenright}{\kern0pt}\ {\isasymle}\ l\ {\isacharasterisk}{\kern0pt}\ dist\ {\isacharparenleft}{\kern0pt}{\isasymnu}\isactrlsub b\ {\isacharquery}{\kern0pt}p{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ v{\isacharparenright}{\kern0pt}\ {\isacharplus}{\kern0pt}\ l\ {\isacharasterisk}{\kern0pt}\ dist\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ v{\isacharparenright}{\kern0pt}\ v{\isachardoublequoteclose}\ \isacommand{{\isachardot}{\kern0pt}}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}dist\ {\isacharparenleft}{\kern0pt}{\isasymnu}\isactrlsub b\ {\isacharquery}{\kern0pt}p{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ v{\isacharparenright}{\kern0pt}\ {\isacharminus}{\kern0pt}\ l\ {\isacharasterisk}{\kern0pt}\ dist\ {\isacharparenleft}{\kern0pt}{\isasymnu}\isactrlsub b\ {\isacharquery}{\kern0pt}p{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ v{\isacharparenright}{\kern0pt}\ {\isasymle}\ l\ {\isacharasterisk}{\kern0pt}\ dist\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ v{\isacharparenright}{\kern0pt}\ v{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}dist\ {\isacharparenleft}{\kern0pt}{\isasymnu}\isactrlsub b\ {\isacharquery}{\kern0pt}p{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ v{\isacharparenright}{\kern0pt}\ {\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}\ {\isacharminus}{\kern0pt}\ l{\isacharparenright}{\kern0pt}\ {\isasymle}\ l\ {\isacharasterisk}{\kern0pt}\ dist\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ v{\isacharparenright}{\kern0pt}\ v{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ argo\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ \ {\isachardoublequoteopen}{\isadigit{2}}\ {\isacharasterisk}{\kern0pt}\ dist\ {\isacharparenleft}{\kern0pt}{\isasymnu}\isactrlsub b\ {\isacharquery}{\kern0pt}p{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ v{\isacharparenright}{\kern0pt}\ {\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharminus}{\kern0pt}l{\isacharparenright}{\kern0pt}\ {\isasymle}\ {\isadigit{2}}\ {\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}l\ {\isacharasterisk}{\kern0pt}\ dist\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ v{\isacharparenright}{\kern0pt}\ v{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ zero{\isacharunderscore}{\kern0pt}le{\isacharunderscore}{\kern0pt}disc\ mult{\isacharunderscore}{\kern0pt}left{\isacharunderscore}{\kern0pt}mono\ \isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isasymle}\ eps\ {\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharminus}{\kern0pt}l{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assms\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ mult{\isacharunderscore}{\kern0pt}left{\isacharunderscore}{\kern0pt}mono\ simp{\isacharcolon}{\kern0pt}\ dist{\isacharunderscore}{\kern0pt}commute\ pos{\isacharunderscore}{\kern0pt}divide{\isacharunderscore}{\kern0pt}le{\isacharunderscore}{\kern0pt}eq{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{finally}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isadigit{2}}\ {\isacharasterisk}{\kern0pt}\ dist\ {\isacharparenleft}{\kern0pt}{\isasymnu}\isactrlsub b\ {\isacharquery}{\kern0pt}p{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ v{\isacharparenright}{\kern0pt}\ {\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}\ {\isacharminus}{\kern0pt}\ l{\isacharparenright}{\kern0pt}\ {\isasymle}\ eps\ {\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}\ {\isacharminus}{\kern0pt}\ l{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isacommand{{\isachardot}{\kern0pt}}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isadigit{2}}\ {\isacharasterisk}{\kern0pt}\ dist\ {\isacharparenleft}{\kern0pt}{\isasymnu}\isactrlsub b\ {\isacharquery}{\kern0pt}p{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ v{\isacharparenright}{\kern0pt}\ {\isasymle}\ eps{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ disc{\isacharunderscore}{\kern0pt}lt{\isacharunderscore}{\kern0pt}one\ mult{\isacharunderscore}{\kern0pt}right{\isacharunderscore}{\kern0pt}le{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}le\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isadigit{2}}\ {\isacharasterisk}{\kern0pt}\ dist\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ v{\isacharparenright}{\kern0pt}\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt\ {\isacharless}{\kern0pt}\ eps{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ dist{\isacharunderscore}{\kern0pt}{\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}opt{\isacharunderscore}{\kern0pt}eps\ assms\ \isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ fastforce\isanewline
-\ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}dist\ {\isacharparenleft}{\kern0pt}{\isasymnu}\isactrlsub b\ {\isacharquery}{\kern0pt}p{\isacharparenright}{\kern0pt}\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt\ {\isasymle}\ dist\ {\isacharparenleft}{\kern0pt}{\isasymnu}\isactrlsub b\ {\isacharquery}{\kern0pt}p{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ v{\isacharparenright}{\kern0pt}\ {\isacharplus}{\kern0pt}\ dist\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ v{\isacharparenright}{\kern0pt}\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ dist{\isacharunderscore}{\kern0pt}triangle\ \isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\ \ \isanewline
-\ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\ \isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ vi{\isacharunderscore}{\kern0pt}policy{\isacharprime}{\kern0pt}{\isacharunderscore}{\kern0pt}opt{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}eps\ {\isachargreater}{\kern0pt}\ {\isadigit{0}}{\isachardoublequoteclose}\ {\isachardoublequoteopen}l\ {\isachargreater}{\kern0pt}\ {\isadigit{0}}{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}dist\ {\isacharparenleft}{\kern0pt}{\isasymnu}\isactrlsub b\ {\isacharparenleft}{\kern0pt}mk{\isacharunderscore}{\kern0pt}stationary{\isacharunderscore}{\kern0pt}det\ {\isacharparenleft}{\kern0pt}vi{\isacharunderscore}{\kern0pt}policy{\isacharprime}{\kern0pt}\ eps\ v{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt\ {\isacharless}{\kern0pt}\ eps{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ vi{\isacharunderscore}{\kern0pt}policy{\isacharprime}{\kern0pt}{\isacharunderscore}{\kern0pt}def\ \isanewline
-\ \ \isacommand{using}\isamarkupfalse%
-\ assms\isanewline
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}induction\ eps\ v\ rule{\isacharcolon}{\kern0pt}\ value{\isacharunderscore}{\kern0pt}iteration{\isachardot}{\kern0pt}induct{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{case}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}\ v{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{then}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}case\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ find{\isacharunderscore}{\kern0pt}policy{\isacharprime}{\kern0pt}{\isacharunderscore}{\kern0pt}error{\isacharunderscore}{\kern0pt}bound\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}subst\ value{\isacharunderscore}{\kern0pt}iteration{\isachardot}{\kern0pt}simps{\isacharparenright}{\kern0pt}\ auto\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{end}\isamarkupfalse%
-\isanewline
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{end}\isamarkupfalse%
-%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\end{isabellebody}%
-\endinput
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diff --git a/thys/MDP-Algorithms/output/document/comment.sty b/thys/MDP-Algorithms/output/document/comment.sty
deleted file mode 100644
--- a/thys/MDP-Algorithms/output/document/comment.sty
+++ /dev/null
@@ -1,278 +0,0 @@
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-% Comment.sty   version 3.6, October 1999
-%
-% Purpose:
-% selectively in/exclude pieces of text: the user can define new
-% comment versions, and each is controlled separately.
-% Special comments can be defined where the user specifies the
-% action that is to be taken with each comment line.
-%
-% Author
-%    Victor Eijkhout
-%    Department of Computer Science
-%    University of Tennessee
-%    107 Ayres Hall
-%    Knoxville TN 37996
-%    USA
-%
-%    victor@eijkhout.net
-%
-% This program is free software; you can redistribute it and/or
-% modify it under the terms of the GNU General Public License
-% as published by the Free Software Foundation; either version 2
-% of the License, or (at your option) any later version.
-% 
-% This program is distributed in the hope that it will be useful,
-% but WITHOUT ANY WARRANTY; without even the implied warranty of
-% MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
-% GNU General Public License for more details.
-%
-% For a copy of the GNU General Public License, write to the 
-% Free Software Foundation, Inc.,
-% 59 Temple Place - Suite 330, Boston, MA  02111-1307, USA,
-% or find it on the net, for instance at
-% http://www.gnu.org/copyleft/gpl.html
-%
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-% This style can be used with plain TeX or LaTeX, and probably
-% most other packages too.
-%
-% Usage: all text included between
-%    \comment ... \endcomment
-% or \begin{comment} ... \end{comment}
-% is discarded. 
-%
-% The opening and closing commands should appear on a line
-% of their own. No starting spaces, nothing after it.
-% This environment should work with arbitrary amounts
-% of comment, and the comment can be arbitrary text.
-%
-% Other `comment' environments are defined by
-% and are selected/deselected with
-% \includecomment{versiona}
-% \excludecoment{versionb}
-%
-% These environments are used as
-% \versiona ... \endversiona
-% or \begin{versiona} ... \end{versiona}
-% with the opening and closing commands again on a line of 
-% their own.
-%
-% LaTeX users note: for an included comment, the
-% \begin and \end lines act as if they don't exist.
-% In particular, they don't imply grouping, so assignments 
-% &c are not local.
-%
-% Special comments are defined as
-% \specialcomment{name}{before commands}{after commands}
-% where the second and third arguments are executed before
-% and after each comment block. You can use this for global
-% formatting commands.
-% To keep definitions &c local, you can include \begingroup
-% in the `before commands' and \endgroup in the `after commands'.
-% ex:
-% \specialcomment{smalltt}
-%     {\begingroup\ttfamily\footnotesize}{\endgroup}
-% You do *not* have to do an additional
-% \includecomment{smalltt}
-% To remove 'smalltt' blocks, give \excludecomment{smalltt}
-% after the definition.
-%
-% Processing comments can apply processing to each line.
-% \processcomment{name}{each-line commands}%
-%    {before commands}{after commands}
-% By defining a control sequence 
-% \def\Thiscomment##1{...} in the before commands the user can
-% specify what is to be done with each comment line.
-% BUG this does not work quite yet BUG
-%
-% Trick for short in/exclude macros (such as \maybe{this snippet}):
-%\includecomment{cond}
-%\newcommand{\maybe}[1]{}
-%\begin{cond}
-%\renewcommand{\maybe}[1]{#1}
-%\end{cond}
-%
-% Basic approach of the implementation:
-% to comment something out, scoop up  every line in verbatim mode
-% as macro argument, then throw it away.
-% For inclusions, in LaTeX the block is written out to
-% a file \CommentCutFile (default "comment.cut"), which is
-% then included.
-% In plain TeX (and other formats) both the opening and
-% closing comands are defined as noop.
-%
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-% Changes in version 3.1
-% - updated author's address
-% - cleaned up some code
-% - trailing contents on \begin{env} line is always discarded
-%  even if you've done \includecomment{env}
-% - comments no longer define grouping!! you can even
-%   \includecomment{env}
-%   \begin{env}
-%   \begin{itemize}
-%   \end{env}
-%  Isn't that something ...
-% - included comments are written to file and input again.
-% Changes in 3.2
-% - \specialcomment brought up to date (thanks to Ivo Welch).
-% Changes in 3.3
-% - updated author's address again
-% - parametrised \CommentCutFile
-% Changes in 3.4
-% - added GNU public license
-% - added \processcomment, because Ivo's fix (above) brought an
-%   inconsistency to light.
-% Changes in 3.5
-% - corrected typo in header.
-% - changed author email
-% - corrected \specialcomment yet again.
-% - fixed excludecomment of an earlier defined environment.
-% Changes in 3.6
-% - The 'cut' file is now written more verbatim, using \meaning;
-%   some people reported having trouble with ISO latin 1, or umlaute.sty.
-% - removed some \newif statements.
-%   Has this suddenly become \outer again?
-%
-% Known bugs:
-% - excludecomment leads to one superfluous space
-% - processcomment leads to a superfluous line break
-%
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-
-\def\makeinnocent#1{\catcode`#1=12 }
-\def\csarg#1#2{\expandafter#1\csname#2\endcsname}
-\def\latexname{lplain}\def\latexename{LaTeX2e}
-\newwrite\CommentStream
-\def\CommentCutFile{comment.cut}
-
-\def\ProcessComment#1% start it all of
-   {\begingroup
-    \def\CurrentComment{#1}%
-    \let\do\makeinnocent \dospecials 
-    \makeinnocent\^^L% and whatever other special cases
-    \endlinechar`\^^M \catcode`\^^M=12 \xComment}
-%\def\ProcessCommentWithArg#1#2% to be used in \leveledcomment
-%   {\begingroup
-%    \def\CurrentComment{#1}%
-%    \let\do\makeinnocent \dospecials 
-%    \makeinnocent\^^L% and whatever other special cases
-%    \endlinechar`\^^M \catcode`\^^M=12 \xComment}
-{\catcode`\^^M=12 \endlinechar=-1 %
- \gdef\xComment#1^^M{%
-    \expandafter\ProcessCommentLine}
- \gdef\ProcessCommentLine#1^^M{\def\test{#1}
-      \csarg\ifx{End\CurrentComment Test}\test
-          \edef\next{\noexpand\EndOfComment{\CurrentComment}}%
-      \else \ThisComment{#1}\let\next\ProcessCommentLine
-      \fi \next}
-}
-
-\def\CSstringmeaning#1{\expandafter\CSgobblearrow\meaning#1}
-\def\CSstringcsnoescape#1{\expandafter\CSgobbleescape\string#1}
-{\escapechar-1
-\expandafter\expandafter\expandafter\gdef
-  \expandafter\expandafter\expandafter\CSgobblearrow
-    \expandafter\string\csname macro:->\endcsname{}
-}
-\def\CSgobbleescape#1{\ifnum`\\=`#1 \else #1\fi}
-\def\WriteCommentLine#1{\def\CStmp{#1}%
-    \immediate\write\CommentStream{\CSstringmeaning\CStmp}}
-
-% 3.1 change: in LaTeX and LaTeX2e prevent grouping
-\if 0%
-\ifx\fmtname\latexename 
-    0%
-\else \ifx\fmtname\latexname 
-          0%
-      \else 
-          1%
-\fi   \fi
-%%%%
-%%%% definitions for LaTeX
-%%%%
-\def\AfterIncludedComment
-   {\immediate\closeout\CommentStream
-    \input{\CommentCutFile}\relax
-    }%
-\def\TossComment{\immediate\closeout\CommentStream}
-\def\BeforeIncludedComment
-   {\immediate\openout\CommentStream=\CommentCutFile
-    \let\ThisComment\WriteCommentLine}
-\def\includecomment
- #1{\message{Include comment '#1'}%
-    \csarg\let{After#1Comment}\AfterIncludedComment
-    \csarg\def{#1}{\BeforeIncludedComment
-        \ProcessComment{#1}}%
-    \CommentEndDef{#1}}
-\long\def\specialcomment
- #1#2#3{\message{Special comment '#1'}%
-    % note: \AfterIncludedComment does \input, so #2 goes here!
-    \csarg\def{After#1Comment}{#2\AfterIncludedComment#3}%
-    \csarg\def{#1}{\BeforeIncludedComment\relax
-          \ProcessComment{#1}}%
-    \CommentEndDef{#1}}
-\long\def\processcomment
- #1#2#3#4{\message{Lines-Processing comment '#1'}%
-    \csarg\def{After#1Comment}{#3\AfterIncludedComment#4}%
-    \csarg\def{#1}{\BeforeIncludedComment#2\relax
-          \ProcessComment{#1}}%
-    \CommentEndDef{#1}}
-\def\leveledcomment
- #1#2{\message{Include comment '#1' up to level '#2'}%
-    %\csname #1IsLeveledCommenttrue\endcsname
-    \csarg\let{After#1Comment}\AfterIncludedComment
-    \csarg\def{#1}{\BeforeIncludedComment
-        \ProcessCommentWithArg{#1}}%
-    \CommentEndDef{#1}}
-\else 
-%%%%
-%%%%plain TeX and other formats
-%%%%
-\def\includecomment
- #1{\message{Including comment '#1'}%
-    \csarg\def{#1}{}%
-    \csarg\def{end#1}{}}
-\long\def\specialcomment
- #1#2#3{\message{Special comment '#1'}%
-    \csarg\def{#1}{\def\ThisComment{}\def\AfterComment{#3}#2%
-           \ProcessComment{#1}}%
-    \CommentEndDef{#1}}
-\fi
-
-%%%%
-%%%% general definition of skipped comment
-%%%%
-\def\excludecomment
- #1{\message{Excluding comment '#1'}%
-    \csarg\def{#1}{\let\AfterComment\relax
-           \def\ThisComment####1{}\ProcessComment{#1}}%
-    \csarg\let{After#1Comment}\TossComment
-    \CommentEndDef{#1}}
-
-\if 0%
-\ifx\fmtname\latexename 
-    0%
-\else \ifx\fmtname\latexname 
-          0%
-      \else 
-          1%
-\fi   \fi
-% latex & latex2e:
-\def\EndOfComment#1{\endgroup\end{#1}%
-    \csname After#1Comment\endcsname}
-\def\CommentEndDef#1{{\escapechar=-1\relax
-    \csarg\xdef{End#1Test}{\string\\end\string\{#1\string\}}%
-    }}
-\else
-% plain & other
-\def\EndOfComment#1{\endgroup\AfterComment}
-\def\CommentEndDef#1{{\escapechar=-1\relax
-    \csarg\xdef{End#1Test}{\string\\end#1}%
-    }}
-\fi
-
-\excludecomment{comment}
-
-\endinput
diff --git a/thys/MDP-Algorithms/output/document/isabelle.sty b/thys/MDP-Algorithms/output/document/isabelle.sty
deleted file mode 100644
--- a/thys/MDP-Algorithms/output/document/isabelle.sty
+++ /dev/null
@@ -1,282 +0,0 @@
-%%
-%% macros for Isabelle generated LaTeX output
-%%
-
-%%% Simple document preparation (based on theory token language and symbols)
-
-% isabelle environments
-
-\newcommand{\isabellecontext}{UNKNOWN}
-\newcommand{\setisabellecontext}[1]{\def\isabellecontext{#1}}
-
-\newcommand{\isastyle}{\UNDEF}
-\newcommand{\isastylett}{\UNDEF}
-\newcommand{\isastyleminor}{\UNDEF}
-\newcommand{\isastyleminortt}{\UNDEF}
-\newcommand{\isastylescript}{\UNDEF}
-\newcommand{\isastyletext}{\normalsize\normalfont\rmfamily}
-\newcommand{\isastyletxt}{\normalfont\rmfamily}
-\newcommand{\isastylecmt}{\normalfont\rmfamily}
-
-\newcommand{\isaspacing}{%
-  \sfcode 42 1000 % .
-  \sfcode 63 1000 % ?
-  \sfcode 33 1000 % !
-  \sfcode 58 1000 % :
-  \sfcode 59 1000 % ;
-  \sfcode 44 1000 % ,
-}
-
-%symbol markup -- \emph achieves decent spacing via italic corrections
-\newcommand{\isamath}[1]{\emph{$#1$}}
-\newcommand{\isatext}[1]{\emph{#1}}
-\DeclareRobustCommand{\isascriptstyle}{\def\isamath##1{##1}\def\isatext##1{\mbox{\isaspacing\isastylescript##1}}}
-\newcommand{\isactrlsub}[1]{\emph{\isascriptstyle${}\sb{#1}$}}
-\newcommand{\isactrlsup}[1]{\emph{\isascriptstyle${}\sp{#1}$}}
-\DeclareRobustCommand{\isactrlbsub}{\emph\bgroup\math{}\sb\bgroup\mbox\bgroup\isaspacing\isastylescript}
-\DeclareRobustCommand{\isactrlesub}{\egroup\egroup\endmath\egroup}
-\DeclareRobustCommand{\isactrlbsup}{\emph\bgroup\math{}\sp\bgroup\mbox\bgroup\isaspacing\isastylescript}
-\DeclareRobustCommand{\isactrlesup}{\egroup\egroup\endmath\egroup}
-\newcommand{\isactrlbold}[1]{{\bfseries\upshape\boldmath#1}}
-
-%blackboard-bold (requires font txmia from pxfonts)
-\DeclareSymbolFont{bbbfont}{U}{txmia}{m}{it}
-\SetSymbolFont{bbbfont}{bold}{U}{txmia}{bx}{it}
-\DeclareMathSymbol{\bbbA}{\mathord}{bbbfont}{129}
-\DeclareMathSymbol{\bbbB}{\mathord}{bbbfont}{130}
-\DeclareMathSymbol{\bbbC}{\mathord}{bbbfont}{131}
-\DeclareMathSymbol{\bbbD}{\mathord}{bbbfont}{132}
-\DeclareMathSymbol{\bbbE}{\mathord}{bbbfont}{133}
-\DeclareMathSymbol{\bbbF}{\mathord}{bbbfont}{134}
-\DeclareMathSymbol{\bbbG}{\mathord}{bbbfont}{135}
-\DeclareMathSymbol{\bbbH}{\mathord}{bbbfont}{136}
-\DeclareMathSymbol{\bbbI}{\mathord}{bbbfont}{137}
-\DeclareMathSymbol{\bbbJ}{\mathord}{bbbfont}{138}
-\DeclareMathSymbol{\bbbK}{\mathord}{bbbfont}{139}
-\DeclareMathSymbol{\bbbL}{\mathord}{bbbfont}{140}
-\DeclareMathSymbol{\bbbM}{\mathord}{bbbfont}{141}
-\DeclareMathSymbol{\bbbN}{\mathord}{bbbfont}{142}
-\DeclareMathSymbol{\bbbO}{\mathord}{bbbfont}{143}
-\DeclareMathSymbol{\bbbP}{\mathord}{bbbfont}{144}
-\DeclareMathSymbol{\bbbQ}{\mathord}{bbbfont}{145}
-\DeclareMathSymbol{\bbbR}{\mathord}{bbbfont}{146}
-\DeclareMathSymbol{\bbbS}{\mathord}{bbbfont}{147}
-\DeclareMathSymbol{\bbbT}{\mathord}{bbbfont}{148}
-\DeclareMathSymbol{\bbbU}{\mathord}{bbbfont}{149}
-\DeclareMathSymbol{\bbbV}{\mathord}{bbbfont}{150}
-\DeclareMathSymbol{\bbbW}{\mathord}{bbbfont}{151}
-\DeclareMathSymbol{\bbbX}{\mathord}{bbbfont}{152}
-\DeclareMathSymbol{\bbbY}{\mathord}{bbbfont}{153}
-\DeclareMathSymbol{\bbbZ}{\mathord}{bbbfont}{154}
-
-\newenvironment{isaantiq}{{\isacharat\isacharbraceleft}}{{\isacharbraceright}}
-
-\newdimen\isa@parindent\newdimen\isa@parskip
-
-\newenvironment{isabellebody}{%
-\isamarkuptrue\par%
-\isa@parindent\parindent\parindent0pt%
-\isa@parskip\parskip\parskip0pt%
-\isaspacing\isastyle}{\par}
-
-\newenvironment{isabellebodytt}{%
-\isamarkuptrue\par%
-\isa@parindent\parindent\parindent0pt%
-\isa@parskip\parskip\parskip0pt%
-\isaspacing\isastylett}{\par}
-
-\newenvironment{isabelle}
-{\begin{trivlist}\begin{isabellebody}\item\relax}
-{\end{isabellebody}\end{trivlist}}
-
-\newenvironment{isabellett}
-{\begin{trivlist}\begin{isabellebodytt}\item\relax}
-{\end{isabellebodytt}\end{trivlist}}
-
-\newcommand{\isa}[1]{\emph{\isaspacing\isastyleminor #1}}
-\newcommand{\isatt}[1]{\emph{\isaspacing\isastyleminortt #1}}
-
-\newcommand{\isaindent}[1]{\hphantom{#1}}
-\newcommand{\isanewline}{\mbox{}\par\mbox{}}
-\newcommand{\isasep}{}
-\newcommand{\isadigit}[1]{#1}
-
-\newcommand{\isachardefaults}{%
-\def\isacharbell{\isamath{\bigbox}}%requires stmaryrd
-\chardef\isacharbang=`\!%
-\chardef\isachardoublequote=`\"%
-\chardef\isachardoublequoteopen=`\"%
-\chardef\isachardoublequoteclose=`\"%
-\chardef\isacharhash=`\#%
-\chardef\isachardollar=`\$%
-\chardef\isacharpercent=`\%%
-\chardef\isacharampersand=`\&%
-\chardef\isacharprime=`\'%
-\chardef\isacharparenleft=`\(%
-\chardef\isacharparenright=`\)%
-\chardef\isacharasterisk=`\*%
-\chardef\isacharplus=`\+%
-\chardef\isacharcomma=`\,%
-\chardef\isacharminus=`\-%
-\chardef\isachardot=`\.%
-\chardef\isacharslash=`\/%
-\chardef\isacharcolon=`\:%
-\chardef\isacharsemicolon=`\;%
-\chardef\isacharless=`\<%
-\chardef\isacharequal=`\=%
-\chardef\isachargreater=`\>%
-\chardef\isacharquery=`\?%
-\chardef\isacharat=`\@%
-\chardef\isacharbrackleft=`\[%
-\chardef\isacharbackslash=`\\%
-\chardef\isacharbrackright=`\]%
-\chardef\isacharcircum=`\^%
-\chardef\isacharunderscore=`\_%
-\def\isacharunderscorekeyword{\_}%
-\chardef\isacharbackquote=`\`%
-\chardef\isacharbackquoteopen=`\`%
-\chardef\isacharbackquoteclose=`\`%
-\chardef\isacharbraceleft=`\{%
-\chardef\isacharbar=`\|%
-\chardef\isacharbraceright=`\}%
-\chardef\isachartilde=`\~%
-\def\isacharverbatimopen{\isacharbraceleft\isacharasterisk}%
-\def\isacharverbatimclose{\isacharasterisk\isacharbraceright}%
-\def\isacartoucheopen{\isatext{\guilsinglleft}}%
-\def\isacartoucheclose{\isatext{\guilsinglright}}%
-}
-
-
-% keyword and section markup
-
-\newcommand{\isakeyword}[1]
-{\emph{\normalfont\bfseries\def\isachardot{.}\def\isacharunderscore{\isacharunderscorekeyword}%
-\def\isacharbraceleft{\{}\def\isacharbraceright{\}}#1}}
-\newcommand{\isacommand}[1]{\isakeyword{#1}}
-
-\newcommand{\isakeywordcontrol}[1]
-{\emph{\normalfont\bfseries\itshape\def\isacharunderscore{\isacharunderscorekeyword}#1\,}}
-
-\newcommand{\isamarkupchapter}[1]{\chapter{#1}}
-\newcommand{\isamarkupsection}[1]{\section{#1}}
-\newcommand{\isamarkupsubsection}[1]{\subsection{#1}}
-\newcommand{\isamarkupsubsubsection}[1]{\subsubsection{#1}}
-\newcommand{\isamarkupparagraph}[1]{\paragraph{#1}}
-\newcommand{\isamarkupsubparagraph}[1]{\subparagraph{#1}}
-
-\newif\ifisamarkup
-\newcommand{\isabeginpar}{\par\ifisamarkup\relax\else\medskip\fi}
-\newcommand{\isaendpar}{\par\medskip}
-\newenvironment{isapar}{\parindent\isa@parindent\parskip\isa@parskip\isabeginpar}{\isaendpar}
-\newenvironment{isamarkuptext}{\par\isastyletext\begin{isapar}}{\end{isapar}}
-\newenvironment{isamarkuptxt}{\par\isastyletxt\begin{isapar}}{\end{isapar}}
-\newcommand{\isamarkupcmt}[1]{{\isastylecmt--- #1}}
-
-
-% index entries
-
-\newcommand{\isaindexdef}[1]{\textbf{#1}}
-\newcommand{\isaindexref}[1]{#1}
-
-
-% styles
-
-\def\isabellestyle#1{\csname isabellestyle#1\endcsname}
-
-\newcommand{\isabellestyledefault}{%
-\def\isastyle{\small\normalfont\ttfamily\slshape}%
-\def\isastylett{\small\normalfont\ttfamily}%
-\def\isastyleminor{\small\normalfont\ttfamily\slshape}%
-\def\isastyleminortt{\small\normalfont\ttfamily}%
-\def\isastylescript{\footnotesize\normalfont\ttfamily\slshape}%
-\isachardefaults%
-}
-\isabellestyledefault
-
-\newcommand{\isabellestylett}{%
-\def\isastyle{\small\normalfont\ttfamily}%
-\def\isastylett{\small\normalfont\ttfamily}%
-\def\isastyleminor{\small\normalfont\ttfamily}%
-\def\isastyleminortt{\small\normalfont\ttfamily}%
-\def\isastylescript{\footnotesize\normalfont\ttfamily}%
-\isachardefaults%
-}
-
-\newcommand{\isabellestyleit}{%
-\def\isastyle{\small\normalfont\itshape}%
-\def\isastylett{\small\normalfont\ttfamily}%
-\def\isastyleminor{\normalfont\itshape}%
-\def\isastyleminortt{\normalfont\ttfamily}%
-\def\isastylescript{\footnotesize\normalfont\itshape}%
-\isachardefaults%
-\def\isacharunderscorekeyword{\mbox{-}}%
-\def\isacharbang{\isamath{!}}%
-\def\isachardoublequote{}%
-\def\isachardoublequoteopen{}%
-\def\isachardoublequoteclose{}%
-\def\isacharhash{\isamath{\#}}%
-\def\isachardollar{\isamath{\$}}%
-\def\isacharpercent{\isamath{\%}}%
-\def\isacharampersand{\isamath{\&}}%
-\def\isacharprime{\isamath{\mskip2mu{'}\mskip-2mu}}%
-\def\isacharparenleft{\isamath{(}}%
-\def\isacharparenright{\isamath{)}}%
-\def\isacharasterisk{\isamath{*}}%
-\def\isacharplus{\isamath{+}}%
-\def\isacharcomma{\isamath{\mathord,}}%
-\def\isacharminus{\isamath{-}}%
-\def\isachardot{\isamath{\mathord.}}%
-\def\isacharslash{\isamath{/}}%
-\def\isacharcolon{\isamath{\mathord:}}%
-\def\isacharsemicolon{\isamath{\mathord;}}%
-\def\isacharless{\isamath{<}}%
-\def\isacharequal{\isamath{=}}%
-\def\isachargreater{\isamath{>}}%
-\def\isacharat{\isamath{@}}%
-\def\isacharbrackleft{\isamath{[}}%
-\def\isacharbackslash{\isamath{\backslash}}%
-\def\isacharbrackright{\isamath{]}}%
-\def\isacharunderscore{\mbox{-}}%
-\def\isacharbraceleft{\isamath{\{}}%
-\def\isacharbar{\isamath{\mid}}%
-\def\isacharbraceright{\isamath{\}}}%
-\def\isachartilde{\isamath{{}\sp{\sim}}}%
-\def\isacharbackquoteopen{\isatext{\guilsinglleft}}%
-\def\isacharbackquoteclose{\isatext{\guilsinglright}}%
-}
-
-\newcommand{\isabellestyleliteral}{%
-\isabellestyleit%
-\def\isacharunderscore{\_}%
-\def\isacharunderscorekeyword{\_}%
-\chardef\isacharbackquoteopen=`\`%
-\chardef\isacharbackquoteclose=`\`%
-}
-
-\newcommand{\isabellestyleliteralunderscore}{%
-\isabellestyleliteral%
-\def\isacharunderscore{\textunderscore}%
-\def\isacharunderscorekeyword{\textunderscore}%
-}
-
-\newcommand{\isabellestylesl}{%
-\isabellestyleit%
-\def\isastyle{\small\normalfont\slshape}%
-\def\isastylett{\small\normalfont\ttfamily}%
-\def\isastyleminor{\normalfont\slshape}%
-\def\isastyleminortt{\normalfont\ttfamily}%
-\def\isastylescript{\footnotesize\normalfont\slshape}%
-}
-
-
-% cancel text
-
-\usepackage[normalem]{ulem}
-\newcommand{\isamarkupcancel}[1]{\isa{\xout{#1}}}
-
-
-% tags
-
-\newcommand{\isafold}[1]{\emph{$\langle\mathord{\mathit{#1}}\rangle$}}
-
-\IfFileExists{isabelletags.sty}{\usepackage{isabelletags}}{}
diff --git a/thys/MDP-Algorithms/output/document/isabellesym.sty b/thys/MDP-Algorithms/output/document/isabellesym.sty
deleted file mode 100644
--- a/thys/MDP-Algorithms/output/document/isabellesym.sty
+++ /dev/null
@@ -1,496 +0,0 @@
-%%
-%% definitions of standard Isabelle symbols
-%%
-
-\newcommand{\isasymzero}{\isamath{\mathbf{0}}}  %requires amssymb
-\newcommand{\isasymone}{\isamath{\mathbf{1}}}  %requires amssymb
-\newcommand{\isasymtwo}{\isamath{\mathbf{2}}}  %requires amssymb
-\newcommand{\isasymthree}{\isamath{\mathbf{3}}}  %requires amssymb
-\newcommand{\isasymfour}{\isamath{\mathbf{4}}}  %requires amssymb
-\newcommand{\isasymfive}{\isamath{\mathbf{5}}}  %requires amssymb
-\newcommand{\isasymsix}{\isamath{\mathbf{6}}}  %requires amssymb
-\newcommand{\isasymseven}{\isamath{\mathbf{7}}}  %requires amssymb
-\newcommand{\isasymeight}{\isamath{\mathbf{8}}}  %requires amssymb
-\newcommand{\isasymnine}{\isamath{\mathbf{9}}}  %requires amssymb
-\newcommand{\isasymA}{\isamath{\mathcal{A}}}
-\newcommand{\isasymB}{\isamath{\mathcal{B}}}
-\newcommand{\isasymC}{\isamath{\mathcal{C}}}
-\newcommand{\isasymD}{\isamath{\mathcal{D}}}
-\newcommand{\isasymE}{\isamath{\mathcal{E}}}
-\newcommand{\isasymF}{\isamath{\mathcal{F}}}
-\newcommand{\isasymG}{\isamath{\mathcal{G}}}
-\newcommand{\isasymH}{\isamath{\mathcal{H}}}
-\newcommand{\isasymI}{\isamath{\mathcal{I}}}
-\newcommand{\isasymJ}{\isamath{\mathcal{J}}}
-\newcommand{\isasymK}{\isamath{\mathcal{K}}}
-\newcommand{\isasymL}{\isamath{\mathcal{L}}}
-\newcommand{\isasymM}{\isamath{\mathcal{M}}}
-\newcommand{\isasymN}{\isamath{\mathcal{N}}}
-\newcommand{\isasymO}{\isamath{\mathcal{O}}}
-\newcommand{\isasymP}{\isamath{\mathcal{P}}}
-\newcommand{\isasymQ}{\isamath{\mathcal{Q}}}
-\newcommand{\isasymR}{\isamath{\mathcal{R}}}
-\newcommand{\isasymS}{\isamath{\mathcal{S}}}
-\newcommand{\isasymT}{\isamath{\mathcal{T}}}
-\newcommand{\isasymU}{\isamath{\mathcal{U}}}
-\newcommand{\isasymV}{\isamath{\mathcal{V}}}
-\newcommand{\isasymW}{\isamath{\mathcal{W}}}
-\newcommand{\isasymX}{\isamath{\mathcal{X}}}
-\newcommand{\isasymY}{\isamath{\mathcal{Y}}}
-\newcommand{\isasymZ}{\isamath{\mathcal{Z}}}
-\newcommand{\isasyma}{\isamath{\mathrm{a}}}
-\newcommand{\isasymb}{\isamath{\mathrm{b}}}
-\newcommand{\isasymc}{\isamath{\mathrm{c}}}
-\newcommand{\isasymd}{\isamath{\mathrm{d}}}
-\newcommand{\isasyme}{\isamath{\mathrm{e}}}
-\newcommand{\isasymf}{\isamath{\mathrm{f}}}
-\newcommand{\isasymg}{\isamath{\mathrm{g}}}
-\newcommand{\isasymh}{\isamath{\mathrm{h}}}
-\newcommand{\isasymi}{\isamath{\mathrm{i}}}
-\newcommand{\isasymj}{\isamath{\mathrm{j}}}
-\newcommand{\isasymk}{\isamath{\mathrm{k}}}
-\newcommand{\isasyml}{\isamath{\mathrm{l}}}
-\newcommand{\isasymm}{\isamath{\mathrm{m}}}
-\newcommand{\isasymn}{\isamath{\mathrm{n}}}
-\newcommand{\isasymo}{\isamath{\mathrm{o}}}
-\newcommand{\isasymp}{\isamath{\mathrm{p}}}
-\newcommand{\isasymq}{\isamath{\mathrm{q}}}
-\newcommand{\isasymr}{\isamath{\mathrm{r}}}
-\newcommand{\isasyms}{\isamath{\mathrm{s}}}
-\newcommand{\isasymt}{\isamath{\mathrm{t}}}
-\newcommand{\isasymu}{\isamath{\mathrm{u}}}
-\newcommand{\isasymv}{\isamath{\mathrm{v}}}
-\newcommand{\isasymw}{\isamath{\mathrm{w}}}
-\newcommand{\isasymx}{\isamath{\mathrm{x}}}
-\newcommand{\isasymy}{\isamath{\mathrm{y}}}
-\newcommand{\isasymz}{\isamath{\mathrm{z}}}
-\newcommand{\isasymAA}{\isamath{\mathfrak{A}}}  %requires eufrak
-\newcommand{\isasymBB}{\isamath{\mathfrak{B}}}  %requires eufrak
-\newcommand{\isasymCC}{\isamath{\mathfrak{C}}}  %requires eufrak
-\newcommand{\isasymDD}{\isamath{\mathfrak{D}}}  %requires eufrak
-\newcommand{\isasymEE}{\isamath{\mathfrak{E}}}  %requires eufrak
-\newcommand{\isasymFF}{\isamath{\mathfrak{F}}}  %requires eufrak
-\newcommand{\isasymGG}{\isamath{\mathfrak{G}}}  %requires eufrak
-\newcommand{\isasymHH}{\isamath{\mathfrak{H}}}  %requires eufrak
-\newcommand{\isasymII}{\isamath{\mathfrak{I}}}  %requires eufrak
-\newcommand{\isasymJJ}{\isamath{\mathfrak{J}}}  %requires eufrak
-\newcommand{\isasymKK}{\isamath{\mathfrak{K}}}  %requires eufrak
-\newcommand{\isasymLL}{\isamath{\mathfrak{L}}}  %requires eufrak
-\newcommand{\isasymMM}{\isamath{\mathfrak{M}}}  %requires eufrak
-\newcommand{\isasymNN}{\isamath{\mathfrak{N}}}  %requires eufrak
-\newcommand{\isasymOO}{\isamath{\mathfrak{O}}}  %requires eufrak
-\newcommand{\isasymPP}{\isamath{\mathfrak{P}}}  %requires eufrak
-\newcommand{\isasymQQ}{\isamath{\mathfrak{Q}}}  %requires eufrak
-\newcommand{\isasymRR}{\isamath{\mathfrak{R}}}  %requires eufrak
-\newcommand{\isasymSS}{\isamath{\mathfrak{S}}}  %requires eufrak
-\newcommand{\isasymTT}{\isamath{\mathfrak{T}}}  %requires eufrak
-\newcommand{\isasymUU}{\isamath{\mathfrak{U}}}  %requires eufrak
-\newcommand{\isasymVV}{\isamath{\mathfrak{V}}}  %requires eufrak
-\newcommand{\isasymWW}{\isamath{\mathfrak{W}}}  %requires eufrak
-\newcommand{\isasymXX}{\isamath{\mathfrak{X}}}  %requires eufrak
-\newcommand{\isasymYY}{\isamath{\mathfrak{Y}}}  %requires eufrak
-\newcommand{\isasymZZ}{\isamath{\mathfrak{Z}}}  %requires eufrak
-\newcommand{\isasymaa}{\isamath{\mathfrak{a}}}  %requires eufrak
-\newcommand{\isasymbb}{\isamath{\mathfrak{b}}}  %requires eufrak
-\newcommand{\isasymcc}{\isamath{\mathfrak{c}}}  %requires eufrak
-\newcommand{\isasymdd}{\isamath{\mathfrak{d}}}  %requires eufrak
-\newcommand{\isasymee}{\isamath{\mathfrak{e}}}  %requires eufrak
-\newcommand{\isasymff}{\isamath{\mathfrak{f}}}  %requires eufrak
-\newcommand{\isasymgg}{\isamath{\mathfrak{g}}}  %requires eufrak
-\newcommand{\isasymhh}{\isamath{\mathfrak{h}}}  %requires eufrak
-\newcommand{\isasymii}{\isamath{\mathfrak{i}}}  %requires eufrak
-\newcommand{\isasymjj}{\isamath{\mathfrak{j}}}  %requires eufrak
-\newcommand{\isasymkk}{\isamath{\mathfrak{k}}}  %requires eufrak
-\newcommand{\isasymll}{\isamath{\mathfrak{l}}}  %requires eufrak
-\newcommand{\isasymmm}{\isamath{\mathfrak{m}}}  %requires eufrak
-\newcommand{\isasymnn}{\isamath{\mathfrak{n}}}  %requires eufrak
-\newcommand{\isasymoo}{\isamath{\mathfrak{o}}}  %requires eufrak
-\newcommand{\isasympp}{\isamath{\mathfrak{p}}}  %requires eufrak
-\newcommand{\isasymqq}{\isamath{\mathfrak{q}}}  %requires eufrak
-\newcommand{\isasymrr}{\isamath{\mathfrak{r}}}  %requires eufrak
-\newcommand{\isasymss}{\isamath{\mathfrak{s}}}  %requires eufrak
-\newcommand{\isasymtt}{\isamath{\mathfrak{t}}}  %requires eufrak
-\newcommand{\isasymuu}{\isamath{\mathfrak{u}}}  %requires eufrak
-\newcommand{\isasymvv}{\isamath{\mathfrak{v}}}  %requires eufrak
-\newcommand{\isasymww}{\isamath{\mathfrak{w}}}  %requires eufrak
-\newcommand{\isasymxx}{\isamath{\mathfrak{x}}}  %requires eufrak
-\newcommand{\isasymyy}{\isamath{\mathfrak{y}}}  %requires eufrak
-\newcommand{\isasymzz}{\isamath{\mathfrak{z}}}  %requires eufrak
-\newcommand{\isasymalpha}{\isamath{\alpha}}
-\newcommand{\isasymbeta}{\isamath{\beta}}
-\newcommand{\isasymgamma}{\isamath{\gamma}}
-\newcommand{\isasymdelta}{\isamath{\delta}}
-\newcommand{\isasymepsilon}{\isamath{\varepsilon}}
-\newcommand{\isasymzeta}{\isamath{\zeta}}
-\newcommand{\isasymeta}{\isamath{\eta}}
-\newcommand{\isasymtheta}{\isamath{\vartheta}}
-\newcommand{\isasymiota}{\isamath{\iota}}
-\newcommand{\isasymkappa}{\isamath{\kappa}}
-\newcommand{\isasymlambda}{\isamath{\lambda}}
-\newcommand{\isasymmu}{\isamath{\mu}}
-\newcommand{\isasymnu}{\isamath{\nu}}
-\newcommand{\isasymxi}{\isamath{\xi}}
-\newcommand{\isasympi}{\isamath{\pi}}
-\newcommand{\isasymrho}{\isamath{\varrho}}
-\newcommand{\isasymsigma}{\isamath{\sigma}}
-\newcommand{\isasymtau}{\isamath{\tau}}
-\newcommand{\isasymupsilon}{\isamath{\upsilon}}
-\newcommand{\isasymphi}{\isamath{\varphi}}
-\newcommand{\isasymchi}{\isamath{\chi}}
-\newcommand{\isasympsi}{\isamath{\psi}}
-\newcommand{\isasymomega}{\isamath{\omega}}
-\newcommand{\isasymGamma}{\isamath{\Gamma}}
-\newcommand{\isasymDelta}{\isamath{\Delta}}
-\newcommand{\isasymTheta}{\isamath{\Theta}}
-\newcommand{\isasymLambda}{\isamath{\Lambda}}
-\newcommand{\isasymXi}{\isamath{\Xi}}
-\newcommand{\isasymPi}{\isamath{\Pi}}
-\newcommand{\isasymSigma}{\isamath{\Sigma}}
-\newcommand{\isasymUpsilon}{\isamath{\Upsilon}}
-\newcommand{\isasymPhi}{\isamath{\Phi}}
-\newcommand{\isasymPsi}{\isamath{\Psi}}
-\newcommand{\isasymOmega}{\isamath{\Omega}}
-\newcommand{\isasymbbbA}{\isamath{\bbbA}}  %requires font txmia from txfonts
-\newcommand{\isasymbool}{\isamath{\bbbB}}  %requires font txmia from txfonts
-\newcommand{\isasymcomplex}{\isamath{\bbbC}}  %requires font txmia from txfonts
-\newcommand{\isasymbbbD}{\isamath{\bbbD}}  %requires font txmia from txfonts
-\newcommand{\isasymbbbE}{\isamath{\bbbE}}  %requires font txmia from txfonts
-\newcommand{\isasymbbbF}{\isamath{\bbbF}}  %requires font txmia from txfonts
-\newcommand{\isasymbbbG}{\isamath{\bbbG}}  %requires font txmia from txfonts
-\newcommand{\isasymbbbH}{\isamath{\bbbH}}  %requires font txmia from txfonts
-\newcommand{\isasymbbbI}{\isamath{\bbbI}}  %requires font txmia from txfonts
-\newcommand{\isasymbbbJ}{\isamath{\bbbJ}}  %requires font txmia from txfonts
-\newcommand{\isasymbbbK}{\isamath{\bbbK}}  %requires font txmia from txfonts
-\newcommand{\isasymbbbL}{\isamath{\bbbL}}  %requires font txmia from txfonts
-\newcommand{\isasymbbbM}{\isamath{\bbbM}}  %requires font txmia from txfonts
-\newcommand{\isasymnat}{\isamath{\bbbN}}  %requires font txmia from txfonts
-\newcommand{\isasymbbbO}{\isamath{\bbbO}}  %requires font txmia from txfonts
-\newcommand{\isasymbbbP}{\isamath{\bbbP}}  %requires font txmia from txfonts
-\newcommand{\isasymrat}{\isamath{\bbbQ}}  %requires font txmia from txfonts
-\newcommand{\isasymreal}{\isamath{\bbbR}}  %requires font txmia from txfonts
-\newcommand{\isasymbbbS}{\isamath{\bbbS}}  %requires font txmia from txfonts
-\newcommand{\isasymbbbT}{\isamath{\bbbT}}  %requires font txmia from txfonts
-\newcommand{\isasymbbbU}{\isamath{\bbbU}}  %requires font txmia from txfonts
-\newcommand{\isasymbbbV}{\isamath{\bbbV}}  %requires font txmia from txfonts
-\newcommand{\isasymbbbW}{\isamath{\bbbW}}  %requires font txmia from txfonts
-\newcommand{\isasymbbbX}{\isamath{\bbbX}}  %requires font txmia from txfonts
-\newcommand{\isasymbbbY}{\isamath{\bbbY}}  %requires font txmia from txfonts
-\newcommand{\isasymint}{\isamath{\bbbZ}}  %requires font txmia from txfonts
-\newcommand{\isasymleftarrow}{\isamath{\leftarrow}}
-\newcommand{\isasymrightarrow}{\isamath{\rightarrow}}
-\newcommand{\isasymlongleftarrow}{\isamath{\longleftarrow}}
-\newcommand{\isasymlongrightarrow}{\isamath{\longrightarrow}}
-\newcommand{\isasymlonglongleftarrow}{\isamath{\xleftarrow{\hphantom{AAA}}}}  %requires amsmath
-\newcommand{\isasymlonglongrightarrow}{\isamath{\xrightarrow{\hphantom{AAA}}}}  %requires amsmath
-\newcommand{\isasymlonglonglongleftarrow}{\isamath{\xleftarrow{\hphantom{AAAA}}}}  %requires amsmath
-\newcommand{\isasymlonglonglongrightarrow}{\isamath{\xrightarrow{\hphantom{AAAA}}}}  %requires amsmath
-\newcommand{\isasymLeftarrow}{\isamath{\Leftarrow}}
-\newcommand{\isasymRightarrow}{\isamath{\Rightarrow}}
-\newcommand{\isasymLongleftarrow}{\isamath{\Longleftarrow}}
-\newcommand{\isasymLongrightarrow}{\isamath{\Longrightarrow}}
-\newcommand{\isasymLleftarrow}{\isamath{\Lleftarrow}}  %requires amssymb
-\newcommand{\isasymRrightarrow}{\isamath{\Rrightarrow}}  %requires amssymb
-\newcommand{\isasymleftrightarrow}{\isamath{\leftrightarrow}}
-\newcommand{\isasymLeftrightarrow}{\isamath{\Leftrightarrow}}
-\newcommand{\isasymlongleftrightarrow}{\isamath{\longleftrightarrow}}
-\newcommand{\isasymLongleftrightarrow}{\isamath{\Longleftrightarrow}}
-\newcommand{\isasymmapsto}{\isamath{\mapsto}}
-\newcommand{\isasymlongmapsto}{\isamath{\longmapsto}}
-\newcommand{\isasymmidarrow}{\isamath{\relbar}}
-\newcommand{\isasymMidarrow}{\isamath{\Relbar}}
-\newcommand{\isasymhookleftarrow}{\isamath{\hookleftarrow}}
-\newcommand{\isasymhookrightarrow}{\isamath{\hookrightarrow}}
-\newcommand{\isasymleftharpoondown}{\isamath{\leftharpoondown}}
-\newcommand{\isasymrightharpoondown}{\isamath{\rightharpoondown}}
-\newcommand{\isasymleftharpoonup}{\isamath{\leftharpoonup}}
-\newcommand{\isasymrightharpoonup}{\isamath{\rightharpoonup}}
-\newcommand{\isasymrightleftharpoons}{\isamath{\rightleftharpoons}}
-\newcommand{\isasymleadsto}{\isamath{\leadsto}}  %requires amssymb
-\newcommand{\isasymdownharpoonleft}{\isamath{\downharpoonleft}}  %requires amssymb
-\newcommand{\isasymdownharpoonright}{\isamath{\downharpoonright}}  %requires amssymb
-\newcommand{\isasymupharpoonleft}{\isamath{\upharpoonleft}}  %requires amssymb
-\newcommand{\isasymupharpoonright}{\isamath{\upharpoonright}}  %requires amssymb
-\newcommand{\isasymrestriction}{\isamath{\restriction}}  %requires amssymb
-\newcommand{\isasymColon}{\isamath{\mathrel{::}}}
-\newcommand{\isasymup}{\isamath{\uparrow}}
-\newcommand{\isasymUp}{\isamath{\Uparrow}}
-\newcommand{\isasymdown}{\isamath{\downarrow}}
-\newcommand{\isasymDown}{\isamath{\Downarrow}}
-\newcommand{\isasymupdown}{\isamath{\updownarrow}}
-\newcommand{\isasymUpdown}{\isamath{\Updownarrow}}
-\newcommand{\isasymlangle}{\isamath{\langle}}
-\newcommand{\isasymrangle}{\isamath{\rangle}}
-\newcommand{\isasymllangle}{\isamath{\langle\mskip-5mu\langle}}
-\newcommand{\isasymrrangle}{\isamath{\rangle\mskip-5mu\rangle}}
-\newcommand{\isasymlceil}{\isamath{\lceil}}
-\newcommand{\isasymrceil}{\isamath{\rceil}}
-\newcommand{\isasymlfloor}{\isamath{\lfloor}}
-\newcommand{\isasymrfloor}{\isamath{\rfloor}}
-\newcommand{\isasymlparr}{\isamath{\mathopen{(\mkern-3.3mu\mid}}}
-\newcommand{\isasymrparr}{\isamath{\mathclose{\mid\mkern-3.3mu)}}}
-\newcommand{\isasymlbrakk}{\isamath{\mathopen{\lbrack\mkern-3mu\lbrack}}}
-\newcommand{\isasymrbrakk}{\isamath{\mathclose{\rbrack\mkern-3mu\rbrack}}}
-\newcommand{\isasymlbrace}{\isamath{\mathopen{\lbrace\mkern-4.3mu\mid}}}
-\newcommand{\isasymrbrace}{\isamath{\mathclose{\mid\mkern-4.3mu\rbrace}}}
-\newcommand{\isasymlblot}{\isamath{{\langle}\mkern -3.5mu{|}}}
-\newcommand{\isasymrblot}{\isamath{{|}\mkern -3.5mu{\rangle}}}
-\newcommand{\isasymguillemotleft}{\isatext{\guillemotleft}}
-\newcommand{\isasymguillemotright}{\isatext{\guillemotright}}
-\newcommand{\isasymbottom}{\isamath{\bot}}
-\newcommand{\isasymtop}{\isamath{\top}}
-\newcommand{\isasymand}{\isamath{\wedge}}
-\newcommand{\isasymAnd}{\isamath{\bigwedge}}
-\newcommand{\isasymor}{\isamath{\vee}}
-\newcommand{\isasymOr}{\isamath{\bigvee}}
-\newcommand{\isasymforall}{\isamath{\forall\,}}
-\newcommand{\isasymexists}{\isamath{\exists\,}}
-\newcommand{\isasymnot}{\isamath{\neg}}
-\newcommand{\isasymnexists}{\isamath{\nexists\,}}  %requires amssymb
-\newcommand{\isasymcircle}{\isamath{\ocircle}}  %requires wasysym
-\newcommand{\isasymbox}{\isamath{\Box}}  %requires amssymb
-\newcommand{\isasymdiamond}{\isamath{\Diamond}}  %requires amssymb
-\newcommand{\isasymdiamondop}{\isamath{\diamond}}
-\newcommand{\isasymsurd}{\isamath{\surd}}
-\newcommand{\isasymturnstile}{\isamath{\vdash}}
-\newcommand{\isasymTurnstile}{\isamath{\models}}
-\newcommand{\isasymtturnstile}{\isamath{\vdash\!\!\!\vdash}}
-\newcommand{\isasymTTurnstile}{\isamath{\mid\!\models}}
-\newcommand{\isasymstileturn}{\isamath{\dashv}}
-\newcommand{\isasymle}{\isamath{\le}}
-\newcommand{\isasymge}{\isamath{\ge}}
-\newcommand{\isasymlless}{\isamath{\ll}}
-\newcommand{\isasymggreater}{\isamath{\gg}}
-\newcommand{\isasymlesssim}{\isamath{\lesssim}}  %requires amssymb
-\newcommand{\isasymgreatersim}{\isamath{\gtrsim}}  %requires amssymb
-\newcommand{\isasymlessapprox}{\isamath{\lessapprox}}  %requires amssymb
-\newcommand{\isasymgreaterapprox}{\isamath{\gtrapprox}}  %requires amssymb
-\newcommand{\isasymin}{\isamath{\in}}
-\newcommand{\isasymnotin}{\isamath{\notin}}
-\newcommand{\isasymsubset}{\isamath{\subset}}
-\newcommand{\isasymsupset}{\isamath{\supset}}
-\newcommand{\isasymsubseteq}{\isamath{\subseteq}}
-\newcommand{\isasymsupseteq}{\isamath{\supseteq}}
-\newcommand{\isasymsqsubset}{\isamath{\sqsubset}}  %requires amssymb
-\newcommand{\isasymsqsupset}{\isamath{\sqsupset}}  %requires amssymb
-\newcommand{\isasymsqsubseteq}{\isamath{\sqsubseteq}}
-\newcommand{\isasymsqsupseteq}{\isamath{\sqsupseteq}}
-\newcommand{\isasyminter}{\isamath{\cap}}
-\newcommand{\isasymInter}{\isamath{\bigcap\,}}
-\newcommand{\isasymunion}{\isamath{\cup}}
-\newcommand{\isasymUnion}{\isamath{\bigcup\,}}
-\newcommand{\isasymsqunion}{\isamath{\sqcup}}
-\newcommand{\isasymSqunion}{\isamath{\bigsqcup\,}}
-\newcommand{\isasymsqinter}{\isamath{\sqcap}}
-\newcommand{\isasymSqinter}{\isamath{\bigsqcap\,}}  %requires stmaryrd
-\newcommand{\isasymsetminus}{\isamath{\setminus}}
-\newcommand{\isasympropto}{\isamath{\propto}}
-\newcommand{\isasymuplus}{\isamath{\uplus}}
-\newcommand{\isasymUplus}{\isamath{\biguplus\,}}
-\newcommand{\isasymnoteq}{\isamath{\not=}}
-\newcommand{\isasymsim}{\isamath{\sim}}
-\newcommand{\isasymdoteq}{\isamath{\doteq}}
-\newcommand{\isasymsimeq}{\isamath{\simeq}}
-\newcommand{\isasymapprox}{\isamath{\approx}}
-\newcommand{\isasymasymp}{\isamath{\asymp}}
-\newcommand{\isasymcong}{\isamath{\cong}}
-\newcommand{\isasymsmile}{\isamath{\smile}}
-\newcommand{\isasymequiv}{\isamath{\equiv}}
-\newcommand{\isasymfrown}{\isamath{\frown}}
-\newcommand{\isasymJoin}{\isamath{\Join}}  %requires amssymb
-\newcommand{\isasymbowtie}{\isamath{\bowtie}}
-\newcommand{\isasymprec}{\isamath{\prec}}
-\newcommand{\isasymsucc}{\isamath{\succ}}
-\newcommand{\isasympreceq}{\isamath{\preceq}}
-\newcommand{\isasymsucceq}{\isamath{\succeq}}
-\newcommand{\isasymparallel}{\isamath{\parallel}}
-\newcommand{\isasymParallel}{\isamath{\bigparallel}}  %requires stmaryrd
-\newcommand{\isasyminterleace}{\isamath{\interleave}}  %requires stmaryrd
-\newcommand{\isasymsslash}{\isamath{\sslash}}  %requires stmaryrd
-\newcommand{\isasymbar}{\isamath{\mid}}
-\newcommand{\isasymbbar}{\isamath{[\mskip-1.5mu]}}
-\newcommand{\isasymplusminus}{\isamath{\pm}}
-\newcommand{\isasymminusplus}{\isamath{\mp}}
-\newcommand{\isasymtimes}{\isamath{\times}}
-\newcommand{\isasymdiv}{\isamath{\div}}
-\newcommand{\isasymcdot}{\isamath{\cdot}}
-\newcommand{\isasymsqdot}{\isamath{\sbox\z@{$\centerdot$}\ht\z@=.33333\ht\z@\vcenter{\box\z@}}}  %requires amssymb
-\newcommand{\isasymstar}{\isamath{\star}}
-\newcommand{\isasymbullet}{\boldmath\isamath{\mathchoice{\displaystyle{\cdot}}{\textstyle{\cdot}}{\scriptstyle{\bullet}}{\scriptscriptstyle{\bullet}}}}
-\newcommand{\isasymcirc}{\isamath{\circ}}
-\newcommand{\isasymdagger}{\isamath{\dagger}}
-\newcommand{\isasymddagger}{\isamath{\ddagger}}
-\newcommand{\isasymlhd}{\isamath{\lhd}}  %requires amssymb
-\newcommand{\isasymrhd}{\isamath{\rhd}}  %requires amssymb
-\newcommand{\isasymunlhd}{\isamath{\unlhd}}  %requires amssymb
-\newcommand{\isasymunrhd}{\isamath{\unrhd}}  %requires amssymb
-\newcommand{\isasymtriangleleft}{\isamath{\triangleleft}}
-\newcommand{\isasymtriangleright}{\isamath{\triangleright}}
-\newcommand{\isasymtriangle}{\isamath{\triangle}}
-\newcommand{\isasymtriangleq}{\isamath{\triangleq}}  %requires amssymb
-\newcommand{\isasymoplus}{\isamath{\oplus}}
-\newcommand{\isasymOplus}{\isamath{\bigoplus\,}}
-\newcommand{\isasymotimes}{\isamath{\otimes}}
-\newcommand{\isasymOtimes}{\isamath{\bigotimes\,}}
-\newcommand{\isasymodot}{\isamath{\odot}}
-\newcommand{\isasymOdot}{\isamath{\bigodot\,}}
-\newcommand{\isasymominus}{\isamath{\ominus}}
-\newcommand{\isasymoslash}{\isamath{\oslash}}
-\newcommand{\isasymdots}{\isamath{\dots}}
-\newcommand{\isasymcdots}{\isamath{\cdots}}
-\newcommand{\isasymSum}{\isamath{\sum\,}}
-\newcommand{\isasymProd}{\isamath{\prod\,}}
-\newcommand{\isasymCoprod}{\isamath{\coprod\,}}
-\newcommand{\isasyminfinity}{\isamath{\infty}}
-\newcommand{\isasymintegral}{\isamath{\int\,}}
-\newcommand{\isasymointegral}{\isamath{\oint\,}}
-\newcommand{\isasymclubsuit}{\isamath{\clubsuit}}
-\newcommand{\isasymdiamondsuit}{\isamath{\diamondsuit}}
-\newcommand{\isasymheartsuit}{\isamath{\heartsuit}}
-\newcommand{\isasymspadesuit}{\isamath{\spadesuit}}
-\newcommand{\isasymaleph}{\isamath{\aleph}}
-\newcommand{\isasymemptyset}{\isamath{\emptyset}}
-\newcommand{\isasymnabla}{\isamath{\nabla}}
-\newcommand{\isasympartial}{\isamath{\partial}}
-\newcommand{\isasymRe}{\isamath{\Re}}
-\newcommand{\isasymIm}{\isamath{\Im}}
-\newcommand{\isasymflat}{\isamath{\flat}}
-\newcommand{\isasymnatural}{\isamath{\natural}}
-\newcommand{\isasymsharp}{\isamath{\sharp}}
-\newcommand{\isasymangle}{\isamath{\angle}}
-\newcommand{\isasymcopyright}{\isatext{\normalfont\rmfamily\copyright}}
-\newcommand{\isasymregistered}{\isatext{\normalfont\rmfamily\textregistered}}
-\newcommand{\isasyminverse}{\isamath{{}^{-1}}}
-\newcommand{\isasymonequarter}{\isatext{\normalfont\rmfamily\textonequarter}}  %requires textcomp
-\newcommand{\isasymonehalf}{\isatext{\normalfont\rmfamily\textonehalf}}  %requires textcomp
-\newcommand{\isasymthreequarters}{\isatext{\normalfont\rmfamily\textthreequarters}}  %requires textcomp
-\newcommand{\isasymordfeminine}{\isatext{\normalfont\rmfamily\textordfeminine}}
-\newcommand{\isasymordmasculine}{\isatext{\normalfont\rmfamily\textordmasculine}}
-\newcommand{\isasymsection}{\isatext{\normalfont\rmfamily\S}}
-\newcommand{\isasymparagraph}{\isatext{\normalfont\rmfamily\P}}
-\newcommand{\isasymexclamdown}{\isatext{\normalfont\rmfamily\textexclamdown}}
-\newcommand{\isasymquestiondown}{\isatext{\normalfont\rmfamily\textquestiondown}}
-\newcommand{\isasymeuro}{\isatext{\euro}}  %requires eurosym
-\newcommand{\isasympounds}{\isamath{\pounds}}
-\newcommand{\isasymyen}{\isatext{\yen}}  %requires amssymb
-\newcommand{\isasymcent}{\isatext{\textcent}}  %requires textcomp
-\newcommand{\isasymcurrency}{\isatext{\textcurrency}} %requires textcomp
-\newcommand{\isasymdegree}{\isatext{\normalfont\rmfamily\textdegree}}  %requires textcomp
-\newcommand{\isasymhyphen}{\isatext{\normalfont\rmfamily-}}
-\newcommand{\isasymamalg}{\isamath{\amalg}}
-\newcommand{\isasymmho}{\isamath{\mho}}  %requires amssymb
-\newcommand{\isasymlozenge}{\isamath{\lozenge}}  %requires amssymb
-\newcommand{\isasymwp}{\isamath{\wp}}
-\newcommand{\isasymwrong}{\isamath{\wr}}
-\newcommand{\isasymacute}{\isatext{\'\relax}}
-\newcommand{\isasymindex}{\isatext{\i}}
-\newcommand{\isasymdieresis}{\isatext{\"\relax}}
-\newcommand{\isasymcedilla}{\isatext{\c\relax}}
-\newcommand{\isasymhungarumlaut}{\isatext{\H\relax}}
-\newcommand{\isasymsome}{\isamath{\epsilon\,}}
-\newcommand{\isasymbind}{\isamath{\mathbin{>\!\!\!>\mkern-6.7mu=}}}
-\newcommand{\isasymthen}{\isamath{\mathbin{>\!\!\!>}}}
-
-%Z notation
-\newcommand{\isaZhbar}[1]{\rlap{\raise.0001ex\hbox{\isamath{-}}}#1}
-\newcommand{\isaZpvbar}[1]{\ooalign{\hfil\isamath{\mapstochar\mkern 5mu}\hfil\cr#1}}
-\newcommand{\isaZfvbar}[1]{\ooalign{\hfil\isamath{\mapstochar\mkern 3mu\mapstochar\mkern 5mu}\hfil\cr#1}}
-\newcommand{\isaZdarrow}[3]{\ooalign{\isamath{#1}\hfil\cr\isamath{\mkern#3mu\isamath{#2}}}}
-\newcommand{\isasymZcomp}{\isamath{\fatsemi}}  %requires stmaryrd
-\newcommand{\isasymZinj}{\isamath{\rightarrowtail}}  %requires amssymb
-\newcommand{\isasymZpinj}{\isaZpvbar{\isamath{\rightarrowtail}}}  %requires amssymb
-\newcommand{\isasymZfinj}{\isaZfvbar{\isasymZinj}}  %requires amssymb
-\newcommand{\isasymZsurj}{\isaZdarrow{\rightarrow}{\rightarrow}{4}}  %requires amssymb
-\newcommand{\isasymZpsurj}{\isaZpvbar{\isasymZsurj}}  %requires amssymb
-\newcommand{\isasymZbij}{\isaZdarrow{\rightarrowtail}{\rightarrow}{5}}  %requires amssymb
-\newcommand{\isasymZpfun}{\isaZpvbar{\isamath{\rightarrow}}}
-\newcommand{\isasymZffun}{\isaZfvbar{\isamath{\rightarrow}}}
-\newcommand{\isasymZdres}{\isamath{\lhd}}  %requires amssymb
-\newcommand{\isasymZndres}{\isaZhbar{\isamath{\lhd}}}  %requires amssymb
-\newcommand{\isasymZrres}{\isamath{\rhd}}  %requires amssymb
-\newcommand{\isasymZnrres}{\isaZhbar{\isamath{\rhd}}}  %requires amssymb
-\newcommand{\isasymZspot}{\isamath{\bullet}}
-\newcommand{\isasymZproject}{\isamath{\upharpoonright}}  %requires amssymb
-\newcommand{\isasymZsemi}{\isatext{\raise 0.66ex\hbox{\oalign{\hfil\isamath{\scriptscriptstyle\mathrm{o}}\hfil\cr\hfil\isamath{\scriptscriptstyle\mathrm{9}}\hfil}}}}
-\newcommand{\isasymZtypecolon}{\isatext{\raise 0.6ex\hbox{\oalign{\hfil\isamath{\scriptscriptstyle\mathrm{o}}\hfil\cr\hfil\isamath{\scriptscriptstyle\mathrm{o}}\hfil}}}}
-\newcommand{\isasymZhide}{\isamath{\backslash}}
-\newcommand{\isasymZcat}{\isatext{\raise 0.8ex\hbox{\isamath{\mathchar\frown}}}}
-\newcommand{\isasymZinbag}{\isatext{\ooalign{\isamath{\sqsubset\mkern-1mu}\cr\isamath{-\mkern-1mu}\cr}}}
-
-\newcommand{\isasymhole}{\isatext{\normalfont\rmfamily\wasylozenge}}  %requires wasysym
-\newcommand{\isasymnewline}{\isatext{\fbox{$\hookleftarrow$}}}
-\newcommand{\isasymcomment}{\isatext{\isastylecmt---}}
-\newcommand{\isasymproof}{\isamath{\,\langle\mathit{proof}\rangle}}
-\newcommand{\isasymopen}{\isatext{\guilsinglleft}}
-\newcommand{\isasymclose}{\isatext{\guilsinglright}}
-\newcommand{\isasymcheckmark}{\isatext{\ding{51}}}  %requires pifont
-\newcommand{\isasymcrossmark}{\isatext{\ding{55}}}  %requires pifont
-\newcommand{\isactrlmarker}{\isatext{\ding{48}}}  %requires pifont
-\newcommand{\isactrltry}{\isakeywordcontrol{try}}
-\newcommand{\isactrlcan}{\isakeywordcontrol{can}}
-\newcommand{\isactrlassert}{\isakeywordcontrol{assert}}
-\newcommand{\isactrlcancel}{\isakeywordcontrol{cancel}}
-\newcommand{\isactrlbinding}{\isakeywordcontrol{binding}}
-\newcommand{\isactrlclass}{\isakeywordcontrol{class}}
-\newcommand{\isactrlclassUNDERSCOREsyntax}{\isakeywordcontrol{class{\isacharunderscore}syntax}}
-\newcommand{\isactrlcommandUNDERSCOREkeyword}{\isakeywordcontrol{command{\isacharunderscore}keyword}}
-\newcommand{\isactrlconst}{\isakeywordcontrol{const}}
-\newcommand{\isactrlconstUNDERSCOREabbrev}{\isakeywordcontrol{const{\isacharunderscore}abbrev}}
-\newcommand{\isactrlconstUNDERSCOREname}{\isakeywordcontrol{const{\isacharunderscore}name}}
-\newcommand{\isactrlconstUNDERSCOREsyntax}{\isakeywordcontrol{const{\isacharunderscore}syntax}}
-\newcommand{\isactrlcontext}{\isakeywordcontrol{context}}
-\newcommand{\isactrlcprop}{\isakeywordcontrol{cprop}}
-\newcommand{\isactrlcterm}{\isakeywordcontrol{cterm}}
-\newcommand{\isactrlctyp}{\isakeywordcontrol{ctyp}}
-\newcommand{\isactrldir}{\isakeywordcontrol{dir}}
-\newcommand{\isactrlfile}{\isakeywordcontrol{file}}
-\newcommand{\isactrlhere}{\isakeywordcontrol{here}}
-\newcommand{\isactrlinstantiate}{\isakeywordcontrol{instantiate}}
-\newcommand{\isactrlkeyword}{\isakeywordcontrol{keyword}}
-\newcommand{\isactrllatex}{\isakeywordcontrol{latex}}
-\newcommand{\isactrllocale}{\isakeywordcontrol{locale}}
-\newcommand{\isactrlmakeUNDERSCOREjudgment}{\isakeywordcontrol{make{\isacharunderscore}judgment}}
-\newcommand{\isactrldestUNDERSCOREjudgment}{\isakeywordcontrol{dest{\isacharunderscore}judgment}}
-\newcommand{\isactrlmakeUNDERSCOREstring}{\isakeywordcontrol{make{\isacharunderscore}string}}
-\newcommand{\isactrlmasterUNDERSCOREdir}{\isakeywordcontrol{master{\isacharunderscore}dir}}
-\newcommand{\isactrlmethod}{\isakeywordcontrol{method}}
-\newcommand{\isactrlnamedUNDERSCOREtheorems}{\isakeywordcontrol{named{\isacharunderscore}theorems}}
-\newcommand{\isactrlnonterminal}{\isakeywordcontrol{nonterminal}}
-\newcommand{\isactrloracleUNDERSCOREname}{\isakeywordcontrol{oracle{\isacharunderscore}name}}
-\newcommand{\isactrlpath}{\isakeywordcontrol{path}}
-\newcommand{\isactrlpathUNDERSCOREbinding}{\isakeywordcontrol{path{\isacharunderscore}binding}}
-\newcommand{\isactrlplugin}{\isakeywordcontrol{plugin}}
-\newcommand{\isactrlprint}{\isakeywordcontrol{print}}
-\newcommand{\isactrlprop}{\isakeywordcontrol{prop}}
-\newcommand{\isactrlscala}{\isakeywordcontrol{scala}}
-\newcommand{\isactrlscalaUNDERSCOREfunction}{\isakeywordcontrol{scala{\isacharunderscore}function}}
-\newcommand{\isactrlscalaUNDERSCOREmethod}{\isakeywordcontrol{scala{\isacharunderscore}method}}
-\newcommand{\isactrlscalaUNDERSCOREobject}{\isakeywordcontrol{scala{\isacharunderscore}object}}
-\newcommand{\isactrlscalaUNDERSCOREtype}{\isakeywordcontrol{scala{\isacharunderscore}type}}
-\newcommand{\isactrlsimproc}{\isakeywordcontrol{simproc}}
-\newcommand{\isactrlsort}{\isakeywordcontrol{sort}}
-\newcommand{\isactrlsyntaxUNDERSCOREconst}{\isakeywordcontrol{syntax{\isacharunderscore}const}}
-\newcommand{\isactrlsystemUNDERSCOREoption}{\isakeywordcontrol{system{\isacharunderscore}option}}
-\newcommand{\isactrlterm}{\isakeywordcontrol{term}}
-\newcommand{\isactrltheory}{\isakeywordcontrol{theory}}
-\newcommand{\isactrltheoryUNDERSCOREcontext}{\isakeywordcontrol{theory{\isacharunderscore}context}}
-\newcommand{\isactrltyp}{\isakeywordcontrol{typ}}
-\newcommand{\isactrltypeUNDERSCOREabbrev}{\isakeywordcontrol{type{\isacharunderscore}abbrev}}
-\newcommand{\isactrltypeUNDERSCOREname}{\isakeywordcontrol{type{\isacharunderscore}name}}
-\newcommand{\isactrltypeUNDERSCOREsyntax}{\isakeywordcontrol{type{\isacharunderscore}syntax}}
-\newcommand{\isactrlundefined}{\isakeywordcontrol{undefined}}
-\newcommand{\isactrltvar}{\isakeywordcontrol{tvar}}
-\newcommand{\isactrlvar}{\isakeywordcontrol{var}}
-\newcommand{\isactrlConst}{\isakeywordcontrol{Const}}
-\newcommand{\isactrlConstUNDERSCORE}{\isakeywordcontrol{Const{\isacharunderscore}}}
-\newcommand{\isactrlConstUNDERSCOREfn}{\isakeywordcontrol{Const{\isacharunderscore}fn}}
-\newcommand{\isactrlType}{\isakeywordcontrol{Type}}
-\newcommand{\isactrlTypeUNDERSCOREfn}{\isakeywordcontrol{Type{\isacharunderscore}fn}}
-
-\newcommand{\isactrlcode}{\isakeywordcontrol{code}}
-\newcommand{\isactrlcomputation}{\isakeywordcontrol{computation}}
-\newcommand{\isactrlcomputationUNDERSCOREconv}{\isakeywordcontrol{computation{\isacharunderscore}conv}}
-\newcommand{\isactrlcomputationUNDERSCOREcheck}{\isakeywordcontrol{computation{\isacharunderscore}check}}
-\newcommand{\isactrlifUNDERSCORElinux}{\isakeywordcontrol{if{\isacharunderscore}linux}}
-\newcommand{\isactrlifUNDERSCOREmacos}{\isakeywordcontrol{if{\isacharunderscore}macos}}
-\newcommand{\isactrlifUNDERSCOREwindows}{\isakeywordcontrol{if{\isacharunderscore}windows}}
-\newcommand{\isactrlifUNDERSCOREunix}{\isakeywordcontrol{if{\isacharunderscore}unix}}
diff --git a/thys/MDP-Algorithms/output/document/isabelletags.sty b/thys/MDP-Algorithms/output/document/isabelletags.sty
deleted file mode 100644
--- a/thys/MDP-Algorithms/output/document/isabelletags.sty
+++ /dev/null
@@ -1,20 +0,0 @@
-%plain TeX version of comment package -- much faster!
-\let\isafmtname\fmtname\def\fmtname{plain}
-\usepackage{comment}
-\let\fmtname\isafmtname
-
-\newcommand{\isakeeptag}[1]%
-{\includecomment{isadelim#1}\includecomment{isatag#1}\csarg\def{isafold#1}{}}
-\newcommand{\isadroptag}[1]%
-{\excludecomment{isadelim#1}\excludecomment{isatag#1}\csarg\def{isafold#1}{}}
-\newcommand{\isafoldtag}[1]%
-{\includecomment{isadelim#1}\excludecomment{isatag#1}\csarg\def{isafold#1}{\isafold{#1}}}
-
-\isakeeptag{ML}
-\isakeeptag{document}
-\isakeeptag{important}
-\isadroptag{invisible}
-\isakeeptag{proof}
-\isakeeptag{theory}
-\isakeeptag{unimportant}
-\isakeeptag{visible}
diff --git a/thys/MDP-Algorithms/output/document/pdfsetup.sty b/thys/MDP-Algorithms/output/document/pdfsetup.sty
deleted file mode 100644
--- a/thys/MDP-Algorithms/output/document/pdfsetup.sty
+++ /dev/null
@@ -1,7 +0,0 @@
-%%
-%% default hyperref setup (both for pdf and dvi output)
-%%
-
-\usepackage{color}
-\definecolor{linkcolor}{rgb}{0,0,0.5}
-\usepackage[colorlinks=true,linkcolor=linkcolor,citecolor=linkcolor,filecolor=linkcolor,urlcolor=linkcolor,pdfpagelabels]{hyperref}
diff --git a/thys/MDP-Algorithms/output/document/railsetup.sty b/thys/MDP-Algorithms/output/document/railsetup.sty
deleted file mode 100644
--- a/thys/MDP-Algorithms/output/document/railsetup.sty
+++ /dev/null
@@ -1,1202 +0,0 @@
-% rail.sty - style file to support railroad diagrams
-%
-% 09-Jul-90 L. Rooijakkers
-% 08-Oct-90 L. Rooijakkers	fixed centering bug when \rail@tmpc<0.
-% 07-Feb-91 L. Rooijakkers	added \railoptions command, indexing
-% 08-Feb-91 L. Rooijakkers	minor fixes
-% 28-Jun-94 K. Barthelmann	turned into LaTeX2e package
-% 08-Dec-96 K. Barthelmann	replaced \@writefile
-% 13-Dec-96 K. Barthelmann	cleanup
-% 22-Feb-98 K. Barthelmann	fixed catcodes of special characters
-% 18-Apr-98 K. Barthelmann	fixed \par handling
-% 19-May-98 J. Olsson		Added new macros to support arrow heads.
-% 26-Jul-98 K. Barthelmann	changed \par to output newlines
-% 02-May-11 M. Wenzel           default setup for Isabelle
-%
-% This style file needs to be used in conjunction with the 'rail'
-% program. Running LaTeX as 'latex file' produces file.rai, which should be
-% processed by Rail with 'rail file'. This produces file.rao, which will
-% be picked up by LaTeX on the next 'latex file' run.
-%
-% LaTeX will warn if there is no file.rao or it's out of date.
-%
-% The macros in this file thus consist of two parts: those that read and
-% write the .rai and .rao files, and those that do the actual formatting
-% of the railroad diagrams.
-
-\NeedsTeXFormat{LaTeX2e}
-\ProvidesPackage{rail}[1998/05/19]
-
-% railroad diagram formatting parameters (user level)
-% all of these are copied into their internal versions by \railinit
-%
-% \railunit : \unitlength within railroad diagrams
-% \railextra : extra length at outside of diagram
-% \railboxheight : height of ovals and frames
-% \railboxskip : vertical space between lines
-% \railboxleft : space to the left of a box
-% \railboxright : space to the right of a box
-% \railovalspace : extra space around contents of oval
-% \railframespace : extra space around contents of frame
-% \railtextleft : space to the left of text
-% \railtextright : space to the right of text
-% \railtextup : space to lift text up
-% \railjoinsize : circle size of join/split arcs
-% \railjoinadjust : space to adjust join
-%
-% \railnamesep : separator between name and rule body
-
-\newlength\railunit
-\newlength\railextra
-\newlength\railboxheight
-\newlength\railboxskip
-\newlength\railboxleft
-\newlength\railboxright
-\newlength\railovalspace
-\newlength\railframespace
-\newlength\railtextleft
-\newlength\railtextright
-\newlength\railtextup
-\newlength\railjoinsize
-\newlength\railjoinadjust
-\newlength\railnamesep
-
-% initialize the parameters
-
-\setlength\railunit{1sp}
-\setlength\railextra{4ex}
-\setlength\railboxleft{1ex}
-\setlength\railboxright{1ex}
-\setlength\railovalspace{2ex}
-\setlength\railframespace{2ex}
-\setlength\railtextleft{1ex}
-\setlength\railtextright{1ex}
-\setlength\railjoinadjust{0pt}
-\setlength\railnamesep{1ex}
-
-\DeclareOption{10pt}{
-  \setlength\railboxheight{16pt}
-  \setlength\railboxskip{24pt}
-  \setlength\railtextup{5pt}
-  \setlength\railjoinsize{16pt}
-}
-\DeclareOption{11pt}{
-  \setlength\railboxheight{16pt}
-  \setlength\railboxskip{24pt}
-  \setlength\railtextup{5pt}
-  \setlength\railjoinsize{16pt}
-}
-\DeclareOption{12pt}{
-  \setlength\railboxheight{20pt}
-  \setlength\railboxskip{28pt}
-  \setlength\railtextup{6pt}
-  \setlength\railjoinsize{20pt}
-}
-
-\ExecuteOptions{10pt}
-\ProcessOptions
-
-% internal versions of the formatting parameters
-%
-% \rail@extra   : \railextra
-% \rail@boxht   : \railboxheight
-% \rail@boxsp   : \railboxskip
-% \rail@boxlf   : \railboxleft
-% \rail@boxrt   : \railboxright
-% \rail@boxhht  : \railboxheight / 2
-% \rail@ovalsp  : \railovalspace
-% \rail@framesp : \railframespace
-% \rail@textlf  : \railtextleft
-% \rail@textrt  : \railtextright
-% \rail@textup  : \railtextup
-% \rail@joinsz  : \railjoinsize
-% \rail@joinhsz : \railjoinsize / 2
-% \rail@joinadj : \railjoinadjust
-%
-% \railinit : internalize all of the parameters.
-
-\newcount\rail@extra
-\newcount\rail@boxht
-\newcount\rail@boxsp
-\newcount\rail@boxlf
-\newcount\rail@boxrt
-\newcount\rail@boxhht
-\newcount\rail@ovalsp
-\newcount\rail@framesp
-\newcount\rail@textlf
-\newcount\rail@textrt
-\newcount\rail@textup
-\newcount\rail@joinsz
-\newcount\rail@joinhsz
-\newcount\rail@joinadj
-
-\newcommand\railinit{
-\rail@extra=\railextra
-\divide\rail@extra by \railunit
-\rail@boxht=\railboxheight
-\divide\rail@boxht by \railunit
-\rail@boxsp=\railboxskip
-\divide\rail@boxsp by \railunit
-\rail@boxlf=\railboxleft
-\divide\rail@boxlf by \railunit
-\rail@boxrt=\railboxright
-\divide\rail@boxrt by \railunit
-\rail@boxhht=\railboxheight
-\divide\rail@boxhht by \railunit
-\divide\rail@boxhht by 2
-\rail@ovalsp=\railovalspace
-\divide\rail@ovalsp by \railunit
-\rail@framesp=\railframespace
-\divide\rail@framesp by \railunit
-\rail@textlf=\railtextleft
-\divide\rail@textlf by \railunit
-\rail@textrt=\railtextright
-\divide\rail@textrt by \railunit
-\rail@textup=\railtextup
-\divide\rail@textup by \railunit
-\rail@joinsz=\railjoinsize
-\divide\rail@joinsz by \railunit
-\rail@joinhsz=\railjoinsize
-\divide\rail@joinhsz by \railunit
-\divide\rail@joinhsz by 2
-\rail@joinadj=\railjoinadjust
-\divide\rail@joinadj by \railunit
-}
-
-\AtBeginDocument{\railinit}
-
-% \rail@param : declarations for list environment
-%
-% \railparam{TEXT} : sets \rail@param to TEXT
-%
-% \rail@reserved : characters reserved for grammar
-
-\newcommand\railparam[1]{
-\def\rail@param{
-  \setlength\leftmargin{0pt}\setlength\rightmargin{0pt}
-  \setlength\labelwidth{0pt}\setlength\labelsep{0pt}
-  \setlength\itemindent{0pt}\setlength\listparindent{0pt}
-  #1
-}
-}
-\railparam{}
-
-\newtoks\rail@reserved
-\rail@reserved={:;|*+?[]()'"}
-
-% \rail@termfont : format setup for terminals
-%
-% \rail@nontfont : format setup for nonterminals
-%
-% \rail@annofont : format setup for annotations
-%
-% \rail@rulefont : format setup for rule names
-%
-% \rail@indexfont : format setup for index entry
-%
-% \railtermfont{TEXT} : set terminal format setup to TEXT
-%
-% \railnontermfont{TEXT} : set nonterminal format setup to TEXT
-%
-% \railannotatefont{TEXT} : set annotation format setup to TEXT
-%
-% \railnamefont{TEXT} : set rule name format setup to TEXT
-%
-% \railindexfont{TEXT} : set index entry format setup to TEXT
-
-\def\rail@termfont{\ttfamily\upshape}
-\def\rail@nontfont{\rmfamily\upshape}
-\def\rail@annofont{\rmfamily\itshape}
-\def\rail@namefont{\rmfamily\itshape}
-\def\rail@indexfont{\rmfamily\itshape}
-
-\newcommand\railtermfont[1]{
-\def\rail@termfont{#1}
-}
-
-\newcommand\railnontermfont[1]{
-\def\rail@nontfont{#1}
-}
-
-\newcommand\railannotatefont[1]{
-\def\rail@annofont{#1}
-}
-
-\newcommand\railnamefont[1]{
-\def\rail@namefont{#1}
-}
-
-\newcommand\railindexfont[1]{
-\def\rail@indexfont{#1}
-}
-
-% railroad read/write macros
-%
-% \begin{rail} TEXT \end{rail} : TEXT is written out to the .rai file,
-%                                as \rail@i{NR}{TEXT}. Then the matching
-%                                \rail@o{NR}{FMT} from the .rao file is
-%                                executed (if defined).
-%
-% \railoptions{OPTIONS} : OPTIONS are written out to the .rai file,
-%                         as \rail@p{OPTIONS}.
-%
-% \railterm{IDENT,IDENT,...} : format IDENT as terminals. writes out
-%                              \rail@t{IDENT} to the .rai file
-%
-% \railalias{IDENT}{TEXT} : format IDENT as TEXT. defines \rail@t@IDENT as
-%                           TEXT.
-%
-% \railtoken{IDENT}{TEXT} : abbreviates \railalias{IDENT}{TEXT}\railterm{IDENT}
-%                           (for backward compatibility)
-%
-% \rail@setcodes : guards special characters
-%
-% \rail@makeother{CHARACTER} : sets \catcode of CHARACTER to "other"
-%                              used inside a loop for \rail@setcodes
-%
-% \rail@nr : railroad diagram counter
-%
-% \ifrail@match : current \rail@i{NR}{TEXT} matches
-%
-% \rail@first : actions to be done first. read in .rao file,
-%               open .rai file if \@filesw true, undefine \rail@first.
-%               executed from \begin{rail}, \railoptions and \railterm.
-%
-% \rail@i{NR}{TEXT} : defines \rail@i@NR as TEXT. written to the .rai
-%                     file by \rail, read from the .rao file by
-%                     \rail@first
-%
-% \rail@t{IDENT} : tells Rail that IDENT is to be custom formatted,
-%                  written to the .rai file by \railterm.
-%
-% \rail@o{NR}{TEXT} : defines \rail@o@NR as TEXT, read from the .rao
-%                     file by \rail@first.
-%
-% \rail@p{OPTIONS} : pass options to rail, written to the .rai file by
-%                    \railoptions
-%
-% \rail@write{TEXT} : write TEXT to the .rai file
-%
-% \rail@warn : warn user for mismatching diagrams
-%
-% \rail@endwarn : either \relax or \rail@warn
-%
-% \ifrail@all : checked at the end of the document
-
-\def\rail@makeother#1{
-  \expandafter\catcode\expandafter`\csname\string #1\endcsname=12
-}
-
-\def\rail@setcodes{
-\let\par=\relax
-\let\\=\relax
-\expandafter\@tfor\expandafter\rail@symbol\expandafter:\expandafter=%
-  \the\rail@reserved
-\do{\expandafter\rail@makeother\rail@symbol}
-}
-
-\newcount\rail@nr
-
-\newif\ifrail@all
-\rail@alltrue
-
-\newif\ifrail@match
-
-\def\rail@first{
-\begingroup
-\makeatletter
-\rail@setcodes
-\InputIfFileExists{\jobname.rao}{}{\PackageInfo{rail}{No file \jobname.rao}}
-\makeatother
-\endgroup
-\if@filesw
-\newwrite\tf@rai
-\immediate\openout\tf@rai=\jobname.rai
-\fi
-\global\let\rail@first=\relax
-}
-
-\long\def\rail@body#1\end{
-{
-\newlinechar=`^^J
-\def\par{\string\par^^J}
-\rail@write{\string\rail@i{\number\rail@nr}{#1}}
-}
-\xdef\rail@i@{#1}
-\end
-}
-
-\newenvironment{rail}{
-\global\advance\rail@nr by 1
-\rail@first
-\begingroup
-\rail@setcodes
-\rail@body
-}{
-\endgroup
-\rail@matchtrue
-\@ifundefined{rail@o@\number\rail@nr}{\rail@matchfalse}{}
-\expandafter\ifx\csname rail@i@\number\rail@nr\endcsname\rail@i@
-\else
-\rail@matchfalse
-\fi
-\ifrail@match
-\csname rail@o@\number\rail@nr\endcsname
-\else
-\PackageWarning{rail}{Railroad diagram {\number\rail@nr} doesn't match}
-\global\let\rail@endwarn=\rail@warn
-\begin{list}{}{\rail@param}
-\rail@begin{1}{}
-\rail@setbox{\bfseries ???}
-\rail@oval
-\rail@end
-\end{list}
-\fi
-}
-
-\newcommand\railoptions[1]{
-\rail@first
-\rail@write{\string\rail@p{#1}}
-}
-
-\newcommand\railterm[1]{
-\rail@first
-\@for\rail@@:=#1\do{
-\rail@write{\string\rail@t{\rail@@}}
-}
-}
-
-\newcommand\railalias[2]{
-\expandafter\def\csname rail@t@#1\endcsname{#2}
-}
-
-\newcommand\railtoken[2]{\railalias{#1}{#2}\railterm{#1}}
-
-\long\def\rail@i#1#2{
-\expandafter\gdef\csname rail@i@#1\endcsname{#2}
-}
-
-\def\rail@o#1#2{
-\expandafter\gdef\csname rail@o@#1\endcsname{
-\begin{list}{}{\rail@param}
-#2
-\end{list}
-}
-}
-
-\def\rail@t#1{}
-
-\def\rail@p#1{}
-
-\long\def\rail@write#1{\@ifundefined{tf@rai}{}{\immediate\write\tf@rai{#1}}}
-
-\def\rail@warn{
-\PackageWarningNoLine{rail}{Railroad diagram(s) may have changed.
-                            Use 'rail' and rerun}
-}
-
-\let\rail@endwarn=\relax
-
-\AtEndDocument{\rail@endwarn}
-
-% index entry macro
-%
-% \rail@index{IDENT} : add index entry for IDENT
-
-\def\rail@index#1{
-\index{\rail@indexfont#1}
-}
-
-% railroad formatting primitives
-%
-% \rail@x : current x
-% \rail@y : current y
-% \rail@ex : current end x
-% \rail@sx : starting x for \rail@cr
-% \rail@rx : rightmost previous x for \rail@cr
-%
-% \rail@tmpa : temporary count
-% \rail@tmpb : temporary count
-% \rail@tmpc : temporary count
-%
-% \rail@put : put at (\rail@x,\rail@y)
-% \rail@vput : put vector at (\rail@x,\rail@y)
-%
-% \rail@eline : end line by drawing from \rail@ex to \rail@x
-%
-% \rail@vreline : end line by drawing a vector from \rail@x to \rail@ex
-%
-% \rail@vleline : end line by drawing a vector from \rail@ex to \rail@x
-%
-% \rail@sety{LEVEL} : set \rail@y to level LEVEL
-
-\newcount\rail@x
-\newcount\rail@y
-\newcount\rail@ex
-\newcount\rail@sx
-\newcount\rail@rx
-
-\newcount\rail@tmpa
-\newcount\rail@tmpb
-\newcount\rail@tmpc
-
-\def\rail@put{\put(\number\rail@x,\number\rail@y)}
-
-\def\rail@vput{\put(\number\rail@ex,\number\rail@y)}
-
-\def\rail@eline{
-\rail@tmpb=\rail@x
-\advance\rail@tmpb by -\rail@ex
-\rail@put{\line(-1,0){\number\rail@tmpb}}
-}
-
-\def\rail@vreline{
-\rail@tmpb=\rail@x
-\advance\rail@tmpb by -\rail@ex
-\rail@vput{\vector(1,0){\number\rail@tmpb}}
-}
-
-\def\rail@vleline{
-\rail@tmpb=\rail@x
-\advance\rail@tmpb by -\rail@ex
-\rail@put{\vector(-1,0){\number\rail@tmpb}}
-}
-
-\def\rail@sety#1{
-\rail@y=#1
-\multiply\rail@y by -\rail@boxsp
-\advance\rail@y by -\rail@boxht
-}
-
-% \rail@begin{HEIGHT}{NAME} : begin a railroad diagram of height HEIGHT
-%
-% \rail@end : end a railroad diagram
-%
-% \rail@expand{IDENT} : expand IDENT
-
-\def\rail@begin#1#2{
-\item
-\begin{minipage}[t]{\linewidth}
-\ifx\@empty#2\else
-{\rail@namefont \rail@expand{#2}}\\*[\railnamesep]
-\fi
-\unitlength=\railunit
-\rail@tmpa=#1
-\multiply\rail@tmpa by \rail@boxsp
-\begin{picture}(0,\number\rail@tmpa)(0,-\number\rail@tmpa)
-\rail@ex=0
-\rail@rx=0
-\rail@x=\rail@extra
-\rail@sx=\rail@x
-\rail@sety{0}
-}
-
-\def\rail@end{
-\advance\rail@x by \rail@extra
-\rail@eline
-\end{picture}
-\end{minipage}
-}
-
-\def\rail@vend{
-\advance\rail@x by \rail@extra
-\rail@vreline
-\end{picture}
-\end{minipage}
-}
-
-\def\rail@expand#1{\@ifundefined{rail@t@#1}{#1}{\csname rail@t@#1\endcsname}}
-
-% \rail@token{TEXT}[ANNOT] : format token TEXT with annotation
-% \rail@ltoken{TEXT}[ANNOT] : format token TEXT with annotation, arrow left
-% \rail@rtoken{TEXT}[ANNOT] : format token TEXT with annotation, arrow right
-%
-% \rail@ctoken{TEXT}[ANNOT] : format token TEXT centered with annotation
-% \rail@lctoken{TEXT}[ANNOT] : format token TEXT centered with annotation, arrow left
-% \rail@rctoken{TEXT}[ANNOT] : format token TEXT centered with annotation, arrow right
-%
-% \rail@nont{TEXT}[ANNOT] : format nonterminal TEXT with annotation
-% \rail@lnont{TEXT}[ANNOT] : format nonterminal TEXT with annotation, arrow left
-% \rail@rnont{TEXT}[ANNOT] : format nonterminal TEXT with annotation. arrow right
-%
-% \rail@cnont{TEXT}[ANNOT] : format nonterminal TEXT centered with annotation
-% \rail@lcnont{TEXT}[ANNOT] : format nonterminal TEXT centered with annotation,
-%                             arrow left
-% \rail@rcnont{TEXT}[ANNOT] : format nonterminal TEXT centered with annotation,
-%                             arrow right
-%
-% \rail@term{TEXT}[ANNOT] : format terminal TEXT with annotation
-% \rail@lterm{TEXT}[ANNOT] : format terminal TEXT with annotation, arrow left
-% \rail@rterm{TEXT}[ANNOT] : format terminal TEXT with annotation, arrow right
-%
-% \rail@cterm{TEXT}[ANNOT] : format terminal TEXT centered with annotation
-% \rail@lcterm{TEXT}[ANNOT] : format terminal TEXT centered with annotation, arrow left
-% \rail@rcterm{TEXT}[ANNOT] : format terminal TEXT centered with annotation,
-%                             arrow right
-%
-% \rail@annote[TEXT] : format TEXT as annotation
-
-\def\rail@token#1[#2]{
-\rail@setbox{%
-{\rail@termfont \rail@expand{#1}}\ifx\@empty#2\else\ {\rail@annofont #2}\fi
-}
-\rail@oval
-}
-
-\def\rail@ltoken#1[#2]{
-\rail@setbox{%
-{\rail@termfont \rail@expand{#1}}\ifx\@empty#2\else\ {\rail@annofont #2}\fi
-}
-\rail@vloval
-}
-
-\def\rail@rtoken#1[#2]{
-\rail@setbox{%
-{\rail@termfont \rail@expand{#1}}\ifx\@empty#2\else\ {\rail@annofont #2}\fi
-}
-\rail@vroval
-}
-
-\def\rail@ctoken#1[#2]{
-\rail@setbox{%
-{\rail@termfont \rail@expand{#1}}\ifx\@empty#2\else\ {\rail@annofont #2}\fi
-}
-\rail@coval
-}
-
-\def\rail@lctoken#1[#2]{
-\rail@setbox{%
-{\rail@termfont \rail@expand{#1}}\ifx\@empty#2\else\ {\rail@annofont #2}\fi
-}
-\rail@vlcoval
-}
-
-\def\rail@rctoken#1[#2]{
-\rail@setbox{%
-{\rail@termfont \rail@expand{#1}}\ifx\@empty#2\else\ {\rail@annofont #2}\fi
-}
-\rail@vrcoval
-}
-
-\def\rail@nont#1[#2]{
-\rail@setbox{%
-{\rail@nontfont \rail@expand{#1}}\ifx\@empty#2\else\ {\rail@annofont #2}\fi
-}
-\rail@frame
-}
-
-\def\rail@lnont#1[#2]{
-\rail@setbox{%
-{\rail@nontfont \rail@expand{#1}}\ifx\@empty#2\else\ {\rail@annofont #2}\fi
-}
-\rail@vlframe
-}
-
-\def\rail@rnont#1[#2]{
-\rail@setbox{%
-{\rail@nontfont \rail@expand{#1}}\ifx\@empty#2\else\ {\rail@annofont #2}\fi
-}
-\rail@vrframe
-}
-
-\def\rail@cnont#1[#2]{
-\rail@setbox{%
-{\rail@nontfont \rail@expand{#1}}\ifx\@empty#2\else\ {\rail@annofont #2}\fi
-}
-\rail@cframe
-}
-
-\def\rail@lcnont#1[#2]{
-\rail@setbox{%
-{\rail@nontfont \rail@expand{#1}}\ifx\@empty#2\else\ {\rail@annofont #2}\fi
-}
-\rail@vlcframe
-}
-
-\def\rail@rcnont#1[#2]{
-\rail@setbox{%
-{\rail@nontfont \rail@expand{#1}}\ifx\@empty#2\else\ {\rail@annofont #2}\fi
-}
-\rail@vrcframe
-}
-
-\def\rail@term#1[#2]{
-\rail@setbox{%
-{\rail@termfont #1}\ifx\@empty#2\else\ {\rail@annofont #2}\fi
-}
-\rail@oval
-}
-
-\def\rail@lterm#1[#2]{
-\rail@setbox{%
-{\rail@termfont #1}\ifx\@empty#2\else\ {\rail@annofont #2}\fi
-}
-\rail@vloval
-}
-
-\def\rail@rterm#1[#2]{
-\rail@setbox{%
-{\rail@termfont #1}\ifx\@empty#2\else\ {\rail@annofont #2}\fi
-}
-\rail@vroval
-}
-
-\def\rail@cterm#1[#2]{
-\rail@setbox{%
-{\rail@termfont #1}\ifx\@empty#2\else\ {\rail@annofont #2}\fi
-}
-\rail@coval
-}
-
-\def\rail@lcterm#1[#2]{
-\rail@setbox{%
-{\rail@termfont #1}\ifx\@empty#2\else\ {\rail@annofont #2}\fi
-}
-\rail@vlcoval
-}
-
-\def\rail@rcterm#1[#2]{
-\rail@setbox{%
-{\rail@termfont #1}\ifx\@empty#2\else\ {\rail@annofont #2}\fi
-}
-\rail@vrcoval
-}
-
-\def\rail@annote[#1]{
-\rail@setbox{\rail@annofont #1}
-\rail@text
-}
-
-% \rail@box : temporary box for \rail@oval and \rail@frame
-%
-% \rail@setbox{TEXT} : set \rail@box to TEXT, set \rail@tmpa to width
-%
-% \rail@oval : format \rail@box of width \rail@tmpa inside an oval
-% \rail@vloval : format \rail@box of width \rail@tmpa inside an oval, vector left
-% \rail@vroval : format \rail@box of width \rail@tmpa inside an oval, vector right
-%
-% \rail@coval : same as \rail@oval, but centered between \rail@x and
-%               \rail@mx
-% \rail@vlcoval : same as \rail@oval, but centered between \rail@x and
-%                 \rail@mx, vector left
-% \rail@vrcoval : same as \rail@oval, but centered between \rail@x and
-%                 \rail@mx, vector right
-%
-% \rail@frame : format \rail@box of width \rail@tmpa inside a frame
-% \rail@vlframe : format \rail@box of width \rail@tmpa inside a frame, vector left
-% \rail@vrframe : format \rail@box of width \rail@tmpa inside a frame, vector right
-%
-% \rail@cframe : same as \rail@frame, but centered between \rail@x and
-%                \rail@mx
-% \rail@vlcframe : same as \rail@frame, but centered between \rail@x and
-%                  \rail@mx, vector left
-% \rail@vrcframe : same as \rail@frame, but centered between \rail@x and
-%                  \rail@mx, vector right
-%
-% \rail@text : format \rail@box of width \rail@tmpa above the line
-
-\newbox\rail@box
-
-\def\rail@setbox#1{
-\setbox\rail@box\hbox{\strut#1}
-\rail@tmpa=\wd\rail@box
-\divide\rail@tmpa by \railunit
-}
-
-\def\rail@oval{
-\advance\rail@x by \rail@boxlf
-\rail@eline
-\advance\rail@tmpa by \rail@ovalsp
-\ifnum\rail@tmpa<\rail@boxht\rail@tmpa=\rail@boxht\fi
-\rail@tmpb=\rail@tmpa
-\divide\rail@tmpb by 2
-\advance\rail@y by -\rail@boxhht
-\rail@put{\makebox(\number\rail@tmpa,\number\rail@boxht){\box\rail@box}}
-\advance\rail@y by \rail@boxhht
-\advance\rail@x by \rail@tmpb
-\rail@put{\oval(\number\rail@tmpa,\number\rail@boxht)}
-\advance\rail@x by \rail@tmpb
-\rail@ex=\rail@x
-\advance\rail@x by \rail@boxrt
-}
-
-\def\rail@vloval{
-\advance\rail@x by \rail@boxlf
-\rail@eline
-\advance\rail@tmpa by \rail@ovalsp
-\ifnum\rail@tmpa<\rail@boxht\rail@tmpa=\rail@boxht\fi
-\rail@tmpb=\rail@tmpa
-\divide\rail@tmpb by 2
-\advance\rail@y by -\rail@boxhht
-\rail@put{\makebox(\number\rail@tmpa,\number\rail@boxht){\box\rail@box}}
-\advance\rail@y by \rail@boxhht
-\advance\rail@x by \rail@tmpb
-\rail@put{\oval(\number\rail@tmpa,\number\rail@boxht)}
-\advance\rail@x by \rail@tmpb
-\rail@ex=\rail@x
-\advance\rail@x by \rail@boxrt
-\rail@vleline
-}
-
-\def\rail@vroval{
-\advance\rail@x by \rail@boxlf
-\rail@vreline
-\advance\rail@tmpa by \rail@ovalsp
-\ifnum\rail@tmpa<\rail@boxht\rail@tmpa=\rail@boxht\fi
-\rail@tmpb=\rail@tmpa
-\divide\rail@tmpb by 2
-\advance\rail@y by -\rail@boxhht
-\rail@put{\makebox(\number\rail@tmpa,\number\rail@boxht){\box\rail@box}}
-\advance\rail@y by \rail@boxhht
-\advance\rail@x by \rail@tmpb
-\rail@put{\oval(\number\rail@tmpa,\number\rail@boxht)}
-\advance\rail@x by \rail@tmpb
-\rail@ex=\rail@x
-\advance\rail@x by \rail@boxrt
-}
-
-\def\rail@coval{
-\rail@tmpb=\rail@tmpa
-\advance\rail@tmpb by \rail@ovalsp
-\ifnum\rail@tmpb<\rail@boxht\rail@tmpb=\rail@boxht\fi
-\advance\rail@tmpb by \rail@boxlf
-\advance\rail@tmpb by \rail@boxrt
-\rail@tmpc=\rail@mx
-\advance\rail@tmpc by -\rail@x
-\advance\rail@tmpc by -\rail@tmpb
-\divide\rail@tmpc by 2
-\ifnum\rail@tmpc>0
-\advance\rail@x by \rail@tmpc
-\fi
-\rail@oval
-}
-
-\def\rail@vlcoval{
-\rail@tmpb=\rail@tmpa
-\advance\rail@tmpb by \rail@ovalsp
-\ifnum\rail@tmpb<\rail@boxht\rail@tmpb=\rail@boxht\fi
-\advance\rail@tmpb by \rail@boxlf
-\advance\rail@tmpb by \rail@boxrt
-\rail@tmpc=\rail@mx
-\advance\rail@tmpc by -\rail@x
-\advance\rail@tmpc by -\rail@tmpb
-\divide\rail@tmpc by 2
-\ifnum\rail@tmpc>0
-\advance\rail@x by \rail@tmpc
-\fi
-\rail@vloval
-}
-
-\def\rail@vrcoval{
-\rail@tmpb=\rail@tmpa
-\advance\rail@tmpb by \rail@ovalsp
-\ifnum\rail@tmpb<\rail@boxht\rail@tmpb=\rail@boxht\fi
-\advance\rail@tmpb by \rail@boxlf
-\advance\rail@tmpb by \rail@boxrt
-\rail@tmpc=\rail@mx
-\advance\rail@tmpc by -\rail@x
-\advance\rail@tmpc by -\rail@tmpb
-\divide\rail@tmpc by 2
-\ifnum\rail@tmpc>0
-\advance\rail@x by \rail@tmpc
-\fi
-\rail@vroval
-}
-
-\def\rail@frame{
-\advance\rail@x by \rail@boxlf
-\rail@eline
-\advance\rail@tmpa by \rail@framesp
-\ifnum\rail@tmpa<\rail@boxht\rail@tmpa=\rail@boxht\fi
-\advance\rail@y by -\rail@boxhht
-\rail@put{\framebox(\number\rail@tmpa,\number\rail@boxht){\box\rail@box}}
-\advance\rail@y by \rail@boxhht
-\advance\rail@x by \rail@tmpa
-\rail@ex=\rail@x
-\advance\rail@x by \rail@boxrt
-}
-
-\def\rail@vlframe{
-\advance\rail@x by \rail@boxlf
-\rail@eline
-\advance\rail@tmpa by \rail@framesp
-\ifnum\rail@tmpa<\rail@boxht\rail@tmpa=\rail@boxht\fi
-\advance\rail@y by -\rail@boxhht
-\rail@put{\framebox(\number\rail@tmpa,\number\rail@boxht){\box\rail@box}}
-\advance\rail@y by \rail@boxhht
-\advance\rail@x by \rail@tmpa
-\rail@ex=\rail@x
-\advance\rail@x by \rail@boxrt
-\rail@vleline
-}
-
-\def\rail@vrframe{
-\advance\rail@x by \rail@boxlf
-\rail@vreline
-\advance\rail@tmpa by \rail@framesp
-\ifnum\rail@tmpa<\rail@boxht\rail@tmpa=\rail@boxht\fi
-\advance\rail@y by -\rail@boxhht
-\rail@put{\framebox(\number\rail@tmpa,\number\rail@boxht){\box\rail@box}}
-\advance\rail@y by \rail@boxhht
-\advance\rail@x by \rail@tmpa
-\rail@ex=\rail@x
-\advance\rail@x by \rail@boxrt
-}
-
-\def\rail@cframe{
-\rail@tmpb=\rail@tmpa
-\advance\rail@tmpb by \rail@framesp
-\ifnum\rail@tmpb<\rail@boxht\rail@tmpb=\rail@boxht\fi
-\advance\rail@tmpb by \rail@boxlf
-\advance\rail@tmpb by \rail@boxrt
-\rail@tmpc=\rail@mx
-\advance\rail@tmpc by -\rail@x
-\advance\rail@tmpc by -\rail@tmpb
-\divide\rail@tmpc by 2
-\ifnum\rail@tmpc>0
-\advance\rail@x by \rail@tmpc
-\fi
-\rail@frame
-}
-
-\def\rail@vlcframe{
-\rail@tmpb=\rail@tmpa
-\advance\rail@tmpb by \rail@framesp
-\ifnum\rail@tmpb<\rail@boxht\rail@tmpb=\rail@boxht\fi
-\advance\rail@tmpb by \rail@boxlf
-\advance\rail@tmpb by \rail@boxrt
-\rail@tmpc=\rail@mx
-\advance\rail@tmpc by -\rail@x
-\advance\rail@tmpc by -\rail@tmpb
-\divide\rail@tmpc by 2
-\ifnum\rail@tmpc>0
-\advance\rail@x by \rail@tmpc
-\fi
-\rail@vlframe
-}
-
-\def\rail@vrcframe{
-\rail@tmpb=\rail@tmpa
-\advance\rail@tmpb by \rail@framesp
-\ifnum\rail@tmpb<\rail@boxht\rail@tmpb=\rail@boxht\fi
-\advance\rail@tmpb by \rail@boxlf
-\advance\rail@tmpb by \rail@boxrt
-\rail@tmpc=\rail@mx
-\advance\rail@tmpc by -\rail@x
-\advance\rail@tmpc by -\rail@tmpb
-\divide\rail@tmpc by 2
-\ifnum\rail@tmpc>0
-\advance\rail@x by \rail@tmpc
-\fi
-\rail@vrframe
-}
-
-\def\rail@text{
-\advance\rail@x by \rail@textlf
-\advance\rail@y by \rail@textup
-\rail@put{\box\rail@box}
-\advance\rail@y by -\rail@textup
-\advance\rail@x by \rail@tmpa
-\advance\rail@x by \rail@textrt
-}
-
-% alternatives
-%
-% \rail@jx \rail@jy : current join point
-%
-% \rail@gx \rail@gy \rail@gex \rail@grx : global versions of \rail@x etc,
-%                                         to pass values over group closings
-%
-% \rail@mx : maximum x so far
-%
-% \rail@sy : starting \rail@y for alternatives
-%
-% \rail@jput : put at (\rail@jx,\rail@jy)
-%
-% \rail@joval[PART] : put \oval[PART] with adjust
-
-\newcount\rail@jx
-\newcount\rail@jy
-
-\newcount\rail@gx
-\newcount\rail@gy
-\newcount\rail@gex
-\newcount\rail@grx
-
-\newcount\rail@sy
-\newcount\rail@mx
-
-\def\rail@jput{
-\put(\number\rail@jx,\number\rail@jy)
-}
-
-\def\rail@joval[#1]{
-\advance\rail@jx by \rail@joinadj
-\rail@jput{\oval(\number\rail@joinsz,\number\rail@joinsz)[#1]}
-\advance\rail@jx by -\rail@joinadj
-}
-
-% \rail@barsplit : incoming split for '|'
-%
-% \rail@plussplit : incoming split for '+'
-%
-
-\def\rail@barsplit{
-\advance\rail@jy by -\rail@joinhsz
-\rail@joval[tr]
-\advance\rail@jx by \rail@joinhsz
-}
-
-\def\rail@plussplit{
-\advance\rail@jy by -\rail@joinhsz
-\advance\rail@jx by \rail@joinsz
-\rail@joval[tl]
-\advance\rail@jx by -\rail@joinhsz
-}
-
-% \rail@alt{SPLIT} : start alternatives with incoming split SPLIT
-%
-
-\def\rail@alt#1{
-\rail@sy=\rail@y
-\rail@jx=\rail@x
-\rail@jy=\rail@y
-\advance\rail@x by \rail@joinsz
-\rail@mx=0
-\let\rail@list=\@empty
-\let\rail@comma=\@empty
-\let\rail@split=#1
-\begingroup
-\rail@sx=\rail@x
-\rail@rx=0
-}
-
-% \rail@nextalt{FIX}{Y} : start next alternative at vertical position Y
-%                         and fix-up FIX
-%
-
-\def\rail@nextalt#1#2{
-\global\rail@gx=\rail@x
-\global\rail@gy=\rail@y
-\global\rail@gex=\rail@ex
-\global\rail@grx=\rail@rx
-\endgroup
-#1
-\ifnum\rail@gx>\rail@mx\rail@mx=\rail@gx\fi
-\ifnum\rail@grx>\rail@mx\rail@mx=\rail@grx\fi
-\edef\rail@list{\rail@list\rail@comma\number\rail@gex:\number\rail@gy}
-\def\rail@comma{,}
-\rail@split
-\let\rail@split=\@empty
-\rail@sety{#2}
-\rail@tmpa=\rail@jy
-\advance\rail@tmpa by -\rail@y
-\advance\rail@tmpa by -\rail@joinhsz
-\rail@jput{\line(0,-1){\number\rail@tmpa}}
-\rail@jy=\rail@y
-\advance\rail@jy by \rail@joinhsz
-\advance\rail@jx by \rail@joinhsz
-\rail@joval[bl]
-\advance\rail@jx by -\rail@joinhsz
-\rail@ex=\rail@x
-\begingroup
-\rail@sx=\rail@x
-\rail@rx=0
-}
-
-% \rail@barjoin : outgoing join for first '|' alternative
-%
-% \rail@plusjoin : outgoing join for first '+' alternative
-%
-% \rail@altjoin : join for subsequent alternative
-%
-
-\def\rail@barjoin{
-\ifnum\rail@y<\rail@sy
-\global\rail@gex=\rail@jx
-\else
-\global\rail@gex=\rail@ex
-\fi
-\advance\rail@jy by -\rail@joinhsz
-\rail@joval[tl]
-\advance\rail@jx by -\rail@joinhsz
-\ifnum\rail@y<\rail@sy
-\rail@altjoin
-\fi
-}
-
-\def\rail@plusjoin{
-\global\rail@gex=\rail@ex
-\advance\rail@jy by -\rail@joinhsz
-\advance\rail@jx by -\rail@joinsz
-\rail@joval[tr]
-\advance\rail@jx by \rail@joinhsz
-}
-
-\def\rail@altjoin{
-\rail@eline
-\rail@tmpa=\rail@jy
-\advance\rail@tmpa by -\rail@y
-\advance\rail@tmpa by -\rail@joinhsz
-\rail@jput{\line(0,-1){\number\rail@tmpa}}
-\rail@jy=\rail@y
-\advance\rail@jy by \rail@joinhsz
-\advance\rail@jx by -\rail@joinhsz
-\rail@joval[br]
-\advance\rail@jx by \rail@joinhsz
-}
-
-% \rail@eltsplit EX:Y; : split EX:Y into \rail@ex \rail@y
-%
-% \rail@endalt{JOIN} : end alternatives with outgoing join JOIN
-
-\def\rail@eltsplit#1:#2;{\rail@ex=#1\rail@y=#2}
-
-\def\rail@endalt#1{
-\global\rail@gx=\rail@x
-\global\rail@gy=\rail@y
-\global\rail@gex=\rail@ex
-\global\rail@grx=\rail@rx
-\endgroup
-\ifnum\rail@gx>\rail@mx\rail@mx=\rail@gx\fi
-\ifnum\rail@grx>\rail@mx\rail@mx=\rail@grx\fi
-\edef\rail@list{\rail@list\rail@comma\number\rail@gex:\number\rail@gy}
-\rail@x=\rail@mx
-\rail@jx=\rail@x
-\rail@jy=\rail@sy
-\advance\rail@jx by \rail@joinsz
-\let\rail@join=#1
-\@for\rail@elt:=\rail@list\do{
-\expandafter\rail@eltsplit\rail@elt;
-\rail@join
-\let\rail@join=\rail@altjoin
-}
-\rail@x=\rail@mx
-\rail@y=\rail@sy
-\rail@ex=\rail@gex
-\advance\rail@x by \rail@joinsz
-}
-
-% \rail@bar : start '|' alternatives
-%
-% \rail@nextbar : next '|' alternative
-%
-% \rail@endbar : end '|' alternatives
-%
-
-\def\rail@bar{
-\rail@alt\rail@barsplit
-}
-
-\def\rail@nextbar{
-\rail@nextalt\relax
-}
-
-\def\rail@endbar{
-\rail@endalt\rail@barjoin
-}
-
-% \rail@plus : start '+' alternatives
-%
-% \rail@nextplus: next '+' alternative
-%
-% \rail@endplus : end '+' alternatives
-%
-
-\def\rail@plus{
-\rail@alt\rail@plussplit
-}
-
-\def\rail@nextplus{
-\rail@nextalt\rail@fixplus
-}
-
-\def\rail@fixplus{
-\ifnum\rail@gy<\rail@sy
-\begingroup
-\rail@x=\rail@gx
-\rail@y=\rail@gy
-\rail@ex=\rail@gex
-\rail@rx=\rail@grx
-\ifnum\rail@x<\rail@rx
-\rail@x=\rail@rx
-\fi
-\rail@eline
-\rail@jx=\rail@x
-\rail@jy=\rail@y
-\advance\rail@jy by \rail@joinhsz
-\rail@joval[br]
-\advance\rail@jx by \rail@joinhsz
-\rail@tmpa=\rail@sy
-\advance\rail@tmpa by -\rail@joinhsz
-\advance\rail@tmpa by -\rail@jy
-\rail@jput{\line(0,1){\number\rail@tmpa}}
-\rail@jy=\rail@sy
-\advance\rail@jy by -\rail@joinhsz
-\advance\rail@jx by \rail@joinhsz
-\rail@joval[tl]
-\advance\rail@jy by \rail@joinhsz
-\global\rail@gx=\rail@jx
-\global\rail@gy=\rail@jy
-\global\rail@gex=\rail@gx
-\global\rail@grx=\rail@rx
-\endgroup
-\fi
-}
-
-\def\rail@endplus{
-\rail@endalt\rail@plusjoin
-}
-
-% \rail@cr{Y} : carriage return to vertical position Y
-
-\def\rail@cr#1{
-\rail@tmpa=\rail@sx
-\advance\rail@tmpa by \rail@joinsz
-\ifnum\rail@x<\rail@tmpa\rail@x=\rail@tmpa\fi
-\rail@eline
-\rail@jx=\rail@x
-\rail@jy=\rail@y
-\advance\rail@x by \rail@joinsz
-\ifnum\rail@x>\rail@rx\rail@rx=\rail@x\fi
-\advance\rail@jy by -\rail@joinhsz
-\rail@joval[tr]
-\advance\rail@jx by \rail@joinhsz
-\rail@sety{#1}
-\rail@tmpa=\rail@jy
-\advance\rail@tmpa by -\rail@y
-\advance\rail@tmpa by -\rail@boxsp
-\advance\rail@tmpa by -\rail@joinhsz
-\rail@jput{\line(0,-1){\number\rail@tmpa}}
-\rail@jy=\rail@y
-\advance\rail@jy by \rail@boxsp
-\advance\rail@jy by \rail@joinhsz
-\advance\rail@jx by -\rail@joinhsz
-\rail@joval[br]
-\advance\rail@jy by -\rail@joinhsz
-\rail@tmpa=\rail@jx
-\advance\rail@tmpa by -\rail@sx
-\advance\rail@tmpa by -\rail@joinhsz
-\rail@jput{\line(-1,0){\number\rail@tmpa}}
-\rail@jx=\rail@sx
-\advance\rail@jx by \rail@joinhsz
-\advance\rail@jy by -\rail@joinhsz
-\rail@joval[tl]
-\advance\rail@jx by -\rail@joinhsz
-\rail@tmpa=\rail@boxsp
-\advance\rail@tmpa by -\rail@joinsz
-\rail@jput{\line(0,-1){\number\rail@tmpa}}
-\advance\rail@jy by -\rail@tmpa
-\advance\rail@jx by \rail@joinhsz
-\rail@joval[bl]
-\rail@x=\rail@jx
-\rail@ex=\rail@x
-}
-
-% default setup for Isabelle
-\newenvironment{railoutput}%
-{\begin{list}{}{\rail@param}\def\rail@expand{\relax}\makeatletter}{\makeatother\end{list}}
-
-\def\rail@termfont{\isabellestyle{tt}}
-\def\rail@nontfont{\isabellestyle{it}}
-\def\rail@namefont{\isabellestyle{it}}
diff --git a/thys/MDP-Algorithms/output/document/root.bib b/thys/MDP-Algorithms/output/document/root.bib
deleted file mode 100644
--- a/thys/MDP-Algorithms/output/document/root.bib
+++ /dev/null
@@ -1,13 +0,0 @@
-@book{Puterman,
-  author    = {Martin L. Puterman},
-  title     = {Markov Decision Processes: Discrete Stochastic Dynamic Programming},
-  series    = {Wiley Series in Probability and Statistics},
-  publisher = {Wiley},
-  year      = {1994},
-  url       = {https://doi.org/10.1002/9780470316887},
-  doi       = {10.1002/9780470316887},
-  isbn      = {978-0-47161977-2},
-  timestamp = {Mon, 22 Jul 2019 15:00:49 +0200},
-  biburl    = {https://dblp.org/rec/books/wi/Puterman94.bib},
-  bibsource = {dblp computer science bibliography, https://dblp.org}
-}
diff --git a/thys/MDP-Algorithms/output/document/root.tex b/thys/MDP-Algorithms/output/document/root.tex
deleted file mode 100644
--- a/thys/MDP-Algorithms/output/document/root.tex
+++ /dev/null
@@ -1,69 +0,0 @@
-\documentclass[11pt,a4paper]{article}
-\usepackage{isabelle,isabellesym}
-
-% further packages required for unusual symbols (see also
-% isabellesym.sty), use only when needed
-
-\usepackage{amssymb, amsmath, amsfonts}
-  %for \<leadsto>, \<box>, \<diamond>, \<sqsupset>, \<mho>, \<Join>,
-  %\<lhd>, \<lesssim>, \<greatersim>, \<lessapprox>, \<greaterapprox>,
-  %\<triangleq>, \<yen>, \<lozenge>
-
-%\usepackage{eurosym}
-  %for \<euro>
-
-\usepackage[only,bigsqcap]{stmaryrd}
-  %for \<Sqinter>
-
-%\usepackage{eufrak}
-  %for \<AA> ... \<ZZ>, \<aa> ... \<zz> (also included in amssymb)
-
-%\usepackage{textcomp}
-  %for \<onequarter>, \<onehalf>, \<threequarters>, \<degree>, \<cent>,
-  %\<currency>
-
-% this should be the last package used
-\usepackage{pdfsetup}
-
-% urls in roman style, theory text in math-similar italics
-\urlstyle{rm}
-\isabellestyle{it}
-
-% for uniform font size
-%\renewcommand{\isastyle}{\isastyleminor}
-
-
-\begin{document}
-
-\title{Verified Algorithms for Solving Markov Decision Processes}
-\author{Maximilian Schäffeler and Mohammad Abdulaziz}
-\maketitle
-
-\abstract{
-We present a formalization of algorithms for solving Markov Decision Processes (MDPs) with formal guarantees on the optimality of their
-solutions.
-In particular we build on our analysis of the Bellman operator for discounted infinite horizon MDPs.
-From the iterator rule on the Bellman operator we directly derive executable value iteration and policy iteration algorithms to iteratively solve finite MDPs.
-We also prove correct optimized versions of value iteration that use matrix splittings to improve the convergence rate. In particular, we formally verify Gauss-Seidel value iteration and modified policy iteration.
-The algorithms are evaluated on two standard examples from the literature, namely, inventory management and gridworld.
-Our formalization covers most of chapter 6 in Puterman's book \cite{Puterman}.
-}
-
-\tableofcontents
-
-% sane default for proof documents
-\parindent 0pt\parskip 0.5ex
-
-% generated text of all theories
-\input{session}
-
-% optional bibliography
-\bibliographystyle{abbrv}
-\bibliography{root}
-
-\end{document}
-
-%%% Local Variables:
-%%% mode: latex
-%%% TeX-master: t
-%%% End:
diff --git a/thys/MDP-Algorithms/output/document/session.tex b/thys/MDP-Algorithms/output/document/session.tex
deleted file mode 100644
--- a/thys/MDP-Algorithms/output/document/session.tex
+++ /dev/null
@@ -1,13 +0,0 @@
-\input{Value_Iteration.tex}
-\input{Policy_Iteration.tex}
-\input{Modified_Policy_Iteration.tex}
-\input{Matrix_Util.tex}
-\input{Blinfun_Matrix.tex}
-\input{Splitting_Methods.tex}
-\input{Algorithms.tex}
-\input{Code_DP.tex}
-\input{Code_Mod.tex}
-\input{Code_Real_Approx_By_Float_Fix.tex}
-\input{Code_Inventory.tex}
-\input{Code_Gridworld.tex}
-\input{Examples.tex}
diff --git a/thys/MDP-Algorithms/output/document/session_graph.pdf b/thys/MDP-Algorithms/output/document/session_graph.pdf
deleted file mode 100644
index b03b483f5536b1b7b1f0209d0e8065bb41570191..e69de29bb2d1d6434b8b29ae775ad8c2e48c5391
GIT binary patch
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diff --git a/thys/MDP-Algorithms/output/outline.pdf b/thys/MDP-Algorithms/output/outline.pdf
deleted file mode 100644
index bdef981cc0a2b882d43dcab50fbe8174bd206d83..e69de29bb2d1d6434b8b29ae775ad8c2e48c5391
GIT binary patch
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Hc$@<O00001

diff --git a/thys/MDP-Algorithms/output/outline/Algorithms.tex b/thys/MDP-Algorithms/output/outline/Algorithms.tex
deleted file mode 100644
--- a/thys/MDP-Algorithms/output/outline/Algorithms.tex
+++ /dev/null
@@ -1,42 +0,0 @@
-%
-\begin{isabellebody}%
-\setisabellecontext{Algorithms}%
-%
-\isadelimtheory
-\isanewline
-\isanewline
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{theory}\isamarkupfalse%
-\ Algorithms\isanewline
-\ \ \isakeyword{imports}\isanewline
-\ \ \ \ Value{\isacharunderscore}{\kern0pt}Iteration\isanewline
-\ \ \ \ Policy{\isacharunderscore}{\kern0pt}Iteration\isanewline
-\ \ \ \ Modified{\isacharunderscore}{\kern0pt}Policy{\isacharunderscore}{\kern0pt}Iteration\isanewline
-\ \ \ \ Splitting{\isacharunderscore}{\kern0pt}Methods\isanewline
-\isakeyword{begin}\isanewline
-\isacommand{end}\isamarkupfalse%
-%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\end{isabellebody}%
-\endinput
-%:%file=Algorithms.tex%:%
-%:%6=1%:%
-%:%7=2%:%
-%:%12=3%:%
-%:%13=3%:%
-%:%14=4%:%
-%:%15=5%:%
-%:%16=6%:%
-%:%17=7%:%
-%:%18=8%:%
-%:%19=9%:%
-%:%20=10%:%
diff --git a/thys/MDP-Algorithms/output/outline/Blinfun_Matrix.tex b/thys/MDP-Algorithms/output/outline/Blinfun_Matrix.tex
deleted file mode 100644
--- a/thys/MDP-Algorithms/output/outline/Blinfun_Matrix.tex
+++ /dev/null
@@ -1,1342 +0,0 @@
-%
-\begin{isabellebody}%
-\setisabellecontext{Blinfun{\isacharunderscore}{\kern0pt}Matrix}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{theory}\isamarkupfalse%
-\ Blinfun{\isacharunderscore}{\kern0pt}Matrix\isanewline
-\ \ \isakeyword{imports}\ \isanewline
-\ \ \ \ {\isachardoublequoteopen}MDP{\isacharminus}{\kern0pt}Rewards{\isachardot}{\kern0pt}Blinfun{\isacharunderscore}{\kern0pt}Util{\isachardoublequoteclose}\ \isanewline
-\ \ \ \ Matrix{\isacharunderscore}{\kern0pt}Util\isanewline
-\isakeyword{begin}%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsection{Bounded Linear Functions and Matrices%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{definition}\isamarkupfalse%
-\ {\isachardoublequoteopen}blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ {\isacharparenleft}{\kern0pt}f\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}b{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}finite\ {\isasymRightarrow}\isactrlsub b\ real{\isacharparenright}{\kern0pt}\ {\isasymRightarrow}\isactrlsub L\ {\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}c{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}finite\ {\isasymRightarrow}\isactrlsub b\ {\isacharunderscore}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ \isanewline
-\ \ matrix\ {\isacharparenleft}{\kern0pt}{\isasymlambda}v{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymchi}\ j{\isachardot}{\kern0pt}\ f\ {\isacharparenleft}{\kern0pt}Bfun\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}{\isachardollar}{\kern0pt}{\isacharparenright}{\kern0pt}\ v{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ j{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ {\isachardoublequoteopen}matrix{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}blinfun\ X\ {\isacharequal}{\kern0pt}\ Blinfun\ {\isacharparenleft}{\kern0pt}{\isasymlambda}v{\isachardot}{\kern0pt}\ Bfun\ {\isacharparenleft}{\kern0pt}{\isasymlambda}i{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}X\ {\isacharasterisk}{\kern0pt}v\ {\isacharparenleft}{\kern0pt}{\isasymchi}\ i{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}apply{\isacharunderscore}{\kern0pt}bfun\ v\ i{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isachardollar}{\kern0pt}\ i{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ plus{\isacharunderscore}{\kern0pt}vec{\isacharunderscore}{\kern0pt}eq{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isasymchi}\ i{\isachardot}{\kern0pt}\ f\ i\ \ {\isacharplus}{\kern0pt}\ g\ i{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymchi}\ i{\isachardot}{\kern0pt}\ f\ i{\isacharparenright}{\kern0pt}\ {\isacharplus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymchi}\ i{\isachardot}{\kern0pt}\ g\ i{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ Finite{\isacharunderscore}{\kern0pt}Cartesian{\isacharunderscore}{\kern0pt}Product{\isachardot}{\kern0pt}plus{\isacharunderscore}{\kern0pt}vec{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ matrix{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}blinfun{\isacharunderscore}{\kern0pt}mult{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}matrix{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}blinfun\ m\ {\isacharparenleft}{\kern0pt}v\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isacharprime}{\kern0pt}c{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}finite\ {\isasymRightarrow}\isactrlsub b\ real{\isacharparenright}{\kern0pt}\ i\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}m\ {\isacharasterisk}{\kern0pt}v\ {\isacharparenleft}{\kern0pt}{\isasymchi}\ i{\isachardot}{\kern0pt}\ v\ i{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isachardollar}{\kern0pt}\ i{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isasymchi}\ i{\isachardot}{\kern0pt}\ c\ {\isacharasterisk}{\kern0pt}\ x\ i{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ c\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isacharparenleft}{\kern0pt}{\isasymchi}\ i{\isachardot}{\kern0pt}\ x\ i{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{for}\ c\ x\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ vector{\isacharunderscore}{\kern0pt}scalar{\isacharunderscore}{\kern0pt}mult{\isacharunderscore}{\kern0pt}def\ scalar{\isacharunderscore}{\kern0pt}mult{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}scaleR{\isacharbrackleft}{\kern0pt}symmetric{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}bounded{\isacharunderscore}{\kern0pt}linear\ {\isacharparenleft}{\kern0pt}{\isasymlambda}v{\isachardot}{\kern0pt}\ bfun{\isachardot}{\kern0pt}Bfun\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}{\isachardollar}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}m\ {\isacharasterisk}{\kern0pt}v\ vec{\isacharunderscore}{\kern0pt}lambda\ {\isacharparenleft}{\kern0pt}apply{\isacharunderscore}{\kern0pt}bfun\ v{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}rule\ bounded{\isacharunderscore}{\kern0pt}linear{\isacharunderscore}{\kern0pt}compose{\isacharbrackleft}{\kern0pt}of\ {\isachardoublequoteopen}{\isasymlambda}x{\isachardot}{\kern0pt}\ bfun{\isachardot}{\kern0pt}Bfun\ {\isacharparenleft}{\kern0pt}{\isasymlambda}y{\isachardot}{\kern0pt}\ x\ {\isachardollar}{\kern0pt}\ y{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}{\isacharbrackright}{\kern0pt}{\isacharcomma}{\kern0pt}\ goal{\isacharunderscore}{\kern0pt}cases{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{case}\isamarkupfalse%
-\ {\isadigit{1}}\isanewline
-\ \ \ \ \isacommand{then}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}case\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ bounded{\isacharunderscore}{\kern0pt}linear{\isacharunderscore}{\kern0pt}bfun{\isacharunderscore}{\kern0pt}nth{\isacharbrackleft}{\kern0pt}of\ id{\isacharcomma}{\kern0pt}\ simplified{\isacharbrackright}{\kern0pt}\ bounded{\isacharunderscore}{\kern0pt}linear{\isacharunderscore}{\kern0pt}ident\ eq{\isacharunderscore}{\kern0pt}id{\isacharunderscore}{\kern0pt}iff\ \isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{case}\isamarkupfalse%
-\ {\isadigit{2}}\isanewline
-\ \ \ \ \isacommand{then}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}case\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ norm{\isacharunderscore}{\kern0pt}vec{\isacharunderscore}{\kern0pt}le{\isacharunderscore}{\kern0pt}norm{\isacharunderscore}{\kern0pt}bfun\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ matrix{\isacharunderscore}{\kern0pt}vector{\isacharunderscore}{\kern0pt}right{\isacharunderscore}{\kern0pt}distrib\ plus{\isacharunderscore}{\kern0pt}vec{\isacharunderscore}{\kern0pt}eq\isanewline
-\ \ \ \ \ \ \ \ \ \ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ bounded{\isacharunderscore}{\kern0pt}linear{\isacharunderscore}{\kern0pt}intro\ bounded{\isacharunderscore}{\kern0pt}linear{\isacharunderscore}{\kern0pt}compose{\isacharbrackleft}{\kern0pt}OF\ matrix{\isacharunderscore}{\kern0pt}vector{\isacharunderscore}{\kern0pt}mul{\isacharunderscore}{\kern0pt}bounded{\isacharunderscore}{\kern0pt}linear{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ Blinfun{\isacharunderscore}{\kern0pt}inverse\ matrix{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}blinfun{\isacharunderscore}{\kern0pt}def\ Bfun{\isacharunderscore}{\kern0pt}inverse{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}mult{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ f\ {\isacharasterisk}{\kern0pt}v\ {\isacharparenleft}{\kern0pt}{\isasymchi}\ i{\isachardot}{\kern0pt}\ apply{\isacharunderscore}{\kern0pt}bfun\ v\ i{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isachardollar}{\kern0pt}\ i\ {\isacharequal}{\kern0pt}\ f\ v\ i{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ f\ {\isacharasterisk}{\kern0pt}v\ {\isacharparenleft}{\kern0pt}{\isasymchi}\ i{\isachardot}{\kern0pt}\ v\ i{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isachardollar}{\kern0pt}\ i\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSum}j{\isasymin}UNIV{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}f\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}v\ j\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ bfun{\isachardot}{\kern0pt}Bfun\ {\isacharparenleft}{\kern0pt}{\isasymlambda}i{\isachardot}{\kern0pt}\ if\ i\ {\isacharequal}{\kern0pt}\ j\ then\ {\isadigit{1}}\ else\ {\isadigit{0}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ i{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}def\ matrix{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ matrix{\isacharunderscore}{\kern0pt}vector{\isacharunderscore}{\kern0pt}mult{\isacharunderscore}{\kern0pt}def\ mult{\isachardot}{\kern0pt}commute\ axis{\isacharunderscore}{\kern0pt}def\ blinfun{\isachardot}{\kern0pt}scaleR{\isacharunderscore}{\kern0pt}right\ vec{\isacharunderscore}{\kern0pt}lambda{\isacharunderscore}{\kern0pt}inverse{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSum}j{\isasymin}UNIV{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}f\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}v\ j\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ bfun{\isachardot}{\kern0pt}Bfun\ {\isacharparenleft}{\kern0pt}{\isasymlambda}i{\isachardot}{\kern0pt}\ if\ i\ {\isacharequal}{\kern0pt}\ j\ then\ {\isadigit{1}}\ else\ {\isadigit{0}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ i{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ intro{\isacharcolon}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}induct{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isacharequal}{\kern0pt}\ f\ {\isacharparenleft}{\kern0pt}{\isasymSum}j{\isasymin}UNIV{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}v\ j\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ bfun{\isachardot}{\kern0pt}Bfun\ {\isacharparenleft}{\kern0pt}{\isasymlambda}i{\isachardot}{\kern0pt}\ if\ i\ {\isacharequal}{\kern0pt}\ j\ then\ {\isadigit{1}}\ else\ {\isadigit{0}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ i{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ blinfun{\isachardot}{\kern0pt}sum{\isacharunderscore}{\kern0pt}right{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isacharequal}{\kern0pt}\ f\ v\ i{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isasymSum}j{\isasymin}UNIV{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}v\ j\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ bfun{\isachardot}{\kern0pt}Bfun\ {\isacharparenleft}{\kern0pt}{\isasymlambda}i{\isachardot}{\kern0pt}\ if\ i\ {\isacharequal}{\kern0pt}\ j\ then\ {\isadigit{1}}\ else\ {\isadigit{0}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ x\ {\isacharequal}{\kern0pt}\ v\ x{\isachardoublequoteclose}\ \isakeyword{for}\ x\isanewline
-\ \ \ \ \isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isasymSum}j{\isasymin}UNIV{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}v\ j\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ bfun{\isachardot}{\kern0pt}Bfun\ {\isacharparenleft}{\kern0pt}{\isasymlambda}i{\isachardot}{\kern0pt}\ if\ i\ {\isacharequal}{\kern0pt}\ j\ then\ {\isadigit{1}}\ else\ {\isadigit{0}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ x\ {\isacharequal}{\kern0pt}\ \isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymSum}j{\isasymin}UNIV{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}v\ j\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ bfun{\isachardot}{\kern0pt}Bfun\ {\isacharparenleft}{\kern0pt}{\isasymlambda}i{\isachardot}{\kern0pt}\ if\ i\ {\isacharequal}{\kern0pt}\ j\ then\ {\isadigit{1}}\ else\ {\isadigit{0}}{\isacharparenright}{\kern0pt}\ x{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ intro{\isacharcolon}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}induct{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSum}j{\isasymin}UNIV{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}v\ j\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isacharparenleft}{\kern0pt}{\isasymlambda}i{\isachardot}{\kern0pt}\ if\ i\ {\isacharequal}{\kern0pt}\ j\ then\ {\isadigit{1}}\ else\ {\isadigit{0}}{\isacharparenright}{\kern0pt}\ x{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}subst\ Bfun{\isacharunderscore}{\kern0pt}inverse{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}metis\ vec{\isacharunderscore}{\kern0pt}bfun\ vec{\isacharunderscore}{\kern0pt}lambda{\isacharunderscore}{\kern0pt}inverse{\isacharbrackleft}{\kern0pt}OF\ UNIV{\isacharunderscore}{\kern0pt}I{\isacharcomma}{\kern0pt}\ symmetric{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharplus}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSum}j{\isasymin}UNIV{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}if\ x\ {\isacharequal}{\kern0pt}\ j\ then\ v\ j\ {\isacharasterisk}{\kern0pt}\ {\isadigit{1}}\ else\ v\ j\ {\isacharasterisk}{\kern0pt}\ {\isadigit{0}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ if{\isacharunderscore}{\kern0pt}distrib\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ sum{\isachardot}{\kern0pt}cong{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSum}j{\isasymin}UNIV{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}if\ x\ {\isacharequal}{\kern0pt}\ j\ then\ v\ j\ else\ {\isadigit{0}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}meson\ more{\isacharunderscore}{\kern0pt}arith{\isacharunderscore}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{6}}{\isacharparenright}{\kern0pt}\ mult{\isacharunderscore}{\kern0pt}zero{\isacharunderscore}{\kern0pt}right{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isacharequal}{\kern0pt}\ v\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \ \ \isacommand{finally}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isacommand{{\isachardot}{\kern0pt}}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ bfun{\isacharunderscore}{\kern0pt}eqI\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ fastforce\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{finally}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isacommand{{\isachardot}{\kern0pt}}\isamarkupfalse%
-\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}mult{\isacharprime}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ f\ {\isacharasterisk}{\kern0pt}v\ v{\isacharparenright}{\kern0pt}\ {\isachardollar}{\kern0pt}\ i\ {\isacharequal}{\kern0pt}\ f\ {\isacharparenleft}{\kern0pt}Bfun\ {\isacharparenleft}{\kern0pt}{\isasymlambda}i{\isachardot}{\kern0pt}\ v\ {\isachardollar}{\kern0pt}\ i{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ i{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ bfun{\isachardot}{\kern0pt}Bfun{\isacharunderscore}{\kern0pt}inverse\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}mult\ vec{\isacharunderscore}{\kern0pt}bfun\ vec{\isacharunderscore}{\kern0pt}nth{\isacharunderscore}{\kern0pt}inverse{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}mult{\isacharprime}{\kern0pt}{\isacharprime}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ f\ {\isacharasterisk}{\kern0pt}v\ v{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymchi}\ i{\isachardot}{\kern0pt}\ f\ {\isacharparenleft}{\kern0pt}Bfun\ {\isacharparenleft}{\kern0pt}{\isasymlambda}i{\isachardot}{\kern0pt}\ v\ {\isachardollar}{\kern0pt}\ i{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ i{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}mult{\isacharprime}{\kern0pt}\ vec{\isacharunderscore}{\kern0pt}lambda{\isacharunderscore}{\kern0pt}unique{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ matrix{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}blinfun{\isacharunderscore}{\kern0pt}inv{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}matrix{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}blinfun\ {\isacharparenleft}{\kern0pt}blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ f{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ f{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ matrix{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}blinfun{\isacharunderscore}{\kern0pt}mult\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}mult\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ blinfun{\isacharunderscore}{\kern0pt}eqI{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}add{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ {\isacharparenleft}{\kern0pt}f\ {\isacharplus}{\kern0pt}\ g{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ f\ {\isacharplus}{\kern0pt}\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ g{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ matrix{\isacharunderscore}{\kern0pt}eq\ \ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}mult{\isacharprime}{\kern0pt}{\isacharprime}{\kern0pt}\ matrix{\isacharunderscore}{\kern0pt}vector{\isacharunderscore}{\kern0pt}mult{\isacharunderscore}{\kern0pt}add{\isacharunderscore}{\kern0pt}rdistrib\ blinfun{\isachardot}{\kern0pt}add{\isacharunderscore}{\kern0pt}left\ plus{\isacharunderscore}{\kern0pt}vec{\isacharunderscore}{\kern0pt}eq{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}diff{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ {\isacharparenleft}{\kern0pt}f\ {\isacharminus}{\kern0pt}\ g{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ f\ {\isacharminus}{\kern0pt}\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ g{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}add\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ add{\isacharunderscore}{\kern0pt}right{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}eq\ diff{\isacharunderscore}{\kern0pt}add{\isacharunderscore}{\kern0pt}cancel{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}scaleR{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ {\isacharparenleft}{\kern0pt}c\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ f{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ c\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ f{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ matrix{\isacharunderscore}{\kern0pt}eq\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}mult{\isacharprime}{\kern0pt}{\isacharprime}{\kern0pt}\ scaleR{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}vector{\isacharunderscore}{\kern0pt}assoc{\isacharbrackleft}{\kern0pt}symmetric{\isacharbrackright}{\kern0pt}\ \isanewline
-\ \ \ \ \ \ blinfun{\isachardot}{\kern0pt}scaleR{\isacharunderscore}{\kern0pt}left\ vector{\isacharunderscore}{\kern0pt}scalar{\isacharunderscore}{\kern0pt}mult{\isacharunderscore}{\kern0pt}def{\isacharbrackleft}{\kern0pt}of\ c{\isacharcomma}{\kern0pt}\ unfolded\ scalar{\isacharunderscore}{\kern0pt}mult{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}scaleR{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\ \isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ matrix{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}blinfun{\isacharunderscore}{\kern0pt}add{\isacharcolon}{\kern0pt}\ \isanewline
-\ \ {\isachardoublequoteopen}matrix{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}blinfun\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}f\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ real{\isacharcircum}{\kern0pt}{\isacharunderscore}{\kern0pt}{\isacharcircum}{\kern0pt}{\isacharunderscore}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharplus}{\kern0pt}\ g{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ matrix{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}blinfun\ f\ {\isacharplus}{\kern0pt}\ matrix{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}blinfun\ g{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ blinfun{\isacharunderscore}{\kern0pt}eqI\ simp{\isacharcolon}{\kern0pt}\ matrix{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}blinfun{\isacharunderscore}{\kern0pt}mult\ blinfun{\isachardot}{\kern0pt}add{\isacharunderscore}{\kern0pt}left\ matrix{\isacharunderscore}{\kern0pt}vector{\isacharunderscore}{\kern0pt}mult{\isacharunderscore}{\kern0pt}add{\isacharunderscore}{\kern0pt}rdistrib{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ matrix{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}blinfun{\isacharunderscore}{\kern0pt}diff{\isacharcolon}{\kern0pt}\ \isanewline
-\ \ {\isachardoublequoteopen}matrix{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}blinfun\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}f\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ real{\isacharcircum}{\kern0pt}{\isacharunderscore}{\kern0pt}{\isacharcircum}{\kern0pt}{\isacharunderscore}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharminus}{\kern0pt}\ g{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ matrix{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}blinfun\ f\ {\isacharminus}{\kern0pt}\ matrix{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}blinfun\ g{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ matrix{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}blinfun{\isacharunderscore}{\kern0pt}add\ diff{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}eq\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ metis%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ matrix{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}blinfun{\isacharunderscore}{\kern0pt}scaleR{\isacharcolon}{\kern0pt}\ \isanewline
-\ \ {\isachardoublequoteopen}matrix{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}blinfun\ {\isacharparenleft}{\kern0pt}c\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isacharparenleft}{\kern0pt}f\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ real{\isacharcircum}{\kern0pt}{\isacharunderscore}{\kern0pt}{\isacharcircum}{\kern0pt}{\isacharunderscore}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ c\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ matrix{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}blinfun\ f{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ blinfun{\isacharunderscore}{\kern0pt}eqI\ simp{\isacharcolon}{\kern0pt}\ matrix{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}blinfun{\isacharunderscore}{\kern0pt}mult\ blinfun{\isachardot}{\kern0pt}scaleR{\isacharunderscore}{\kern0pt}left\ \isanewline
-\ \ \ \ \ \ matrix{\isacharunderscore}{\kern0pt}vector{\isacharunderscore}{\kern0pt}mult{\isacharunderscore}{\kern0pt}add{\isacharunderscore}{\kern0pt}rdistrib\ scaleR{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}vector{\isacharunderscore}{\kern0pt}assoc{\isacharbrackleft}{\kern0pt}symmetric{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ matrix{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}blinfun{\isacharunderscore}{\kern0pt}comp{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}matrix{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}blinfun\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}m{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ real{\isacharcircum}{\kern0pt}{\isacharunderscore}{\kern0pt}{\isacharcircum}{\kern0pt}{\isacharunderscore}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharasterisk}{\kern0pt}{\isacharasterisk}{\kern0pt}\ n{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}blinfun\ m{\isacharparenright}{\kern0pt}\ o\isactrlsub L\ {\isacharparenleft}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}blinfun\ n{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ blinfun{\isacharunderscore}{\kern0pt}eqI\ simp{\isacharcolon}{\kern0pt}\ matrix{\isacharunderscore}{\kern0pt}vector{\isacharunderscore}{\kern0pt}mul{\isacharunderscore}{\kern0pt}assoc{\isacharbrackleft}{\kern0pt}symmetric{\isacharbrackright}{\kern0pt}\ matrix{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}blinfun{\isacharunderscore}{\kern0pt}mult{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}comp{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ {\isacharparenleft}{\kern0pt}f\ o\isactrlsub L\ g{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ f{\isacharparenright}{\kern0pt}\ {\isacharasterisk}{\kern0pt}{\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ g{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ matrix{\isacharunderscore}{\kern0pt}eq\ apply{\isacharunderscore}{\kern0pt}bfun{\isacharunderscore}{\kern0pt}inverse\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}mult{\isacharprime}{\kern0pt}{\isacharprime}{\kern0pt}\ matrix{\isacharunderscore}{\kern0pt}vector{\isacharunderscore}{\kern0pt}mul{\isacharunderscore}{\kern0pt}assoc{\isacharbrackleft}{\kern0pt}symmetric{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}id{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ id{\isacharunderscore}{\kern0pt}blinfun\ {\isacharequal}{\kern0pt}\ mat\ {\isadigit{1}}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ Bfun{\isacharunderscore}{\kern0pt}inverse\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}mult{\isacharprime}{\kern0pt}{\isacharprime}{\kern0pt}\ matrix{\isacharunderscore}{\kern0pt}eq{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ matrix{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}blinfun{\isacharunderscore}{\kern0pt}id{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}matrix{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}blinfun\ {\isacharparenleft}{\kern0pt}mat\ {\isadigit{1}}\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isacharparenleft}{\kern0pt}real\ {\isacharcircum}{\kern0pt}{\isacharunderscore}{\kern0pt}{\isacharcircum}{\kern0pt}{\isacharunderscore}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ id{\isacharunderscore}{\kern0pt}blinfun{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ blinfun{\isacharunderscore}{\kern0pt}eqI\ simp{\isacharcolon}{\kern0pt}\ matrix{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}blinfun{\isacharunderscore}{\kern0pt}mult{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ matrix{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}blinfun{\isacharunderscore}{\kern0pt}inv\isactrlsub L{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}invertible\ m{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}matrix{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}blinfun\ {\isacharparenleft}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}inv\ {\isacharparenleft}{\kern0pt}m\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ real{\isacharcircum}{\kern0pt}{\isacharunderscore}{\kern0pt}{\isacharcircum}{\kern0pt}{\isacharunderscore}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ inv\isactrlsub L\ {\isacharparenleft}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}blinfun\ m{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ {\isachardoublequoteopen}invertible\isactrlsub L\ {\isacharparenleft}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}blinfun\ m{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}m\ {\isacharasterisk}{\kern0pt}{\isacharasterisk}{\kern0pt}\ matrix{\isacharunderscore}{\kern0pt}inv\ m\ {\isacharequal}{\kern0pt}\ mat\ {\isadigit{1}}{\isachardoublequoteclose}\ {\isachardoublequoteopen}matrix{\isacharunderscore}{\kern0pt}inv\ m\ {\isacharasterisk}{\kern0pt}{\isacharasterisk}{\kern0pt}\ m\ {\isacharequal}{\kern0pt}\ mat\ {\isadigit{1}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assms\ \isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ matrix{\isacharunderscore}{\kern0pt}inv{\isacharunderscore}{\kern0pt}right\ matrix{\isacharunderscore}{\kern0pt}inv{\isacharunderscore}{\kern0pt}left{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}matrix{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}blinfun\ {\isacharparenleft}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}inv\ m{\isacharparenright}{\kern0pt}\ o\isactrlsub L\ matrix{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}blinfun\ m\ {\isacharequal}{\kern0pt}\ id{\isacharunderscore}{\kern0pt}blinfun{\isachardoublequoteclose}\ \isanewline
-\ \ \ \ {\isachardoublequoteopen}matrix{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}blinfun\ m\ o\isactrlsub L\ matrix{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}blinfun\ {\isacharparenleft}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}inv\ m{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ id{\isacharunderscore}{\kern0pt}blinfun{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ matrix{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}blinfun{\isacharunderscore}{\kern0pt}id\ matrix{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}blinfun{\isacharunderscore}{\kern0pt}comp{\isacharbrackleft}{\kern0pt}symmetric{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}matrix{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}blinfun\ {\isacharparenleft}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}inv\ m{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ inv\isactrlsub L\ {\isacharparenleft}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}blinfun\ m{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ {\isachardoublequoteopen}invertible\isactrlsub L\ {\isacharparenleft}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}blinfun\ m{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ intro{\isacharcolon}{\kern0pt}\ inv\isactrlsub L{\isacharunderscore}{\kern0pt}I{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}inverse{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}invertible\isactrlsub L\ X{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}invertible\ {\isacharparenleft}{\kern0pt}blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ {\isacharparenleft}{\kern0pt}X\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}b{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}finite\ {\isasymRightarrow}\isactrlsub b\ real{\isacharparenright}{\kern0pt}\ {\isasymRightarrow}\isactrlsub L\ {\isacharprime}{\kern0pt}c{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}finite\ {\isasymRightarrow}\isactrlsub b\ real{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isanewline
-\ \ \ \ {\isachardoublequoteopen}blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ {\isacharparenleft}{\kern0pt}inv\isactrlsub L\ X{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ matrix{\isacharunderscore}{\kern0pt}inv\ {\isacharparenleft}{\kern0pt}blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ X{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}X\ o\isactrlsub L\ inv\isactrlsub L\ X\ {\isacharequal}{\kern0pt}\ id{\isacharunderscore}{\kern0pt}blinfun{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ assms{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ X\ {\isacharasterisk}{\kern0pt}{\isacharasterisk}{\kern0pt}\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ {\isacharparenleft}{\kern0pt}inv\isactrlsub L\ X{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ mat\ {\isadigit{1}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}comp\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}id{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}inv\isactrlsub L\ X\ o\isactrlsub L\ X\ {\isacharequal}{\kern0pt}\ id{\isacharunderscore}{\kern0pt}blinfun{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ assms{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isadigit{2}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ {\isacharparenleft}{\kern0pt}inv\isactrlsub L\ X{\isacharparenright}{\kern0pt}\ {\isacharasterisk}{\kern0pt}{\isacharasterisk}{\kern0pt}\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ {\isacharparenleft}{\kern0pt}X{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ mat\ {\isadigit{1}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}comp\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}id{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}invertible\ {\isacharparenleft}{\kern0pt}blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ X{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isadigit{1}}{\isachardoublequoteclose}\ invertible{\isacharunderscore}{\kern0pt}def\ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ {\isacharparenleft}{\kern0pt}inv\isactrlsub L\ X{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ matrix{\isacharunderscore}{\kern0pt}inv\ {\isacharparenleft}{\kern0pt}blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ X{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isadigit{1}}\ {\isadigit{2}}\ matrix{\isacharunderscore}{\kern0pt}inv{\isacharunderscore}{\kern0pt}right\ matrix{\isacharunderscore}{\kern0pt}mul{\isacharunderscore}{\kern0pt}assoc\ matrix{\isacharunderscore}{\kern0pt}mul{\isacharunderscore}{\kern0pt}lid\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}inv{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ {\isacharparenleft}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}blinfun\ f{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ f{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ matrix{\isacharunderscore}{\kern0pt}eq\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}mult{\isacharprime}{\kern0pt}{\isacharprime}{\kern0pt}\ matrix{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}blinfun{\isacharunderscore}{\kern0pt}mult\ bfun{\isachardot}{\kern0pt}Bfun{\isacharunderscore}{\kern0pt}inverse{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ invertible{\isacharunderscore}{\kern0pt}invertible\isactrlsub L{\isacharunderscore}{\kern0pt}I{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}invertible\ {\isacharparenleft}{\kern0pt}blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ f{\isacharparenright}{\kern0pt}\ {\isasymLongrightarrow}\ invertible\isactrlsub L\ f{\isachardoublequoteclose}\isanewline
-\ \ {\isachardoublequoteopen}invertible\isactrlsub L\ {\isacharparenleft}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}blinfun\ X{\isacharparenright}{\kern0pt}\ {\isasymLongrightarrow}\ invertible\ {\isacharparenleft}{\kern0pt}X\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ real{\isacharcircum}{\kern0pt}{\isacharunderscore}{\kern0pt}{\isacharcircum}{\kern0pt}{\isacharunderscore}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ matrix{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}blinfun{\isacharunderscore}{\kern0pt}inv\isactrlsub L{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}inverse{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}\ matrix{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}blinfun{\isacharunderscore}{\kern0pt}inv\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}inv\ \isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ metis{\isacharplus}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ bounded{\isacharunderscore}{\kern0pt}linear{\isacharunderscore}{\kern0pt}blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}bounded{\isacharunderscore}{\kern0pt}linear\ {\isacharparenleft}{\kern0pt}blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}a\ {\isasymRightarrow}\isactrlsub b\ real{\isacharparenright}{\kern0pt}\ {\isasymRightarrow}\isactrlsub L\ {\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}b\ {\isasymRightarrow}\isactrlsub b\ real{\isacharparenright}{\kern0pt}\ {\isasymRightarrow}\ real{\isacharcircum}{\kern0pt}{\isacharprime}{\kern0pt}a{\isacharcircum}{\kern0pt}{\isacharprime}{\kern0pt}b{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}intro\ bounded{\isacharunderscore}{\kern0pt}linear{\isacharunderscore}{\kern0pt}intro{\isacharbrackleft}{\kern0pt}of\ {\isacharunderscore}{\kern0pt}\ {\isachardoublequoteopen}real\ CARD{\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}a{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}finite{\isacharparenright}{\kern0pt}\ {\isacharasterisk}{\kern0pt}\ real\ CARD{\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}b{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}finite{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymAnd}x\ y{\isachardot}{\kern0pt}\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ {\isacharparenleft}{\kern0pt}x\ {\isacharplus}{\kern0pt}\ y{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ x\ {\isacharplus}{\kern0pt}\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ y{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}add\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}scaleR{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymAnd}r\ x{\isachardot}{\kern0pt}\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ {\isacharparenleft}{\kern0pt}r\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ x{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ r\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}def\ matrix{\isacharunderscore}{\kern0pt}def\ blinfun{\isachardot}{\kern0pt}scaleR{\isacharunderscore}{\kern0pt}left\ vec{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}iff{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isacharasterisk}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymAnd}j{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymlambda}i{\isachardot}{\kern0pt}\ if\ i\ {\isacharequal}{\kern0pt}\ j\ then\ {\isadigit{1}}{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}real\ else\ {\isadigit{0}}{\isacharparenright}{\kern0pt}\ {\isasymin}\ bfun{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isacharasterisk}{\kern0pt}{\isacharasterisk}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymAnd}j{\isachardot}{\kern0pt}\ norm\ {\isacharparenleft}{\kern0pt}Bfun\ {\isacharparenleft}{\kern0pt}{\isasymlambda}i{\isachardot}{\kern0pt}\ if\ i\ {\isacharequal}{\kern0pt}\ j\ then\ {\isadigit{1}}{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}real\ else\ {\isadigit{0}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isadigit{1}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ Bfun{\isacharunderscore}{\kern0pt}inverse{\isacharbrackleft}{\kern0pt}OF\ {\isacharasterisk}{\kern0pt}{\isacharbrackright}{\kern0pt}\ norm{\isacharunderscore}{\kern0pt}bfun{\isacharunderscore}{\kern0pt}def{\isacharprime}{\kern0pt}\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ cSup{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}maximum\ {\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}norm\ {\isacharparenleft}{\kern0pt}blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ x{\isacharparenright}{\kern0pt}\ {\isasymle}\ norm\ x\ {\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}real\ CARD{\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}a{\isacharparenright}{\kern0pt}\ {\isacharasterisk}{\kern0pt}\ real\ CARD{\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}b{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{for}\ x\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}a\ {\isasymRightarrow}\isactrlsub b\ real{\isacharparenright}{\kern0pt}\ {\isasymRightarrow}\isactrlsub L\ {\isacharprime}{\kern0pt}b\ {\isasymRightarrow}\isactrlsub b\ real{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}norm\ {\isacharparenleft}{\kern0pt}blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ x{\isacharparenright}{\kern0pt}\ {\isasymle}\ {\isacharparenleft}{\kern0pt}{\isasymSum}i{\isasymin}UNIV{\isachardot}{\kern0pt}\ {\isasymSum}ia{\isasymin}UNIV{\isachardot}{\kern0pt}\ {\isasymbar}{\isacharparenleft}{\kern0pt}x\ {\isacharparenleft}{\kern0pt}bfun{\isachardot}{\kern0pt}Bfun\ {\isacharparenleft}{\kern0pt}{\isasymlambda}i{\isachardot}{\kern0pt}\ if\ i\ {\isacharequal}{\kern0pt}\ ia\ then\ {\isadigit{1}}\ else\ {\isadigit{0}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ i{\isasymbar}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ norm{\isacharunderscore}{\kern0pt}vec{\isacharunderscore}{\kern0pt}def\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}def\ matrix{\isacharunderscore}{\kern0pt}def\ axis{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ vec{\isacharunderscore}{\kern0pt}lambda{\isacharunderscore}{\kern0pt}inverse\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ order{\isachardot}{\kern0pt}trans{\isacharbrackleft}{\kern0pt}OF\ L{\isadigit{2}}{\isacharunderscore}{\kern0pt}set{\isacharunderscore}{\kern0pt}le{\isacharunderscore}{\kern0pt}sum{\isacharunderscore}{\kern0pt}abs{\isacharbrackright}{\kern0pt}\ order{\isachardot}{\kern0pt}trans{\isacharbrackleft}{\kern0pt}OF\ sum{\isacharunderscore}{\kern0pt}mono{\isacharbrackleft}{\kern0pt}OF\ L{\isadigit{2}}{\isacharunderscore}{\kern0pt}set{\isacharunderscore}{\kern0pt}le{\isacharunderscore}{\kern0pt}sum{\isacharunderscore}{\kern0pt}abs{\isacharbrackright}{\kern0pt}{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isasymle}\ {\isacharparenleft}{\kern0pt}{\isasymSum}i{\isasymin}{\isacharparenleft}{\kern0pt}UNIV{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}{\isacharprime}{\kern0pt}b\ set{\isacharparenright}{\kern0pt}{\isachardot}{\kern0pt}\ {\isasymSum}ia{\isasymin}{\isacharparenleft}{\kern0pt}UNIV\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isacharprime}{\kern0pt}a\ set{\isacharparenright}{\kern0pt}{\isachardot}{\kern0pt}\ norm\ x{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ norm{\isacharunderscore}{\kern0pt}blinfun\ abs{\isacharunderscore}{\kern0pt}le{\isacharunderscore}{\kern0pt}norm{\isacharunderscore}{\kern0pt}bfun\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}fastforce\ simp{\isacharcolon}{\kern0pt}\ {\isacharasterisk}{\kern0pt}{\isacharasterisk}{\kern0pt}\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ sum{\isacharunderscore}{\kern0pt}mono\ intro{\isacharcolon}{\kern0pt}\ order{\isachardot}{\kern0pt}trans{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isacharequal}{\kern0pt}\ norm\ x\ {\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}real\ CARD{\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}a{\isacharparenright}{\kern0pt}\ {\isacharasterisk}{\kern0pt}\ real\ CARD{\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}b{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \isacommand{finally}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isacommand{{\isachardot}{\kern0pt}}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ summable{\isacharunderscore}{\kern0pt}blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}summable\ {\isacharparenleft}{\kern0pt}f\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ nat\ {\isasymRightarrow}\ {\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}c{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}finite\ {\isasymRightarrow}\isactrlsub b\ {\isacharunderscore}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymRightarrow}\isactrlsub L\ {\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}c\ {\isasymRightarrow}\isactrlsub b\ {\isacharunderscore}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}summable\ {\isacharparenleft}{\kern0pt}{\isasymlambda}i{\isachardot}{\kern0pt}\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ {\isacharparenleft}{\kern0pt}f\ i{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ assms\ bounded{\isacharunderscore}{\kern0pt}linear{\isachardot}{\kern0pt}summable\ bounded{\isacharunderscore}{\kern0pt}linear{\isacharunderscore}{\kern0pt}blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{abbreviation}\isamarkupfalse%
-\ {\isachardoublequoteopen}nonneg{\isacharunderscore}{\kern0pt}blinfun\ Q\ {\isasymequiv}\ {\isadigit{0}}\ {\isasymle}\ {\isacharparenleft}{\kern0pt}blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ Q{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ nonneg{\isacharunderscore}{\kern0pt}blinfun{\isacharunderscore}{\kern0pt}mono{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}nonneg{\isacharunderscore}{\kern0pt}blinfun\ Q\ {\isasymLongrightarrow}\ u\ {\isasymle}\ v\ {\isasymLongrightarrow}\ Q\ u\ {\isasymle}\ Q\ v{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ nonneg{\isacharunderscore}{\kern0pt}mat{\isacharunderscore}{\kern0pt}mono{\isacharbrackleft}{\kern0pt}of\ {\isachardoublequoteopen}blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ Q{\isachardoublequoteclose}\ {\isachardoublequoteopen}vec{\isacharunderscore}{\kern0pt}lambda\ u{\isachardoublequoteclose}\ {\isachardoublequoteopen}vec{\isacharunderscore}{\kern0pt}lambda\ v{\isachardoublequoteclose}{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}fastforce\ simp{\isacharcolon}{\kern0pt}\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}mult{\isacharprime}{\kern0pt}{\isacharprime}{\kern0pt}\ apply{\isacharunderscore}{\kern0pt}bfun{\isacharunderscore}{\kern0pt}inverse\ Finite{\isacharunderscore}{\kern0pt}Cartesian{\isacharunderscore}{\kern0pt}Product{\isachardot}{\kern0pt}less{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}vec{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ nonneg{\isacharunderscore}{\kern0pt}blinfun{\isacharunderscore}{\kern0pt}nonneg{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}nonneg{\isacharunderscore}{\kern0pt}blinfun\ Q\ {\isasymLongrightarrow}\ {\isadigit{0}}\ {\isasymle}\ v\ {\isasymLongrightarrow}\ {\isadigit{0}}\ {\isasymle}\ Q\ v{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ nonneg{\isacharunderscore}{\kern0pt}blinfun{\isacharunderscore}{\kern0pt}mono\ blinfun{\isachardot}{\kern0pt}zero{\isacharunderscore}{\kern0pt}right\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ metis%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ nonneg{\isacharunderscore}{\kern0pt}id{\isacharunderscore}{\kern0pt}blinfun{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}nonneg{\isacharunderscore}{\kern0pt}blinfun\ id{\isacharunderscore}{\kern0pt}blinfun{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}id{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ norm{\isacharunderscore}{\kern0pt}nonneg{\isacharunderscore}{\kern0pt}blinfun{\isacharunderscore}{\kern0pt}one{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}{\isadigit{0}}\ {\isasymle}\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ X{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}norm\ X\ {\isacharequal}{\kern0pt}\ norm\ {\isacharparenleft}{\kern0pt}blinfun{\isacharunderscore}{\kern0pt}apply\ X\ {\isadigit{1}}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ norm{\isacharunderscore}{\kern0pt}blinfun{\isacharunderscore}{\kern0pt}mono{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}one\ assms\ nonneg{\isacharunderscore}{\kern0pt}blinfun{\isacharunderscore}{\kern0pt}nonneg{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ matrix{\isacharunderscore}{\kern0pt}le{\isacharunderscore}{\kern0pt}norm{\isacharunderscore}{\kern0pt}mono{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}{\isadigit{0}}\ {\isasymle}\ {\isacharparenleft}{\kern0pt}blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ C{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isakeyword{and}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ C{\isacharparenright}{\kern0pt}\ {\isasymle}\ {\isacharparenleft}{\kern0pt}blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ D{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}norm\ C\ {\isasymle}\ norm\ D{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isadigit{0}}\ {\isasymle}\ C\ {\isadigit{1}}{\isachardoublequoteclose}\ {\isachardoublequoteopen}{\isadigit{0}}\ {\isasymle}\ D\ {\isadigit{1}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assms\ zero{\isacharunderscore}{\kern0pt}le{\isacharunderscore}{\kern0pt}one\ \isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}fastforce\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ nonneg{\isacharunderscore}{\kern0pt}blinfun{\isacharunderscore}{\kern0pt}nonneg{\isacharparenright}{\kern0pt}{\isacharplus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymAnd}v{\isachardot}{\kern0pt}\ v\ {\isasymge}\ {\isadigit{0}}\ {\isasymLongrightarrow}\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ C\ {\isacharasterisk}{\kern0pt}v\ v\ {\isasymle}\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ D\ {\isacharasterisk}{\kern0pt}v\ v{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assms\ nonneg{\isacharunderscore}{\kern0pt}mat{\isacharunderscore}{\kern0pt}mono{\isacharbrackleft}{\kern0pt}of\ {\isachardoublequoteopen}blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ D\ {\isacharminus}{\kern0pt}\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ C{\isachardoublequoteclose}{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}fastforce\ simp{\isacharcolon}{\kern0pt}\ matrix{\isacharunderscore}{\kern0pt}vector{\isacharunderscore}{\kern0pt}mult{\isacharunderscore}{\kern0pt}diff{\isacharunderscore}{\kern0pt}rdistrib{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isacharasterisk}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymAnd}v{\isachardot}{\kern0pt}\ v\ {\isasymge}\ {\isadigit{0}}\ {\isasymLongrightarrow}\ C\ v\ {\isasymle}\ D\ v{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ less{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}vec{\isacharunderscore}{\kern0pt}def\ less{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}bfun{\isacharunderscore}{\kern0pt}def\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}mult{\isacharbrackleft}{\kern0pt}symmetric{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assms{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}\ assms{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}\ {\isacartoucheopen}{\isadigit{0}}\ {\isasymle}\ C\ {\isadigit{1}}{\isacartoucheclose}\ {\isacartoucheopen}{\isadigit{0}}\ {\isasymle}\ D\ {\isadigit{1}}{\isacartoucheclose}\ less{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}bfunD{\isacharbrackleft}{\kern0pt}OF\ {\isacharasterisk}{\kern0pt}{\isacharcomma}{\kern0pt}\ of\ {\isadigit{1}}{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}fastforce\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ cSUP{\isacharunderscore}{\kern0pt}mono\ simp{\isacharcolon}{\kern0pt}\ norm{\isacharunderscore}{\kern0pt}nonneg{\isacharunderscore}{\kern0pt}blinfun{\isacharunderscore}{\kern0pt}one\ norm{\isacharunderscore}{\kern0pt}bfun{\isacharunderscore}{\kern0pt}def{\isacharprime}{\kern0pt}\ less{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}bfun{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}matpow{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ {\isacharparenleft}{\kern0pt}X\ {\isacharcircum}{\kern0pt}{\isacharcircum}{\kern0pt}\ i{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ matpow\ {\isacharparenleft}{\kern0pt}blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ X{\isacharparenright}{\kern0pt}\ i{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}induction\ i{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}id\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}comp\ blinfunpow{\isacharunderscore}{\kern0pt}assoc\ simp\ del{\isacharcolon}{\kern0pt}\ blinfunpow{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ nonneg{\isacharunderscore}{\kern0pt}blinfun{\isacharunderscore}{\kern0pt}iff{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}nonneg{\isacharunderscore}{\kern0pt}blinfun\ X\ {\isasymlongleftrightarrow}\ {\isacharparenleft}{\kern0pt}{\isasymforall}v{\isasymge}{\isadigit{0}}{\isachardot}{\kern0pt}\ X\ v\ {\isasymge}\ {\isadigit{0}}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ nonneg{\isacharunderscore}{\kern0pt}mat{\isacharunderscore}{\kern0pt}iff{\isacharbrackleft}{\kern0pt}of\ {\isachardoublequoteopen}blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ X{\isachardoublequoteclose}{\isacharbrackright}{\kern0pt}\ nonneg{\isacharunderscore}{\kern0pt}blinfun{\isacharunderscore}{\kern0pt}nonneg\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}mult{\isacharprime}{\kern0pt}{\isacharprime}{\kern0pt}\ bfun{\isachardot}{\kern0pt}Bfun{\isacharunderscore}{\kern0pt}inverse\ less{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}vec{\isacharunderscore}{\kern0pt}def\ less{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}bfun{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ blinfun{\isacharunderscore}{\kern0pt}apply{\isacharunderscore}{\kern0pt}mono{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isadigit{0}}{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}real{\isacharcircum}{\kern0pt}{\isacharunderscore}{\kern0pt}{\isacharcircum}{\kern0pt}{\isacharunderscore}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymle}\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ X\ {\isasymLongrightarrow}\ {\isadigit{0}}\ {\isasymle}\ v\ {\isasymLongrightarrow}\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ X\ {\isasymle}\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ Y\ {\isasymLongrightarrow}\ X\ v\ {\isasymle}\ Y\ v{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ blinfun{\isachardot}{\kern0pt}diff{\isacharunderscore}{\kern0pt}left\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}diff\ diff{\isacharunderscore}{\kern0pt}ge{\isacharunderscore}{\kern0pt}{\isadigit{0}}{\isacharunderscore}{\kern0pt}iff{\isacharunderscore}{\kern0pt}ge\ nonneg{\isacharunderscore}{\kern0pt}blinfun{\isacharunderscore}{\kern0pt}nonneg{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-%
-\isadelimtheory
-\isanewline
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{end}\isamarkupfalse%
-%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\end{isabellebody}%
-\endinput
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diff --git a/thys/MDP-Algorithms/output/outline/Code_DP.tex b/thys/MDP-Algorithms/output/outline/Code_DP.tex
deleted file mode 100644
--- a/thys/MDP-Algorithms/output/outline/Code_DP.tex
+++ /dev/null
@@ -1,3010 +0,0 @@
-%
-\begin{isabellebody}%
-\setisabellecontext{Code{\isacharunderscore}{\kern0pt}DP}%
-%
-\isadelimtheory
-\isanewline
-\isanewline
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{theory}\isamarkupfalse%
-\ Code{\isacharunderscore}{\kern0pt}DP\isanewline
-\ \ \isakeyword{imports}\ \isanewline
-\ \ \ \ Value{\isacharunderscore}{\kern0pt}Iteration\isanewline
-\ \ \ \ Policy{\isacharunderscore}{\kern0pt}Iteration\isanewline
-\ \ \ \ Modified{\isacharunderscore}{\kern0pt}Policy{\isacharunderscore}{\kern0pt}Iteration\isanewline
-\ \ \ \ Splitting{\isacharunderscore}{\kern0pt}Methods\isanewline
-\isanewline
-{\isachardoublequoteopen}HOL{\isacharminus}{\kern0pt}Library{\isachardot}{\kern0pt}Code{\isacharunderscore}{\kern0pt}Target{\isacharunderscore}{\kern0pt}Numeral{\isachardoublequoteclose}\isanewline
-{\isachardoublequoteopen}Gauss{\isacharunderscore}{\kern0pt}Jordan{\isachardot}{\kern0pt}Code{\isacharunderscore}{\kern0pt}Generation{\isacharunderscore}{\kern0pt}IArrays{\isachardoublequoteclose}\isanewline
-\isakeyword{begin}%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsection{Code Generation for MDP Algorithms%
-}
-\isamarkuptrue%
-%
-\isamarkupsubsection{Least Argmax%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{lemma}\isamarkupfalse%
-\ least{\isacharunderscore}{\kern0pt}list{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}sorted\ xs{\isachardoublequoteclose}\ {\isachardoublequoteopen}{\isasymexists}x\ {\isasymin}\ set\ xs{\isachardot}{\kern0pt}\ P\ x{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}LEAST\ x\ {\isasymin}\ set\ xs{\isachardot}{\kern0pt}\ P\ x{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ the\ {\isacharparenleft}{\kern0pt}find\ P\ xs{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ assms\isanewline
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}induction\ xs{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{case}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}Cons\ a\ xs{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}case\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}cases\ {\isachardoublequoteopen}P\ a{\isachardoublequoteclose}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{case}\isamarkupfalse%
-\ False\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}LEAST\ x\ {\isasymin}\ set\ {\isacharparenleft}{\kern0pt}a\ {\isacharhash}{\kern0pt}\ xs{\isacharparenright}{\kern0pt}{\isachardot}{\kern0pt}\ P\ x{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}LEAST\ x\ {\isasymin}\ set\ xs{\isachardot}{\kern0pt}\ P\ x{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ False\ Cons{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\ metis\isanewline
-\ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ False\ Cons\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ intro{\isacharcolon}{\kern0pt}\ Least{\isacharunderscore}{\kern0pt}equality{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ {\isachardoublequoteopen}least{\isacharunderscore}{\kern0pt}enum\ P\ {\isacharequal}{\kern0pt}\ the\ {\isacharparenleft}{\kern0pt}find\ P\ {\isacharparenleft}{\kern0pt}sorted{\isacharunderscore}{\kern0pt}list{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}set\ {\isacharparenleft}{\kern0pt}UNIV\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}b{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}finite{\isacharcomma}{\kern0pt}\ linorder{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ set{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ least{\isacharunderscore}{\kern0pt}enum{\isacharunderscore}{\kern0pt}eq{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymexists}x{\isachardot}{\kern0pt}\ P\ x\ {\isasymLongrightarrow}\ least{\isacharunderscore}{\kern0pt}enum\ P\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}LEAST\ x{\isachardot}{\kern0pt}\ P\ x{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ least{\isacharunderscore}{\kern0pt}list{\isacharbrackleft}{\kern0pt}symmetric{\isacharbrackright}{\kern0pt}\ least{\isacharunderscore}{\kern0pt}enum{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ {\isachardoublequoteopen}least{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}list\ f\ init\ xs\ {\isacharequal}{\kern0pt}\ \isanewline
-\ \ foldl\ {\isacharparenleft}{\kern0pt}{\isasymlambda}{\isacharparenleft}{\kern0pt}am{\isacharcomma}{\kern0pt}\ m{\isacharparenright}{\kern0pt}\ x{\isachardot}{\kern0pt}\ if\ f\ x\ {\isachargreater}{\kern0pt}\ m\ then\ {\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ f\ x{\isacharparenright}{\kern0pt}\ else\ {\isacharparenleft}{\kern0pt}am{\isacharcomma}{\kern0pt}\ m{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ init\ xs{\isachardoublequoteclose}\ \ \isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ snd{\isacharunderscore}{\kern0pt}least{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}list{\isacharcolon}{\kern0pt}\ \isanewline
-\ \ {\isachardoublequoteopen}snd\ {\isacharparenleft}{\kern0pt}least{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}list\ f\ {\isacharparenleft}{\kern0pt}n{\isacharcomma}{\kern0pt}\ f\ n{\isacharparenright}{\kern0pt}\ xs{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}MAX\ x\ {\isasymin}\ insert\ n\ {\isacharparenleft}{\kern0pt}set\ xs{\isacharparenright}{\kern0pt}{\isachardot}{\kern0pt}\ f\ x{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ least{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}list{\isacharunderscore}{\kern0pt}def\isanewline
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}induction\ xs\ arbitrary{\isacharcolon}{\kern0pt}\ n{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{case}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}Cons\ a\ xs{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{then}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}case\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}cases\ {\isachardoublequoteopen}xs\ {\isacharequal}{\kern0pt}\ {\isacharbrackleft}{\kern0pt}{\isacharbrackright}{\kern0pt}{\isachardoublequoteclose}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}fastforce\ simp{\isacharcolon}{\kern0pt}\ max{\isachardot}{\kern0pt}assoc{\isacharbrackleft}{\kern0pt}symmetric{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharplus}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ least{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}list{\isacharunderscore}{\kern0pt}snd{\isacharunderscore}{\kern0pt}fst{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}snd\ {\isacharparenleft}{\kern0pt}least{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}list\ f\ {\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ f\ x{\isacharparenright}{\kern0pt}\ xs{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ f\ {\isacharparenleft}{\kern0pt}fst\ {\isacharparenleft}{\kern0pt}least{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}list\ f\ {\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ f\ x{\isacharparenright}{\kern0pt}\ xs{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}induction\ xs\ arbitrary{\isacharcolon}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ least{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}list{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ fst{\isacharunderscore}{\kern0pt}least{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}list{\isacharcolon}{\kern0pt}\ \isanewline
-\ \ \isakeyword{fixes}\ f\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharunderscore}{\kern0pt}\ {\isasymRightarrow}\ {\isacharunderscore}{\kern0pt}\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ linorder{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}sorted\ {\isacharparenleft}{\kern0pt}n{\isacharhash}{\kern0pt}xs{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}fst\ {\isacharparenleft}{\kern0pt}least{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}list\ f\ {\isacharparenleft}{\kern0pt}n{\isacharcomma}{\kern0pt}\ f\ n{\isacharparenright}{\kern0pt}\ xs{\isacharparenright}{\kern0pt}\ \isanewline
-\ \ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}LEAST\ x{\isachardot}{\kern0pt}\ is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max\ f\ {\isacharparenleft}{\kern0pt}{\isasymlambda}x{\isachardot}{\kern0pt}\ x\ {\isasymin}\ insert\ n\ {\isacharparenleft}{\kern0pt}set\ xs{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ x{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ least{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}list{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \isacommand{using}\isamarkupfalse%
-\ assms\ \isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}induction\ xs\ arbitrary{\isacharcolon}{\kern0pt}\ n{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{case}\isamarkupfalse%
-\ Nil\isanewline
-\ \ \isacommand{then}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}case\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}def\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ Least{\isacharunderscore}{\kern0pt}equality{\isacharbrackleft}{\kern0pt}symmetric{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{case}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}Cons\ a\ xs{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}sorted\ {\isacharparenleft}{\kern0pt}a{\isacharhash}{\kern0pt}xs{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{then}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}case\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}cases\ {\isachardoublequoteopen}f\ a\ {\isachargreater}{\kern0pt}\ f\ n{\isachardoublequoteclose}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{case}\isamarkupfalse%
-\ True\isanewline
-\ \ \ \ \isacommand{then}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}fastforce\ simp{\isacharcolon}{\kern0pt}\ is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}def\ Cons{\isachardot}{\kern0pt}IH{\isacharbrackleft}{\kern0pt}OF\ {\isacartoucheopen}sorted\ {\isacharparenleft}{\kern0pt}a{\isacharhash}{\kern0pt}xs{\isacharparenright}{\kern0pt}{\isacartoucheclose}{\isacharbrackright}{\kern0pt}\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ cong{\isacharbrackleft}{\kern0pt}of\ Least{\isacharcomma}{\kern0pt}\ OF\ refl{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{case}\isamarkupfalse%
-\ False\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}LEAST\ b{\isachardot}{\kern0pt}\ is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max\ f\ {\isacharparenleft}{\kern0pt}{\isasymlambda}x{\isachardot}{\kern0pt}\ x\ {\isasymin}\ insert\ n\ {\isacharparenleft}{\kern0pt}set\ {\isacharparenleft}{\kern0pt}a\ {\isacharhash}{\kern0pt}\ xs{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ b{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}LEAST\ b{\isachardot}{\kern0pt}\ is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max\ f\ {\isacharparenleft}{\kern0pt}{\isasymlambda}x{\isachardot}{\kern0pt}\ x\ {\isasymin}\ {\isacharparenleft}{\kern0pt}set\ {\isacharparenleft}{\kern0pt}n\ {\isacharhash}{\kern0pt}\ xs{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ b{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}cases\ {\isachardoublequoteopen}is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max\ f\ {\isacharparenleft}{\kern0pt}{\isasymlambda}x{\isachardot}{\kern0pt}\ x\ {\isasymin}\ set\ {\isacharparenleft}{\kern0pt}n\ {\isacharhash}{\kern0pt}a{\isacharhash}{\kern0pt}\ xs{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ a{\isachardoublequoteclose}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{case}\isamarkupfalse%
-\ True\isanewline
-\ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}LEAST\ b{\isachardot}{\kern0pt}\ is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max\ f\ {\isacharparenleft}{\kern0pt}{\isasymlambda}x{\isachardot}{\kern0pt}\ x\ {\isasymin}\ \ {\isacharparenleft}{\kern0pt}set\ {\isacharparenleft}{\kern0pt}n{\isacharhash}{\kern0pt}a\ {\isacharhash}{\kern0pt}\ xs{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ b{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ n{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ Cons\ False\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}fastforce\ simp{\isacharcolon}{\kern0pt}\ is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}linorder\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ Least{\isacharunderscore}{\kern0pt}equality{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ False\ True\ Cons\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}fastforce\ simp{\isacharcolon}{\kern0pt}\ is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}linorder\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ Least{\isacharunderscore}{\kern0pt}equality{\isacharbrackleft}{\kern0pt}symmetric{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \isacommand{case}\isamarkupfalse%
-\ False\isanewline
-\ \ \ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}fastforce\ simp{\isacharcolon}{\kern0pt}\ not{\isacharunderscore}{\kern0pt}less\ is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}linorder\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ cong{\isacharbrackleft}{\kern0pt}of\ Least{\isacharcomma}{\kern0pt}\ OF\ refl{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ False\ Cons\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp\ add{\isacharcolon}{\kern0pt}\ Cons{\isachardot}{\kern0pt}IH{\isacharbrackleft}{\kern0pt}OF\ {\isacartoucheopen}sorted\ {\isacharparenleft}{\kern0pt}a{\isacharhash}{\kern0pt}xs{\isacharparenright}{\kern0pt}{\isacartoucheclose}{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ {\isachardoublequoteopen}least{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}enum\ f\ X\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}\isanewline
-\ \ let\ xs\ {\isacharequal}{\kern0pt}\ sorted{\isacharunderscore}{\kern0pt}list{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}set\ {\isacharparenleft}{\kern0pt}X\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharunderscore}{\kern0pt}\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}finite{\isacharcomma}{\kern0pt}\ linorder{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ set{\isacharparenright}{\kern0pt}\ in\isanewline
-\ \ fst\ {\isacharparenleft}{\kern0pt}least{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}list\ f\ {\isacharparenleft}{\kern0pt}hd\ xs{\isacharcomma}{\kern0pt}\ f\ {\isacharparenleft}{\kern0pt}hd\ xs{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}tl\ xs{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ {\isachardoublequoteopen}least{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}enum\ f\ X\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}\isanewline
-\ \ let\ xs\ {\isacharequal}{\kern0pt}\ sorted{\isacharunderscore}{\kern0pt}list{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}set\ {\isacharparenleft}{\kern0pt}X\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharunderscore}{\kern0pt}\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}finite{\isacharcomma}{\kern0pt}\ linorder{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ set{\isacharparenright}{\kern0pt}\ in\isanewline
-\ \ {\isacharparenleft}{\kern0pt}least{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}list\ f\ {\isacharparenleft}{\kern0pt}hd\ xs{\isacharcomma}{\kern0pt}\ f\ {\isacharparenleft}{\kern0pt}hd\ xs{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}tl\ xs{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ least{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}enum{\isacharunderscore}{\kern0pt}correct{\isacharcolon}{\kern0pt}\ \isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}X\ {\isasymnoteq}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}\isanewline
-\ \ {\isacharparenleft}{\kern0pt}least{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}enum\ {\isacharparenleft}{\kern0pt}f\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isacharunderscore}{\kern0pt}\ {\isasymRightarrow}\ {\isacharparenleft}{\kern0pt}{\isacharunderscore}{\kern0pt}\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ linorder{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ X{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}LEAST\ x{\isachardot}{\kern0pt}\ is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max\ f\ {\isacharparenleft}{\kern0pt}{\isasymlambda}x{\isachardot}{\kern0pt}\ x\ {\isasymin}\ X{\isacharparenright}{\kern0pt}\ x{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isacharasterisk}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}xs\ {\isasymnoteq}\ {\isacharbrackleft}{\kern0pt}{\isacharbrackright}{\kern0pt}\ {\isasymLongrightarrow}\ {\isacharparenleft}{\kern0pt}x\ {\isacharequal}{\kern0pt}\ hd\ xs\ {\isasymor}\ x\ {\isasymin}\ set\ {\isacharparenleft}{\kern0pt}tl\ xs{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymlongleftrightarrow}\ x\ {\isasymin}\ set\ xs{\isachardoublequoteclose}\ \isakeyword{for}\ x\ xs\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ list{\isachardot}{\kern0pt}set{\isacharunderscore}{\kern0pt}sel\ list{\isachardot}{\kern0pt}exhaust{\isacharunderscore}{\kern0pt}sel\ set{\isacharunderscore}{\kern0pt}ConsD\ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ least{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}enum{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ \ assms\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ Let{\isacharunderscore}{\kern0pt}def\ fst{\isacharunderscore}{\kern0pt}least{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}list\ {\isacharasterisk}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ least{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}enum{\isacharunderscore}{\kern0pt}correct{\isadigit{1}}{\isacharcolon}{\kern0pt}\ \isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}X\ {\isasymnoteq}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\ \isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}fst\ {\isacharparenleft}{\kern0pt}least{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}enum\ {\isacharparenleft}{\kern0pt}f\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isacharunderscore}{\kern0pt}\ {\isasymRightarrow}\ {\isacharparenleft}{\kern0pt}{\isacharunderscore}{\kern0pt}\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ linorder{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ X{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}LEAST\ x{\isachardot}{\kern0pt}\ is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max\ f\ {\isacharparenleft}{\kern0pt}{\isasymlambda}x{\isachardot}{\kern0pt}\ x\ {\isasymin}\ X{\isacharparenright}{\kern0pt}\ x{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isacharasterisk}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}xs\ {\isasymnoteq}\ {\isacharbrackleft}{\kern0pt}{\isacharbrackright}{\kern0pt}\ {\isasymLongrightarrow}\ {\isacharparenleft}{\kern0pt}x\ {\isacharequal}{\kern0pt}\ hd\ xs\ {\isasymor}\ x\ {\isasymin}\ set\ {\isacharparenleft}{\kern0pt}tl\ xs{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymlongleftrightarrow}\ x\ {\isasymin}\ set\ xs{\isachardoublequoteclose}\ \isakeyword{for}\ x\ xs\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ list{\isachardot}{\kern0pt}set{\isacharunderscore}{\kern0pt}sel\ list{\isachardot}{\kern0pt}exhaust{\isacharunderscore}{\kern0pt}sel\ set{\isacharunderscore}{\kern0pt}ConsD\ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assms\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ least{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}enum{\isacharunderscore}{\kern0pt}def\ Let{\isacharunderscore}{\kern0pt}def\ fst{\isacharunderscore}{\kern0pt}least{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}list\ {\isacharasterisk}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ least{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}enum{\isacharunderscore}{\kern0pt}correct{\isadigit{2}}{\isacharcolon}{\kern0pt}\ \isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}X\ {\isasymnoteq}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}snd\ {\isacharparenleft}{\kern0pt}least{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}enum\ {\isacharparenleft}{\kern0pt}f\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isacharunderscore}{\kern0pt}\ {\isasymRightarrow}\ {\isacharparenleft}{\kern0pt}{\isacharunderscore}{\kern0pt}\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ linorder{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ X{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}MAX\ x\ {\isasymin}\ X{\isachardot}{\kern0pt}\ f\ x{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isacharasterisk}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}xs\ {\isasymnoteq}\ {\isacharbrackleft}{\kern0pt}{\isacharbrackright}{\kern0pt}\ {\isasymLongrightarrow}\ insert\ {\isacharparenleft}{\kern0pt}hd\ xs{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}set\ {\isacharparenleft}{\kern0pt}tl\ xs{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ set\ xs{\isachardoublequoteclose}\ \isakeyword{for}\ xs\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ list{\isachardot}{\kern0pt}exhaust{\isacharunderscore}{\kern0pt}sel\ list{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isadigit{5}}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assms\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ least{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}enum{\isacharunderscore}{\kern0pt}def\ Let{\isacharunderscore}{\kern0pt}def\ snd{\isacharunderscore}{\kern0pt}least{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}list\ {\isacharasterisk}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Functions as Vectors%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{typedef}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}a{\isacharcomma}{\kern0pt}\ {\isacharprime}{\kern0pt}b{\isacharparenright}{\kern0pt}\ Fun\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}UNIV\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}a\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}b{\isacharparenright}{\kern0pt}\ set{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ blast%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{setup{\isacharunderscore}{\kern0pt}lifting}\isamarkupfalse%
-\ type{\isacharunderscore}{\kern0pt}definition{\isacharunderscore}{\kern0pt}Fun\isanewline
-\isanewline
-\isacommand{lift{\isacharunderscore}{\kern0pt}definition}\isamarkupfalse%
-\ to{\isacharunderscore}{\kern0pt}Fun\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}a\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}b{\isacharparenright}{\kern0pt}\ {\isasymRightarrow}\ {\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}a{\isacharcomma}{\kern0pt}\ {\isacharprime}{\kern0pt}b{\isacharparenright}{\kern0pt}\ Fun{\isachardoublequoteclose}\ \isakeyword{is}\ id%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{{\isachardot}{\kern0pt}}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ {\isachardoublequoteopen}fun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}vec\ {\isacharparenleft}{\kern0pt}v\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}a{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}finite{\isacharcomma}{\kern0pt}\ {\isacharprime}{\kern0pt}b{\isacharparenright}{\kern0pt}\ Fun{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ vec{\isacharunderscore}{\kern0pt}lambda\ {\isacharparenleft}{\kern0pt}Rep{\isacharunderscore}{\kern0pt}Fun\ v{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{lift{\isacharunderscore}{\kern0pt}definition}\isamarkupfalse%
-\ vec{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}fun\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}b{\isacharcircum}{\kern0pt}{\isacharprime}{\kern0pt}a\ {\isasymRightarrow}\ {\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}a{\isacharcomma}{\kern0pt}\ {\isacharprime}{\kern0pt}b{\isacharparenright}{\kern0pt}\ Fun{\isachardoublequoteclose}\ \isakeyword{is}\ vec{\isacharunderscore}{\kern0pt}nth%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{{\isachardot}{\kern0pt}}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ Fun{\isacharunderscore}{\kern0pt}inverse{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}Rep{\isacharunderscore}{\kern0pt}Fun\ {\isacharparenleft}{\kern0pt}Abs{\isacharunderscore}{\kern0pt}Fun\ f{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ f{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ Abs{\isacharunderscore}{\kern0pt}Fun{\isacharunderscore}{\kern0pt}inverse\ \isacommand{by}\isamarkupfalse%
-\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lift{\isacharunderscore}{\kern0pt}definition}\isamarkupfalse%
-\ zero{\isacharunderscore}{\kern0pt}Fun\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}a{\isacharcomma}{\kern0pt}\ {\isacharprime}{\kern0pt}b{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}zero{\isacharparenright}{\kern0pt}\ Fun{\isachardoublequoteclose}\ \isakeyword{is}\ {\isadigit{0}}%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{{\isachardot}{\kern0pt}}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{code{\isacharunderscore}{\kern0pt}datatype}\isamarkupfalse%
-\ vec{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}fun\isanewline
-\isanewline
-\isacommand{lemmas}\isamarkupfalse%
-\ vec{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}fun{\isachardot}{\kern0pt}rep{\isacharunderscore}{\kern0pt}eq{\isacharbrackleft}{\kern0pt}code{\isacharbrackright}{\kern0pt}\isanewline
-\isanewline
-\isacommand{instantiation}\isamarkupfalse%
-\ Fun\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isacharparenleft}{\kern0pt}enum{\isacharcomma}{\kern0pt}\ equal{\isacharparenright}{\kern0pt}\ equal\isanewline
-\isakeyword{begin}\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ {\isachardoublequoteopen}equal{\isacharunderscore}{\kern0pt}Fun\ {\isacharparenleft}{\kern0pt}f\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}a{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}enum{\isacharcomma}{\kern0pt}\ {\isacharprime}{\kern0pt}b{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}equal{\isacharparenright}{\kern0pt}\ Fun{\isacharparenright}{\kern0pt}\ g\ {\isacharequal}{\kern0pt}\ \ {\isacharparenleft}{\kern0pt}Rep{\isacharunderscore}{\kern0pt}Fun\ f\ {\isacharequal}{\kern0pt}\ Rep{\isacharunderscore}{\kern0pt}Fun\ g{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isacommand{instance}\isamarkupfalse%
-\ \isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ standard\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ equal{\isacharunderscore}{\kern0pt}Fun{\isacharunderscore}{\kern0pt}def\ Rep{\isacharunderscore}{\kern0pt}Fun{\isacharunderscore}{\kern0pt}inject{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isacommand{end}\isamarkupfalse%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Bounded Functions as Vectors%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{lemma}\isamarkupfalse%
-\ Bfun{\isacharunderscore}{\kern0pt}inverse{\isacharunderscore}{\kern0pt}fin{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}apply{\isacharunderscore}{\kern0pt}bfun\ {\isacharparenleft}{\kern0pt}Bfun\ {\isacharparenleft}{\kern0pt}f\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isacharprime}{\kern0pt}c\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ finite\ {\isasymRightarrow}\ {\isacharunderscore}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ f{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ finite\ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}fastforce\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ Bfun{\isacharunderscore}{\kern0pt}inverse\ simp{\isacharcolon}{\kern0pt}\ bfun{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ {\isachardoublequoteopen}bfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}vec\ {\isacharparenleft}{\kern0pt}v\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}a{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}finite{\isacharparenright}{\kern0pt}\ {\isasymRightarrow}\isactrlsub b\ {\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}b{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}metric{\isacharunderscore}{\kern0pt}space{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ vec{\isacharunderscore}{\kern0pt}lambda\ v{\isachardoublequoteclose}\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ {\isachardoublequoteopen}vec{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}bfun\ v\ {\isacharequal}{\kern0pt}\ Bfun\ {\isacharparenleft}{\kern0pt}vec{\isacharunderscore}{\kern0pt}nth\ v{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{code{\isacharunderscore}{\kern0pt}datatype}\isamarkupfalse%
-\ vec{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}bfun\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ apply{\isacharunderscore}{\kern0pt}bfun{\isacharunderscore}{\kern0pt}vec{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}bfun{\isacharbrackleft}{\kern0pt}code{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}apply{\isacharunderscore}{\kern0pt}bfun\ {\isacharparenleft}{\kern0pt}vec{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}bfun\ f{\isacharparenright}{\kern0pt}\ x\ {\isacharequal}{\kern0pt}\ f\ {\isachardollar}{\kern0pt}\ x{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ vec{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}bfun{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ {\isacharbrackleft}{\kern0pt}code{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isadigit{0}}\ {\isacharequal}{\kern0pt}\ vec{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}bfun\ {\isadigit{0}}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ vec{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}bfun{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{IArrays with Lengths in the Type%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{typedef}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}s\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ mod{\isacharunderscore}{\kern0pt}type{\isacharcomma}{\kern0pt}\ {\isacharprime}{\kern0pt}a{\isacharparenright}{\kern0pt}\ iarray{\isacharunderscore}{\kern0pt}type\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}arr\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isacharprime}{\kern0pt}a\ iarray{\isachardot}{\kern0pt}\ IArray{\isachardot}{\kern0pt}length\ arr\ {\isacharequal}{\kern0pt}\ CARD{\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}s{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ someI{\isacharunderscore}{\kern0pt}ex{\isacharbrackleft}{\kern0pt}OF\ Ex{\isacharunderscore}{\kern0pt}list{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}length{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ exI{\isacharbrackleft}{\kern0pt}of\ {\isacharunderscore}{\kern0pt}\ {\isachardoublequoteopen}IArray\ {\isacharparenleft}{\kern0pt}SOME\ xs{\isachardot}{\kern0pt}\ length\ xs\ {\isacharequal}{\kern0pt}\ CARD{\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}s{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{setup{\isacharunderscore}{\kern0pt}lifting}\isamarkupfalse%
-\ type{\isacharunderscore}{\kern0pt}definition{\isacharunderscore}{\kern0pt}iarray{\isacharunderscore}{\kern0pt}type\isanewline
-\isanewline
-\isacommand{lift{\isacharunderscore}{\kern0pt}definition}\isamarkupfalse%
-\ fun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}iarray{\isacharunderscore}{\kern0pt}t\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}s{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}{\isacharbraceleft}{\kern0pt}mod{\isacharunderscore}{\kern0pt}type{\isacharbraceright}{\kern0pt}\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a{\isacharparenright}{\kern0pt}\ {\isasymRightarrow}\ {\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}s{\isacharcomma}{\kern0pt}\ {\isacharprime}{\kern0pt}a{\isacharparenright}{\kern0pt}\ iarray{\isacharunderscore}{\kern0pt}type{\isachardoublequoteclose}\ \isakeyword{is}\ {\isachardoublequoteopen}{\isasymlambda}f{\isachardot}{\kern0pt}\ IArray{\isachardot}{\kern0pt}of{\isacharunderscore}{\kern0pt}fun\ {\isacharparenleft}{\kern0pt}{\isasymlambda}s{\isachardot}{\kern0pt}\ f\ {\isacharparenleft}{\kern0pt}from{\isacharunderscore}{\kern0pt}nat\ s{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}CARD{\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}s{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lift{\isacharunderscore}{\kern0pt}definition}\isamarkupfalse%
-\ iarray{\isacharunderscore}{\kern0pt}t{\isacharunderscore}{\kern0pt}sub\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}s{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}mod{\isacharunderscore}{\kern0pt}type{\isacharcomma}{\kern0pt}\ {\isacharprime}{\kern0pt}a{\isacharparenright}{\kern0pt}\ iarray{\isacharunderscore}{\kern0pt}type\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}s\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\ \isakeyword{is}\ {\isachardoublequoteopen}{\isasymlambda}v\ x{\isachardot}{\kern0pt}\ IArray{\isachardot}{\kern0pt}sub\ v\ {\isacharparenleft}{\kern0pt}to{\isacharunderscore}{\kern0pt}nat\ x{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{{\isachardot}{\kern0pt}}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{lift{\isacharunderscore}{\kern0pt}definition}\isamarkupfalse%
-\ iarray{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}vec\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}s{\isacharcomma}{\kern0pt}\ {\isacharprime}{\kern0pt}a{\isacharparenright}{\kern0pt}\ iarray{\isacharunderscore}{\kern0pt}type\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a{\isacharcircum}{\kern0pt}{\isacharprime}{\kern0pt}s{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}{\isacharbraceleft}{\kern0pt}mod{\isacharunderscore}{\kern0pt}type{\isacharcomma}{\kern0pt}\ finite{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\ \isanewline
-\ \ \isakeyword{is}\ {\isachardoublequoteopen}{\isasymlambda}v{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymchi}\ s{\isachardot}{\kern0pt}\ IArray{\isachardot}{\kern0pt}sub\ v\ {\isacharparenleft}{\kern0pt}to{\isacharunderscore}{\kern0pt}nat\ s{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{{\isachardot}{\kern0pt}}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{lift{\isacharunderscore}{\kern0pt}definition}\isamarkupfalse%
-\ vec{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}iarray\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isacharcircum}{\kern0pt}{\isacharprime}{\kern0pt}s{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}{\isacharbraceleft}{\kern0pt}mod{\isacharunderscore}{\kern0pt}type{\isacharcomma}{\kern0pt}\ finite{\isacharbraceright}{\kern0pt}\ {\isasymRightarrow}\ {\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}s{\isacharcomma}{\kern0pt}\ {\isacharprime}{\kern0pt}a{\isacharparenright}{\kern0pt}\ iarray{\isacharunderscore}{\kern0pt}type{\isachardoublequoteclose}\ \isanewline
-\ \ \isakeyword{is}\ {\isachardoublequoteopen}{\isasymlambda}v{\isachardot}{\kern0pt}\ IArray{\isachardot}{\kern0pt}of{\isacharunderscore}{\kern0pt}fun\ {\isacharparenleft}{\kern0pt}{\isasymlambda}s{\isachardot}{\kern0pt}\ v\ {\isachardollar}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}from{\isacharunderscore}{\kern0pt}nat\ s{\isacharparenright}{\kern0pt}\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isacharprime}{\kern0pt}s{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}CARD{\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}s{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ length{\isacharunderscore}{\kern0pt}iarray{\isacharunderscore}{\kern0pt}type\ {\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}length\ {\isacharparenleft}{\kern0pt}IArray{\isachardot}{\kern0pt}list{\isacharunderscore}{\kern0pt}of\ {\isacharparenleft}{\kern0pt}Rep{\isacharunderscore}{\kern0pt}iarray{\isacharunderscore}{\kern0pt}type\ {\isacharparenleft}{\kern0pt}v{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}s{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}{\isacharbraceleft}{\kern0pt}mod{\isacharunderscore}{\kern0pt}type{\isacharbraceright}{\kern0pt}{\isacharcomma}{\kern0pt}\ {\isacharprime}{\kern0pt}a{\isacharparenright}{\kern0pt}\ iarray{\isacharunderscore}{\kern0pt}type{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ CARD{\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}s{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ Rep{\isacharunderscore}{\kern0pt}iarray{\isacharunderscore}{\kern0pt}type\ \isacommand{by}\isamarkupfalse%
-\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ iarray{\isacharunderscore}{\kern0pt}t{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}iff{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}v\ {\isacharequal}{\kern0pt}\ w{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymforall}x{\isachardot}{\kern0pt}\ iarray{\isacharunderscore}{\kern0pt}t{\isacharunderscore}{\kern0pt}sub\ v\ x\ {\isacharequal}{\kern0pt}\ iarray{\isacharunderscore}{\kern0pt}t{\isacharunderscore}{\kern0pt}sub\ w\ x{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ iarray{\isacharunderscore}{\kern0pt}t{\isacharunderscore}{\kern0pt}sub{\isachardot}{\kern0pt}rep{\isacharunderscore}{\kern0pt}eq\ IArray{\isachardot}{\kern0pt}sub{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ Rep{\isacharunderscore}{\kern0pt}iarray{\isacharunderscore}{\kern0pt}type{\isacharunderscore}{\kern0pt}inject\ iarray{\isacharunderscore}{\kern0pt}exhaust{\isadigit{2}}\ length{\isacharunderscore}{\kern0pt}iarray{\isacharunderscore}{\kern0pt}type\ list{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}iff{\isacharunderscore}{\kern0pt}nth{\isacharunderscore}{\kern0pt}eq\ to{\isacharunderscore}{\kern0pt}nat{\isacharunderscore}{\kern0pt}from{\isacharunderscore}{\kern0pt}nat{\isacharunderscore}{\kern0pt}id{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ iarray{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}vec{\isacharunderscore}{\kern0pt}inv{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}iarray{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}vec\ {\isacharparenleft}{\kern0pt}vec{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}iarray\ v{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ v{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ to{\isacharunderscore}{\kern0pt}nat{\isacharunderscore}{\kern0pt}less{\isacharunderscore}{\kern0pt}card\ iarray{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}vec{\isachardot}{\kern0pt}rep{\isacharunderscore}{\kern0pt}eq\ vec{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}iarray{\isachardot}{\kern0pt}rep{\isacharunderscore}{\kern0pt}eq\ vec{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}iff{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ vec{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}iarray{\isacharunderscore}{\kern0pt}inv{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}vec{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}iarray\ {\isacharparenleft}{\kern0pt}iarray{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}vec\ v{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ v{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ to{\isacharunderscore}{\kern0pt}nat{\isacharunderscore}{\kern0pt}less{\isacharunderscore}{\kern0pt}card\ iarray{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}vec{\isachardot}{\kern0pt}rep{\isacharunderscore}{\kern0pt}eq\ vec{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}iarray{\isachardot}{\kern0pt}rep{\isacharunderscore}{\kern0pt}eq\ iarray{\isacharunderscore}{\kern0pt}t{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}iff\ iarray{\isacharunderscore}{\kern0pt}t{\isacharunderscore}{\kern0pt}sub{\isachardot}{\kern0pt}rep{\isacharunderscore}{\kern0pt}eq{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{code{\isacharunderscore}{\kern0pt}datatype}\isamarkupfalse%
-\ iarray{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}vec\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ vec{\isacharunderscore}{\kern0pt}nth{\isacharunderscore}{\kern0pt}iarray{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}vec{\isacharbrackleft}{\kern0pt}code{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}vec{\isacharunderscore}{\kern0pt}nth\ {\isacharparenleft}{\kern0pt}iarray{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}vec\ v{\isacharparenright}{\kern0pt}\ x\ {\isacharequal}{\kern0pt}\ iarray{\isacharunderscore}{\kern0pt}t{\isacharunderscore}{\kern0pt}sub\ v\ x{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ iarray{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}vec{\isachardot}{\kern0pt}rep{\isacharunderscore}{\kern0pt}eq\ iarray{\isacharunderscore}{\kern0pt}t{\isacharunderscore}{\kern0pt}sub{\isachardot}{\kern0pt}rep{\isacharunderscore}{\kern0pt}eq{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ vec{\isacharunderscore}{\kern0pt}lambda{\isacharunderscore}{\kern0pt}iarray{\isacharunderscore}{\kern0pt}t{\isacharbrackleft}{\kern0pt}code{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}vec{\isacharunderscore}{\kern0pt}lambda\ v\ {\isacharequal}{\kern0pt}\ iarray{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}vec\ {\isacharparenleft}{\kern0pt}fun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}iarray{\isacharunderscore}{\kern0pt}t\ v{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ iarray{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}vec{\isachardot}{\kern0pt}rep{\isacharunderscore}{\kern0pt}eq\ fun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}iarray{\isacharunderscore}{\kern0pt}t{\isachardot}{\kern0pt}rep{\isacharunderscore}{\kern0pt}eq\ to{\isacharunderscore}{\kern0pt}nat{\isacharunderscore}{\kern0pt}less{\isacharunderscore}{\kern0pt}card{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ zero{\isacharunderscore}{\kern0pt}iarray{\isacharbrackleft}{\kern0pt}code{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isadigit{0}}\ {\isacharequal}{\kern0pt}\ iarray{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}vec\ {\isacharparenleft}{\kern0pt}fun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}iarray{\isacharunderscore}{\kern0pt}t\ {\isadigit{0}}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ iarray{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}vec{\isachardot}{\kern0pt}rep{\isacharunderscore}{\kern0pt}eq\ fun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}iarray{\isacharunderscore}{\kern0pt}t{\isachardot}{\kern0pt}rep{\isacharunderscore}{\kern0pt}eq\ to{\isacharunderscore}{\kern0pt}nat{\isacharunderscore}{\kern0pt}less{\isacharunderscore}{\kern0pt}card\ vec{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}iff{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Value Iteration%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{locale}\isamarkupfalse%
-\ vi{\isacharunderscore}{\kern0pt}code\ {\isacharequal}{\kern0pt}\ \isanewline
-\ \ MDP{\isacharunderscore}{\kern0pt}ord\ A\ K\ r\ l\ \isakeyword{for}\ A\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}s{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}mod{\isacharunderscore}{\kern0pt}type\ {\isasymRightarrow}\ {\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}a{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}{\isacharbraceleft}{\kern0pt}finite{\isacharcomma}{\kern0pt}\ wellorder{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ set{\isachardoublequoteclose}\ \isanewline
-\ \ \isakeyword{and}\ K\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}s{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}{\isacharbraceleft}{\kern0pt}finite{\isacharcomma}{\kern0pt}\ mod{\isacharunderscore}{\kern0pt}type{\isacharbraceright}{\kern0pt}\ {\isasymtimes}\ {\isacharprime}{\kern0pt}a{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}{\isacharbraceleft}{\kern0pt}finite{\isacharcomma}{\kern0pt}\ wellorder{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}s\ pmf{\isachardoublequoteclose}\ \isakeyword{and}\ r\ l\isanewline
-\isakeyword{begin}\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ {\isachardoublequoteopen}vi{\isacharunderscore}{\kern0pt}test\ {\isacharparenleft}{\kern0pt}v{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}{\isacharprime}{\kern0pt}s{\isasymRightarrow}\isactrlsub b\ real{\isacharparenright}{\kern0pt}\ v{\isacharprime}{\kern0pt}\ eps\ {\isacharequal}{\kern0pt}\ {\isadigit{2}}\ {\isacharasterisk}{\kern0pt}\ l\ {\isacharasterisk}{\kern0pt}\ dist\ v\ v{\isacharprime}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{partial{\isacharunderscore}{\kern0pt}function}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}tailrec{\isacharparenright}{\kern0pt}\ value{\isacharunderscore}{\kern0pt}iteration{\isacharunderscore}{\kern0pt}partial\ \isakeyword{where}\ {\isacharbrackleft}{\kern0pt}code{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}value{\isacharunderscore}{\kern0pt}iteration{\isacharunderscore}{\kern0pt}partial\ eps\ v\ {\isacharequal}{\kern0pt}\ \isanewline
-\ \ {\isacharparenleft}{\kern0pt}let\ v{\isacharprime}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isasymL}\isactrlsub b\ v\ in\isanewline
-\ \ {\isacharparenleft}{\kern0pt}if\ {\isadigit{2}}\ {\isacharasterisk}{\kern0pt}\ l\ {\isacharasterisk}{\kern0pt}\ dist\ v\ v{\isacharprime}{\kern0pt}\ {\isacharless}{\kern0pt}\ eps\ {\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}\ {\isacharminus}{\kern0pt}\ l{\isacharparenright}{\kern0pt}\ then\ v{\isacharprime}{\kern0pt}\ else\ {\isacharparenleft}{\kern0pt}value{\isacharunderscore}{\kern0pt}iteration{\isacharunderscore}{\kern0pt}partial\ eps\ v{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ vi{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}partial{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}eps\ {\isachargreater}{\kern0pt}\ {\isadigit{0}}\ {\isasymLongrightarrow}\ value{\isacharunderscore}{\kern0pt}iteration{\isacharunderscore}{\kern0pt}partial\ eps\ v\ {\isacharequal}{\kern0pt}\ value{\isacharunderscore}{\kern0pt}iteration\ eps\ v{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}induction\ eps\ v\ rule{\isacharcolon}{\kern0pt}\ value{\isacharunderscore}{\kern0pt}iteration{\isachardot}{\kern0pt}induct{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{case}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}\ eps\ v{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{then}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}case\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ Let{\isacharunderscore}{\kern0pt}def\ value{\isacharunderscore}{\kern0pt}iteration{\isachardot}{\kern0pt}simps\ value{\isacharunderscore}{\kern0pt}iteration{\isacharunderscore}{\kern0pt}partial{\isachardot}{\kern0pt}simps{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ {\isachardoublequoteopen}L{\isacharunderscore}{\kern0pt}det\ d\ {\isacharequal}{\kern0pt}\ L\ {\isacharparenleft}{\kern0pt}mk{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\ d{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ code{\isacharunderscore}{\kern0pt}L{\isacharunderscore}{\kern0pt}det\ {\isacharbrackleft}{\kern0pt}code{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}L{\isacharunderscore}{\kern0pt}det\ d\ {\isacharparenleft}{\kern0pt}vec{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}bfun\ v{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ vec{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}bfun\ {\isacharparenleft}{\kern0pt}{\isasymchi}\ s{\isachardot}{\kern0pt}\ L\isactrlsub a\ {\isacharparenleft}{\kern0pt}d\ s{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}vec{\isacharunderscore}{\kern0pt}nth\ v{\isacharparenright}{\kern0pt}\ s{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ L{\isacharunderscore}{\kern0pt}det{\isacharunderscore}{\kern0pt}def\ vec{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}bfun{\isacharunderscore}{\kern0pt}def\ L{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}L\isactrlsub a{\isacharunderscore}{\kern0pt}det{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ code{\isacharunderscore}{\kern0pt}{\isasymL}\isactrlsub b\ {\isacharbrackleft}{\kern0pt}code{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymL}\isactrlsub b\ {\isacharparenleft}{\kern0pt}vec{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}bfun\ v{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ vec{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}bfun\ {\isacharparenleft}{\kern0pt}{\isasymchi}\ s{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}MAX\ a\ {\isasymin}\ A\ s{\isachardot}{\kern0pt}\ r\ {\isacharparenleft}{\kern0pt}s{\isacharcomma}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isacharplus}{\kern0pt}\ l\ {\isacharasterisk}{\kern0pt}\ measure{\isacharunderscore}{\kern0pt}pmf{\isachardot}{\kern0pt}expectation\ {\isacharparenleft}{\kern0pt}K\ {\isacharparenleft}{\kern0pt}s{\isacharcomma}{\kern0pt}\ a{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}vec{\isacharunderscore}{\kern0pt}nth\ v{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ vec{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}bfun{\isacharunderscore}{\kern0pt}def\ {\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}fin{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}det\ A{\isacharunderscore}{\kern0pt}ne\ cSup{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}Max{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ code{\isacharunderscore}{\kern0pt}value{\isacharunderscore}{\kern0pt}iteration{\isacharbrackleft}{\kern0pt}code{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}value{\isacharunderscore}{\kern0pt}iteration\ eps\ {\isacharparenleft}{\kern0pt}vec{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}bfun\ v{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ \isanewline
-\ \ {\isacharparenleft}{\kern0pt}if\ eps\ {\isasymle}\ {\isadigit{0}}\ then\ {\isasymL}\isactrlsub b\ {\isacharparenleft}{\kern0pt}vec{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}bfun\ v{\isacharparenright}{\kern0pt}\ else\ value{\isacharunderscore}{\kern0pt}iteration{\isacharunderscore}{\kern0pt}partial\ eps\ {\isacharparenleft}{\kern0pt}vec{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}bfun\ v{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ value{\isacharunderscore}{\kern0pt}iteration{\isachardot}{\kern0pt}simps\ vi{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}partial{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lift{\isacharunderscore}{\kern0pt}definition}\isamarkupfalse%
-\ find{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}impl\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}s\ {\isasymRightarrow}\isactrlsub b\ real{\isacharparenright}{\kern0pt}\ {\isasymRightarrow}\ {\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}s{\isacharcomma}{\kern0pt}\ {\isacharprime}{\kern0pt}a{\isacharparenright}{\kern0pt}\ Fun{\isachardoublequoteclose}\ \isakeyword{is}\ {\isachardoublequoteopen}{\isasymlambda}v{\isachardot}{\kern0pt}\ find{\isacharunderscore}{\kern0pt}policy{\isacharprime}{\kern0pt}\ v{\isachardoublequoteclose}%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{{\isachardot}{\kern0pt}}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ code{\isacharunderscore}{\kern0pt}find{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}impl{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}find{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}impl\ v\ {\isacharequal}{\kern0pt}\ vec{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}fun\ {\isacharparenleft}{\kern0pt}{\isasymchi}\ s{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}LEAST\ x{\isachardot}{\kern0pt}\ x\ {\isasymin}\ opt{\isacharunderscore}{\kern0pt}acts\ v\ s{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ vec{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}fun{\isacharunderscore}{\kern0pt}def\ find{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}impl{\isacharunderscore}{\kern0pt}def\ find{\isacharunderscore}{\kern0pt}policy{\isacharprime}{\kern0pt}{\isacharunderscore}{\kern0pt}def\ Abs{\isacharunderscore}{\kern0pt}Fun{\isacharunderscore}{\kern0pt}inject{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ code{\isacharunderscore}{\kern0pt}find{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}impl{\isacharunderscore}{\kern0pt}opt{\isacharbrackleft}{\kern0pt}code{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}find{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}impl\ v\ {\isacharequal}{\kern0pt}\ vec{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}fun\ {\isacharparenleft}{\kern0pt}{\isasymchi}\ s{\isachardot}{\kern0pt}\ least{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}enum\ {\isacharparenleft}{\kern0pt}{\isasymlambda}a{\isachardot}{\kern0pt}\ L\isactrlsub a\ a\ v\ s{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}A\ s{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ is{\isacharunderscore}{\kern0pt}opt{\isacharunderscore}{\kern0pt}act{\isacharunderscore}{\kern0pt}def\ code{\isacharunderscore}{\kern0pt}find{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}impl\ least{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}enum{\isacharunderscore}{\kern0pt}correct{\isacharbrackleft}{\kern0pt}OF\ A{\isacharunderscore}{\kern0pt}ne{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ code{\isacharunderscore}{\kern0pt}vi{\isacharunderscore}{\kern0pt}policy{\isacharprime}{\kern0pt}{\isacharbrackleft}{\kern0pt}code{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}vi{\isacharunderscore}{\kern0pt}policy{\isacharprime}{\kern0pt}\ eps\ v\ {\isacharequal}{\kern0pt}\ Rep{\isacharunderscore}{\kern0pt}Fun\ {\isacharparenleft}{\kern0pt}find{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}impl\ {\isacharparenleft}{\kern0pt}value{\isacharunderscore}{\kern0pt}iteration\ eps\ v{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ vi{\isacharunderscore}{\kern0pt}policy{\isacharprime}{\kern0pt}{\isacharunderscore}{\kern0pt}def\ find{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}impl{\isacharunderscore}{\kern0pt}def\ \isacommand{by}\isamarkupfalse%
-\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Policy Iteration%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{partial{\isacharunderscore}{\kern0pt}function}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}tailrec{\isacharparenright}{\kern0pt}\ policy{\isacharunderscore}{\kern0pt}iteration{\isacharunderscore}{\kern0pt}partial\ \isakeyword{where}\ {\isacharbrackleft}{\kern0pt}code{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}policy{\isacharunderscore}{\kern0pt}iteration{\isacharunderscore}{\kern0pt}partial\ d\ {\isacharequal}{\kern0pt}\ \isanewline
-\ \ {\isacharparenleft}{\kern0pt}let\ d{\isacharprime}{\kern0pt}\ {\isacharequal}{\kern0pt}\ policy{\isacharunderscore}{\kern0pt}step\ d\ in\ if\ d\ {\isacharequal}{\kern0pt}\ d{\isacharprime}{\kern0pt}\ then\ d\ else\ policy{\isacharunderscore}{\kern0pt}iteration{\isacharunderscore}{\kern0pt}partial\ d{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ pi{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}partial{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}d\ {\isasymin}\ D\isactrlsub D\ {\isasymLongrightarrow}\ policy{\isacharunderscore}{\kern0pt}iteration{\isacharunderscore}{\kern0pt}partial\ d\ {\isacharequal}{\kern0pt}\ policy{\isacharunderscore}{\kern0pt}iteration\ d{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}induction\ d\ rule{\isacharcolon}{\kern0pt}\ policy{\isacharunderscore}{\kern0pt}iteration{\isachardot}{\kern0pt}induct{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{case}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}\ d{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{then}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}case\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ Let{\isacharunderscore}{\kern0pt}def\ is{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det{\isacharunderscore}{\kern0pt}pi\ policy{\isacharunderscore}{\kern0pt}step{\isacharunderscore}{\kern0pt}def\ policy{\isacharunderscore}{\kern0pt}iteration{\isacharunderscore}{\kern0pt}partial{\isachardot}{\kern0pt}simps{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ {\isachardoublequoteopen}P{\isacharunderscore}{\kern0pt}mat\ d\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymchi}\ i\ j{\isachardot}{\kern0pt}\ pmf\ {\isacharparenleft}{\kern0pt}K\ {\isacharparenleft}{\kern0pt}i{\isacharcomma}{\kern0pt}\ Rep{\isacharunderscore}{\kern0pt}Fun\ d\ i{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ j{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ {\isachardoublequoteopen}r{\isacharunderscore}{\kern0pt}vec{\isacharprime}{\kern0pt}\ d\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymchi}\ i{\isachardot}{\kern0pt}\ r{\isacharparenleft}{\kern0pt}i{\isacharcomma}{\kern0pt}\ Rep{\isacharunderscore}{\kern0pt}Fun\ d\ i{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{lift{\isacharunderscore}{\kern0pt}definition}\isamarkupfalse%
-\ policy{\isacharunderscore}{\kern0pt}eval{\isacharprime}{\kern0pt}\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}s{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}{\isacharbraceleft}{\kern0pt}mod{\isacharunderscore}{\kern0pt}type{\isacharcomma}{\kern0pt}\ finite{\isacharbraceright}{\kern0pt}{\isacharcomma}{\kern0pt}\ {\isacharprime}{\kern0pt}a{\isacharparenright}{\kern0pt}\ Fun\ {\isasymRightarrow}\ {\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}s\ {\isasymRightarrow}\isactrlsub b\ real{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{is}\ {\isachardoublequoteopen}policy{\isacharunderscore}{\kern0pt}eval{\isachardoublequoteclose}%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{{\isachardot}{\kern0pt}}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ mat{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}blinfun{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}mat\ {\isadigit{1}}\ {\isacharminus}{\kern0pt}\ l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isacharparenleft}{\kern0pt}P{\isacharunderscore}{\kern0pt}mat\ {\isacharparenleft}{\kern0pt}vec{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}fun\ d{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ {\isacharparenleft}{\kern0pt}id{\isacharunderscore}{\kern0pt}blinfun\ {\isacharminus}{\kern0pt}\ l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isasymP}\isactrlsub {\isadigit{1}}\ {\isacharparenleft}{\kern0pt}mk{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\ {\isacharparenleft}{\kern0pt}vec{\isacharunderscore}{\kern0pt}nth\ d{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}diff\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}id\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}scaleR\isanewline
-\ \ \isacommand{unfolding}\isamarkupfalse%
-\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}def\ P{\isacharunderscore}{\kern0pt}mat{\isacharunderscore}{\kern0pt}def\ {\isasymP}\isactrlsub {\isadigit{1}}{\isachardot}{\kern0pt}rep{\isacharunderscore}{\kern0pt}eq\ K{\isacharunderscore}{\kern0pt}st{\isacharunderscore}{\kern0pt}def\ push{\isacharunderscore}{\kern0pt}exp{\isacharunderscore}{\kern0pt}def\ matrix{\isacharunderscore}{\kern0pt}def\ axis{\isacharunderscore}{\kern0pt}def\ vec{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}fun{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ if{\isacharunderscore}{\kern0pt}distrib\ mk{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det{\isacharunderscore}{\kern0pt}def\ integral{\isacharunderscore}{\kern0pt}measure{\isacharunderscore}{\kern0pt}pmf{\isacharbrackleft}{\kern0pt}of\ UNIV{\isacharbrackright}{\kern0pt}\ \ pmf{\isacharunderscore}{\kern0pt}expectation{\isacharunderscore}{\kern0pt}bind{\isacharbrackleft}{\kern0pt}of\ UNIV{\isacharbrackright}{\kern0pt}\ pmf{\isacharunderscore}{\kern0pt}bind\ cong{\isacharcolon}{\kern0pt}\ if{\isacharunderscore}{\kern0pt}cong{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}vec{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}policy{\isacharunderscore}{\kern0pt}eval{\isacharprime}{\kern0pt}\ {\isacharparenleft}{\kern0pt}vec{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}fun\ d{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ vec{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}bfun\ {\isacharparenleft}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}inv\ {\isacharparenleft}{\kern0pt}mat\ {\isadigit{1}}\ {\isacharminus}{\kern0pt}\ l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isacharparenleft}{\kern0pt}P{\isacharunderscore}{\kern0pt}mat\ {\isacharparenleft}{\kern0pt}vec{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}fun\ d{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharasterisk}{\kern0pt}v\ {\isacharparenleft}{\kern0pt}r{\isacharunderscore}{\kern0pt}vec{\isacharprime}{\kern0pt}\ {\isacharparenleft}{\kern0pt}vec{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}fun\ d{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}d\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}Rep{\isacharunderscore}{\kern0pt}Fun\ {\isacharparenleft}{\kern0pt}vec{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}fun\ d{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}vec{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}bfun\ {\isacharparenleft}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}inv\ {\isacharparenleft}{\kern0pt}mat\ {\isadigit{1}}\ {\isacharminus}{\kern0pt}\ l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isacharparenleft}{\kern0pt}P{\isacharunderscore}{\kern0pt}mat\ {\isacharparenleft}{\kern0pt}vec{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}fun\ d{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharasterisk}{\kern0pt}v\ {\isacharparenleft}{\kern0pt}r{\isacharunderscore}{\kern0pt}vec{\isacharprime}{\kern0pt}\ {\isacharparenleft}{\kern0pt}vec{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}fun\ d{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ matrix{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}blinfun\ {\isacharparenleft}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}inv\ {\isacharparenleft}{\kern0pt}mat\ {\isadigit{1}}\ {\isacharminus}{\kern0pt}\ l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isacharparenleft}{\kern0pt}P{\isacharunderscore}{\kern0pt}mat\ {\isacharparenleft}{\kern0pt}vec{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}fun\ d{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}vec{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}bfun\ {\isacharparenleft}{\kern0pt}r{\isacharunderscore}{\kern0pt}vec{\isacharprime}{\kern0pt}\ {\isacharparenleft}{\kern0pt}vec{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}fun\ d{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ matrix{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}blinfun{\isacharunderscore}{\kern0pt}mult\ vec{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}bfun{\isacharunderscore}{\kern0pt}def\ r{\isacharunderscore}{\kern0pt}vec{\isacharprime}{\kern0pt}{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isacharequal}{\kern0pt}\ matrix{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}blinfun\ {\isacharparenleft}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}inv\ {\isacharparenleft}{\kern0pt}blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ {\isacharparenleft}{\kern0pt}id{\isacharunderscore}{\kern0pt}blinfun\ {\isacharminus}{\kern0pt}\ l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isasymP}\isactrlsub {\isadigit{1}}\ {\isacharparenleft}{\kern0pt}mk{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\ {\isacharquery}{\kern0pt}d{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}r{\isacharunderscore}{\kern0pt}dec\isactrlsub b\ {\isacharparenleft}{\kern0pt}mk{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\ {\isacharquery}{\kern0pt}d{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ mat{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}blinfun\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ \ r{\isacharunderscore}{\kern0pt}vec{\isacharprime}{\kern0pt}{\isacharunderscore}{\kern0pt}def\ vec{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}bfun{\isacharunderscore}{\kern0pt}def\ vec{\isacharunderscore}{\kern0pt}lambda{\isacharunderscore}{\kern0pt}inverse\ r{\isacharunderscore}{\kern0pt}dec\isactrlsub b{\isacharunderscore}{\kern0pt}def\ vec{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}fun{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isacharequal}{\kern0pt}\ inv\isactrlsub L\ {\isacharparenleft}{\kern0pt}id{\isacharunderscore}{\kern0pt}blinfun\ {\isacharminus}{\kern0pt}\ l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isasymP}\isactrlsub {\isadigit{1}}\ {\isacharparenleft}{\kern0pt}mk{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\ {\isacharquery}{\kern0pt}d{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}r{\isacharunderscore}{\kern0pt}dec\isactrlsub b\ {\isacharparenleft}{\kern0pt}mk{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\ {\isacharquery}{\kern0pt}d{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}inverse{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isacharbrackleft}{\kern0pt}symmetric{\isacharbrackright}{\kern0pt}\ invertible\isactrlsub L{\isacharunderscore}{\kern0pt}inf{\isacharunderscore}{\kern0pt}sum\ matrix{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}blinfun{\isacharunderscore}{\kern0pt}inv{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{finally}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}vec{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}bfun\ {\isacharparenleft}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}inv\ {\isacharparenleft}{\kern0pt}mat\ {\isadigit{1}}\ {\isacharminus}{\kern0pt}\ l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isacharparenleft}{\kern0pt}P{\isacharunderscore}{\kern0pt}mat\ {\isacharparenleft}{\kern0pt}vec{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}fun\ d{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharasterisk}{\kern0pt}v\ {\isacharparenleft}{\kern0pt}r{\isacharunderscore}{\kern0pt}vec{\isacharprime}{\kern0pt}\ {\isacharparenleft}{\kern0pt}vec{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}fun\ d{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ inv\isactrlsub L\ {\isacharparenleft}{\kern0pt}id{\isacharunderscore}{\kern0pt}blinfun\ {\isacharminus}{\kern0pt}\ l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isasymP}\isactrlsub {\isadigit{1}}\ {\isacharparenleft}{\kern0pt}mk{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\ {\isacharquery}{\kern0pt}d{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}r{\isacharunderscore}{\kern0pt}dec\isactrlsub b\ {\isacharparenleft}{\kern0pt}mk{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\ {\isacharquery}{\kern0pt}d{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isacommand{{\isachardot}{\kern0pt}}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ {\isasymnu}{\isacharunderscore}{\kern0pt}stationary\ policy{\isacharunderscore}{\kern0pt}eval{\isacharprime}{\kern0pt}{\isachardot}{\kern0pt}rep{\isacharunderscore}{\kern0pt}eq\ policy{\isacharunderscore}{\kern0pt}eval{\isacharunderscore}{\kern0pt}def\ inv\isactrlsub L{\isacharunderscore}{\kern0pt}inf{\isacharunderscore}{\kern0pt}sum\ blincomp{\isacharunderscore}{\kern0pt}scaleR{\isacharunderscore}{\kern0pt}right{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}vec{\isacharunderscore}{\kern0pt}opt{\isacharbrackleft}{\kern0pt}code{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}policy{\isacharunderscore}{\kern0pt}eval{\isacharprime}{\kern0pt}\ {\isacharparenleft}{\kern0pt}vec{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}fun\ d{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ vec{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}bfun\ {\isacharparenleft}{\kern0pt}Matrix{\isacharunderscore}{\kern0pt}To{\isacharunderscore}{\kern0pt}IArray{\isachardot}{\kern0pt}iarray{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}vec\ {\isacharparenleft}{\kern0pt}Matrix{\isacharunderscore}{\kern0pt}To{\isacharunderscore}{\kern0pt}IArray{\isachardot}{\kern0pt}vec{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}iarray\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}fst\ {\isacharparenleft}{\kern0pt}Gauss{\isacharunderscore}{\kern0pt}Jordan{\isacharunderscore}{\kern0pt}PA\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}mat\ {\isadigit{1}}\ {\isacharminus}{\kern0pt}\ l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isacharparenleft}{\kern0pt}P{\isacharunderscore}{\kern0pt}mat\ {\isacharparenleft}{\kern0pt}vec{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}fun\ d{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharasterisk}{\kern0pt}v\ {\isacharparenleft}{\kern0pt}r{\isacharunderscore}{\kern0pt}vec{\isacharprime}{\kern0pt}\ {\isacharparenleft}{\kern0pt}vec{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}fun\ d{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}vec\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ mat{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}blinfun\ matrix{\isacharunderscore}{\kern0pt}inv{\isacharunderscore}{\kern0pt}Gauss{\isacharunderscore}{\kern0pt}Jordan{\isacharunderscore}{\kern0pt}PA\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}inverse{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}\ invertible\isactrlsub L{\isacharunderscore}{\kern0pt}inf{\isacharunderscore}{\kern0pt}sum\ iarray{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}vec{\isacharunderscore}{\kern0pt}vec{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}iarray{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lift{\isacharunderscore}{\kern0pt}definition}\isamarkupfalse%
-\ policy{\isacharunderscore}{\kern0pt}improvement{\isacharprime}{\kern0pt}\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}s{\isacharcomma}{\kern0pt}\ {\isacharprime}{\kern0pt}a{\isacharparenright}{\kern0pt}\ Fun\ {\isasymRightarrow}\ {\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}s\ {\isasymRightarrow}\isactrlsub b\ real{\isacharparenright}{\kern0pt}\ {\isasymRightarrow}\ {\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}s{\isacharcomma}{\kern0pt}\ {\isacharprime}{\kern0pt}a{\isacharparenright}{\kern0pt}\ Fun{\isachardoublequoteclose}\ \isanewline
-\ \ \isakeyword{is}\ policy{\isacharunderscore}{\kern0pt}improvement%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{{\isachardot}{\kern0pt}}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ {\isacharbrackleft}{\kern0pt}code{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}policy{\isacharunderscore}{\kern0pt}improvement{\isacharprime}{\kern0pt}\ {\isacharparenleft}{\kern0pt}vec{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}fun\ d{\isacharparenright}{\kern0pt}\ v\ {\isacharequal}{\kern0pt}\ vec{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}fun\ {\isacharparenleft}{\kern0pt}{\isasymchi}\ s{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}if\ is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max\ {\isacharparenleft}{\kern0pt}{\isasymlambda}a{\isachardot}{\kern0pt}\ L\isactrlsub a\ a\ v\ s{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymlambda}a{\isachardot}{\kern0pt}\ a\ {\isasymin}\ A\ s{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}d\ {\isachardollar}{\kern0pt}\ s{\isacharparenright}{\kern0pt}\ then\ d\ {\isachardollar}{\kern0pt}\ s\ else\ LEAST\ x{\isachardot}{\kern0pt}\ is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max\ {\isacharparenleft}{\kern0pt}{\isasymlambda}a{\isachardot}{\kern0pt}\ L\isactrlsub a\ a\ v\ s{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymlambda}a{\isachardot}{\kern0pt}\ a\ {\isasymin}\ A\ s{\isacharparenright}{\kern0pt}\ x{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ is{\isacharunderscore}{\kern0pt}opt{\isacharunderscore}{\kern0pt}act{\isacharunderscore}{\kern0pt}def\ policy{\isacharunderscore}{\kern0pt}improvement{\isacharprime}{\kern0pt}{\isacharunderscore}{\kern0pt}def\ vec{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}fun{\isacharunderscore}{\kern0pt}def\ vec{\isacharunderscore}{\kern0pt}lambda{\isacharunderscore}{\kern0pt}inverse\ policy{\isacharunderscore}{\kern0pt}improvement{\isacharunderscore}{\kern0pt}def\ Abs{\isacharunderscore}{\kern0pt}Fun{\isacharunderscore}{\kern0pt}inject{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lift{\isacharunderscore}{\kern0pt}definition}\isamarkupfalse%
-\ policy{\isacharunderscore}{\kern0pt}step{\isacharprime}{\kern0pt}\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}s{\isacharcomma}{\kern0pt}\ {\isacharprime}{\kern0pt}a{\isacharparenright}{\kern0pt}\ Fun\ {\isasymRightarrow}\ {\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}s{\isacharcomma}{\kern0pt}\ {\isacharprime}{\kern0pt}a{\isacharparenright}{\kern0pt}\ Fun{\isachardoublequoteclose}\ \isanewline
-\ \ \isakeyword{is}\ policy{\isacharunderscore}{\kern0pt}step%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{{\isachardot}{\kern0pt}}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ {\isacharbrackleft}{\kern0pt}code{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}policy{\isacharunderscore}{\kern0pt}step{\isacharprime}{\kern0pt}\ d\ {\isacharequal}{\kern0pt}\ policy{\isacharunderscore}{\kern0pt}improvement{\isacharprime}{\kern0pt}\ d\ {\isacharparenleft}{\kern0pt}policy{\isacharunderscore}{\kern0pt}eval{\isacharprime}{\kern0pt}\ d{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ policy{\isacharunderscore}{\kern0pt}step{\isacharprime}{\kern0pt}{\isacharunderscore}{\kern0pt}def\ policy{\isacharunderscore}{\kern0pt}step{\isacharunderscore}{\kern0pt}def\ policy{\isacharunderscore}{\kern0pt}improvement{\isacharprime}{\kern0pt}{\isacharunderscore}{\kern0pt}def\ policy{\isacharunderscore}{\kern0pt}eval{\isacharprime}{\kern0pt}{\isacharunderscore}{\kern0pt}def\ apply{\isacharunderscore}{\kern0pt}bfun{\isacharunderscore}{\kern0pt}inverse{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lift{\isacharunderscore}{\kern0pt}definition}\isamarkupfalse%
-\ policy{\isacharunderscore}{\kern0pt}iteration{\isacharunderscore}{\kern0pt}partial{\isacharprime}{\kern0pt}\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}s{\isacharcomma}{\kern0pt}\ {\isacharprime}{\kern0pt}a{\isacharparenright}{\kern0pt}\ Fun\ {\isasymRightarrow}\ {\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}s{\isacharcomma}{\kern0pt}\ {\isacharprime}{\kern0pt}a{\isacharparenright}{\kern0pt}\ Fun{\isachardoublequoteclose}\ \isanewline
-\ \ \isakeyword{is}\ policy{\isacharunderscore}{\kern0pt}iteration{\isacharunderscore}{\kern0pt}partial%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{{\isachardot}{\kern0pt}}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ {\isacharbrackleft}{\kern0pt}code{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}policy{\isacharunderscore}{\kern0pt}iteration{\isacharunderscore}{\kern0pt}partial{\isacharprime}{\kern0pt}\ d\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}let\ d{\isacharprime}{\kern0pt}\ {\isacharequal}{\kern0pt}\ policy{\isacharunderscore}{\kern0pt}step{\isacharprime}{\kern0pt}\ d\ in\ if\ d\ {\isacharequal}{\kern0pt}\ d{\isacharprime}{\kern0pt}\ then\ d\ else\ policy{\isacharunderscore}{\kern0pt}iteration{\isacharunderscore}{\kern0pt}partial{\isacharprime}{\kern0pt}\ d{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ policy{\isacharunderscore}{\kern0pt}iteration{\isacharunderscore}{\kern0pt}partial{\isacharprime}{\kern0pt}{\isachardot}{\kern0pt}rep{\isacharunderscore}{\kern0pt}eq\ policy{\isacharunderscore}{\kern0pt}step{\isacharprime}{\kern0pt}{\isachardot}{\kern0pt}rep{\isacharunderscore}{\kern0pt}eq\ Let{\isacharunderscore}{\kern0pt}def\ policy{\isacharunderscore}{\kern0pt}iteration{\isacharunderscore}{\kern0pt}partial{\isachardot}{\kern0pt}simps\ Rep{\isacharunderscore}{\kern0pt}Fun{\isacharunderscore}{\kern0pt}inject{\isacharbrackleft}{\kern0pt}symmetric{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lift{\isacharunderscore}{\kern0pt}definition}\isamarkupfalse%
-\ policy{\isacharunderscore}{\kern0pt}iteration{\isacharprime}{\kern0pt}\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}s{\isacharcomma}{\kern0pt}\ {\isacharprime}{\kern0pt}a{\isacharparenright}{\kern0pt}\ Fun\ {\isasymRightarrow}\ {\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}s{\isacharcomma}{\kern0pt}\ {\isacharprime}{\kern0pt}a{\isacharparenright}{\kern0pt}\ Fun{\isachardoublequoteclose}\ \isakeyword{is}\ policy{\isacharunderscore}{\kern0pt}iteration%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{{\isachardot}{\kern0pt}}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ code{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}iteration{\isacharprime}{\kern0pt}{\isacharbrackleft}{\kern0pt}code{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}policy{\isacharunderscore}{\kern0pt}iteration{\isacharprime}{\kern0pt}\ d\ {\isacharequal}{\kern0pt}\isanewline
-\ \ {\isacharparenleft}{\kern0pt}if\ Rep{\isacharunderscore}{\kern0pt}Fun\ d\ {\isasymnotin}\ D\isactrlsub D\ then\ d\ else\ \ {\isacharparenleft}{\kern0pt}policy{\isacharunderscore}{\kern0pt}iteration{\isacharunderscore}{\kern0pt}partial{\isacharprime}{\kern0pt}\ d{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ transfer\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ pi{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}partial{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ code{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}iteration{\isacharbrackleft}{\kern0pt}code{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}policy{\isacharunderscore}{\kern0pt}iteration\ d\ {\isacharequal}{\kern0pt}\ Rep{\isacharunderscore}{\kern0pt}Fun\ {\isacharparenleft}{\kern0pt}policy{\isacharunderscore}{\kern0pt}iteration{\isacharprime}{\kern0pt}\ {\isacharparenleft}{\kern0pt}vec{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}fun\ {\isacharparenleft}{\kern0pt}vec{\isacharunderscore}{\kern0pt}lambda\ d{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp\ add{\isacharcolon}{\kern0pt}\ vec{\isacharunderscore}{\kern0pt}lambda{\isacharunderscore}{\kern0pt}inverse\ policy{\isacharunderscore}{\kern0pt}iteration{\isacharprime}{\kern0pt}{\isachardot}{\kern0pt}rep{\isacharunderscore}{\kern0pt}eq\ vec{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}fun{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Gauss-Seidel Iteration%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{partial{\isacharunderscore}{\kern0pt}function}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}tailrec{\isacharparenright}{\kern0pt}\ gs{\isacharunderscore}{\kern0pt}iteration{\isacharunderscore}{\kern0pt}partial\ \isakeyword{where}\isanewline
-\ \ {\isacharbrackleft}{\kern0pt}code{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}gs{\isacharunderscore}{\kern0pt}iteration{\isacharunderscore}{\kern0pt}partial\ eps\ v\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}\isanewline
-\ \ let\ v{\isacharprime}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}fun\isactrlsub b\ v{\isacharparenright}{\kern0pt}\ in\ \isanewline
-\ \ {\isacharparenleft}{\kern0pt}if\ {\isadigit{2}}\ {\isacharasterisk}{\kern0pt}\ l\ {\isacharasterisk}{\kern0pt}\ dist\ v\ v{\isacharprime}{\kern0pt}\ {\isacharless}{\kern0pt}\ eps\ {\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}\ {\isacharminus}{\kern0pt}\ l{\isacharparenright}{\kern0pt}\ then\ v{\isacharprime}{\kern0pt}\ else\ gs{\isacharunderscore}{\kern0pt}iteration{\isacharunderscore}{\kern0pt}partial\ eps\ v{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ gs{\isacharunderscore}{\kern0pt}iteration{\isacharunderscore}{\kern0pt}partial{\isacharunderscore}{\kern0pt}eq{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}eps\ {\isachargreater}{\kern0pt}\ {\isadigit{0}}\ {\isasymLongrightarrow}\ gs{\isacharunderscore}{\kern0pt}iteration{\isacharunderscore}{\kern0pt}partial\ eps\ v\ {\isacharequal}{\kern0pt}\ gs{\isacharunderscore}{\kern0pt}iteration\ eps\ v{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}induction\ eps\ v\ rule{\isacharcolon}{\kern0pt}\ gs{\isacharunderscore}{\kern0pt}iteration{\isachardot}{\kern0pt}induct{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ gs{\isacharunderscore}{\kern0pt}iteration{\isacharunderscore}{\kern0pt}partial{\isachardot}{\kern0pt}simps\ Let{\isacharunderscore}{\kern0pt}def\ gs{\isacharunderscore}{\kern0pt}iteration{\isachardot}{\kern0pt}simps{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ gs{\isacharunderscore}{\kern0pt}iteration{\isacharunderscore}{\kern0pt}code{\isacharunderscore}{\kern0pt}opt{\isacharbrackleft}{\kern0pt}code{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}gs{\isacharunderscore}{\kern0pt}iteration\ eps\ v\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}if\ eps\ {\isasymle}\ {\isadigit{0}}\ then\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}fun\isactrlsub b\ v\ else\ gs{\isacharunderscore}{\kern0pt}iteration{\isacharunderscore}{\kern0pt}partial\ eps\ v{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ gs{\isacharunderscore}{\kern0pt}iteration{\isacharunderscore}{\kern0pt}partial{\isacharunderscore}{\kern0pt}eq\ gs{\isacharunderscore}{\kern0pt}iteration{\isachardot}{\kern0pt}simps{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ {\isachardoublequoteopen}vec{\isacharunderscore}{\kern0pt}upd\ v\ i\ x\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymchi}\ j{\isachardot}{\kern0pt}\ if\ i\ {\isacharequal}{\kern0pt}\ j\ then\ x\ else\ v\ {\isachardollar}{\kern0pt}\ j{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}fold{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}GS{\isacharunderscore}{\kern0pt}rec\ v\ {\isacharequal}{\kern0pt}\ foldl\ {\isacharparenleft}{\kern0pt}{\isasymlambda}v\ s{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}vec{\isacharunderscore}{\kern0pt}upd\ v\ s\ {\isacharparenleft}{\kern0pt}GS{\isacharunderscore}{\kern0pt}iter{\isacharunderscore}{\kern0pt}max\ v\ s{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ v\ {\isacharparenleft}{\kern0pt}sorted{\isacharunderscore}{\kern0pt}list{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}set\ UNIV{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}vec{\isacharunderscore}{\kern0pt}lambda\ {\isacharparenleft}{\kern0pt}foldl\ {\isacharparenleft}{\kern0pt}{\isasymlambda}v\ s{\isachardot}{\kern0pt}\ v{\isacharparenleft}{\kern0pt}s\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}iter\ v\ s{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}{\isachardollar}{\kern0pt}{\isacharparenright}{\kern0pt}\ v{\isacharparenright}{\kern0pt}\ xs{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ foldl\ {\isacharparenleft}{\kern0pt}{\isasymlambda}v\ s{\isachardot}{\kern0pt}\ vec{\isacharunderscore}{\kern0pt}upd\ v\ s\ {\isacharparenleft}{\kern0pt}GS{\isacharunderscore}{\kern0pt}iter{\isacharunderscore}{\kern0pt}max\ v\ s{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ v\ xs{\isachardoublequoteclose}\ \isakeyword{for}\ xs\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}induction\ xs\ arbitrary{\isacharcolon}{\kern0pt}\ v{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{case}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}Cons\ a\ xs{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}case\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ vec{\isacharunderscore}{\kern0pt}upd{\isacharunderscore}{\kern0pt}def{\isacharbrackleft}{\kern0pt}of\ v{\isacharbrackright}{\kern0pt}\ Cons{\isacharbrackleft}{\kern0pt}symmetric{\isacharbrackright}{\kern0pt}\ eq{\isacharunderscore}{\kern0pt}commute\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}iter{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}iter{\isacharunderscore}{\kern0pt}max\ cong{\isacharcolon}{\kern0pt}\ if{\isacharunderscore}{\kern0pt}cong{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}def\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}fun{\isacharunderscore}{\kern0pt}code{\isacharprime}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}fun{\isacharunderscore}{\kern0pt}code{\isacharprime}{\kern0pt}{\isacharprime}{\kern0pt}{\isacharprime}{\kern0pt}{\isacharprime}{\kern0pt}{\isacharbrackleft}{\kern0pt}code{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}fun\isactrlsub b\ {\isacharparenleft}{\kern0pt}vec{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}bfun\ v{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ vec{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}bfun\ {\isacharparenleft}{\kern0pt}foldl\ {\isacharparenleft}{\kern0pt}{\isasymlambda}v\ s{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}vec{\isacharunderscore}{\kern0pt}upd\ v\ s\ {\isacharparenleft}{\kern0pt}GS{\isacharunderscore}{\kern0pt}iter{\isacharunderscore}{\kern0pt}max\ v\ s{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ v\ {\isacharparenleft}{\kern0pt}sorted{\isacharunderscore}{\kern0pt}list{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}set\ UNIV{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp\ add{\isacharcolon}{\kern0pt}\ \ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}fold{\isacharbrackleft}{\kern0pt}symmetric{\isacharbrackright}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}elem\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}fun\isactrlsub b{\isachardot}{\kern0pt}rep{\isacharunderscore}{\kern0pt}eq\ vec{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}bfun{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ GS{\isacharunderscore}{\kern0pt}iter{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}code\ {\isacharbrackleft}{\kern0pt}code{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}GS{\isacharunderscore}{\kern0pt}iter{\isacharunderscore}{\kern0pt}max\ v\ s\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}MAX\ a\ {\isasymin}\ A\ s{\isachardot}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}iter\ a\ v\ s{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ A{\isacharunderscore}{\kern0pt}ne\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}iter{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}def\ cSup{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}Max{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lift{\isacharunderscore}{\kern0pt}definition}\isamarkupfalse%
-\ opt{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}gs{\isacharprime}{\kern0pt}{\isacharprime}{\kern0pt}\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}s\ {\isasymRightarrow}\isactrlsub b\ real{\isacharparenright}{\kern0pt}\ {\isasymRightarrow}\ {\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}s{\isacharcomma}{\kern0pt}\ {\isacharprime}{\kern0pt}a{\isacharparenright}{\kern0pt}\ Fun{\isachardoublequoteclose}\ \isakeyword{is}\ opt{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}gs%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{{\isachardot}{\kern0pt}}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{declare}\isamarkupfalse%
-\ opt{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}gs{\isacharprime}{\kern0pt}{\isacharprime}{\kern0pt}{\isachardot}{\kern0pt}rep{\isacharunderscore}{\kern0pt}eq{\isacharbrackleft}{\kern0pt}symmetric{\isacharcomma}{\kern0pt}\ code{\isacharbrackright}{\kern0pt}\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}am{\isacharunderscore}{\kern0pt}code{\isacharprime}{\kern0pt}{\isacharunderscore}{\kern0pt}prod{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}am{\isacharunderscore}{\kern0pt}code{\isacharprime}{\kern0pt}\ v\ d\ {\isacharequal}{\kern0pt}\ \isanewline
-\ \ {\isacharparenleft}{\kern0pt}{\isasymlambda}s{\isacharprime}{\kern0pt}{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}\isanewline
-\ \ \ \ let\ {\isacharparenleft}{\kern0pt}v{\isacharprime}{\kern0pt}{\isacharcomma}{\kern0pt}\ d{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ foldl\ {\isacharparenleft}{\kern0pt}{\isasymlambda}{\isacharparenleft}{\kern0pt}v{\isacharcomma}{\kern0pt}d{\isacharparenright}{\kern0pt}\ s{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}v{\isacharparenleft}{\kern0pt}s\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}GS{\isacharunderscore}{\kern0pt}iter{\isacharunderscore}{\kern0pt}max\ {\isacharparenleft}{\kern0pt}vec{\isacharunderscore}{\kern0pt}lambda\ v{\isacharparenright}{\kern0pt}\ s{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharcomma}{\kern0pt}\ \ d{\isacharparenleft}{\kern0pt}s\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}iter{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max\ {\isacharparenleft}{\kern0pt}vec{\isacharunderscore}{\kern0pt}lambda\ v{\isacharparenright}{\kern0pt}\ s{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}vec{\isacharunderscore}{\kern0pt}nth\ v{\isacharcomma}{\kern0pt}\ d{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}sorted{\isacharunderscore}{\kern0pt}list{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}set\ UNIV{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ in\ {\isacharparenleft}{\kern0pt}v{\isacharprime}{\kern0pt}\ s{\isacharprime}{\kern0pt}{\isacharcomma}{\kern0pt}\ d{\isacharprime}{\kern0pt}\ s{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isasymlambda}x{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}f\ x{\isacharcomma}{\kern0pt}\ g\ x{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenleft}{\kern0pt}y\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}z{\isacharcomma}{\kern0pt}\ w{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymlambda}x{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}f{\isacharparenleft}{\kern0pt}y\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ z{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ x{\isacharcomma}{\kern0pt}\ {\isacharparenleft}{\kern0pt}g{\isacharparenleft}{\kern0pt}y\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ w{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ x{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{for}\ f\ g\ y\ z\ w\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{2}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}{\isachardollar}{\kern0pt}{\isacharparenright}{\kern0pt}\ f{\isacharparenright}{\kern0pt}{\isacharparenleft}{\kern0pt}a\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ y{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isachardollar}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}vec{\isacharunderscore}{\kern0pt}lambda\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}vec{\isacharunderscore}{\kern0pt}nth\ f{\isacharparenright}{\kern0pt}{\isacharparenleft}{\kern0pt}a\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ y{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{for}\ f\ a\ y\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}foldl\ {\isacharparenleft}{\kern0pt}{\isasymlambda}vd\ s{\isachardot}{\kern0pt}\ vd{\isacharparenleft}{\kern0pt}s\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}GS{\isacharunderscore}{\kern0pt}iter{\isacharunderscore}{\kern0pt}max\ {\isacharparenleft}{\kern0pt}{\isasymchi}\ s{\isachardot}{\kern0pt}\ fst\ {\isacharparenleft}{\kern0pt}vd\ s{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ s{\isacharcomma}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}iter{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max\ {\isacharparenleft}{\kern0pt}{\isasymchi}\ s{\isachardot}{\kern0pt}\ fst\ {\isacharparenleft}{\kern0pt}vd\ s{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ s{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymlambda}s{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}v\ {\isachardollar}{\kern0pt}\ s{\isacharcomma}{\kern0pt}\ d\ s{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ xs\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymlambda}s{\isacharprime}{\kern0pt}{\isachardot}{\kern0pt}\ let\ {\isacharparenleft}{\kern0pt}v{\isacharprime}{\kern0pt}{\isacharcomma}{\kern0pt}\ d{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ foldl\ {\isacharparenleft}{\kern0pt}{\isasymlambda}{\isacharparenleft}{\kern0pt}v{\isacharcomma}{\kern0pt}\ d{\isacharparenright}{\kern0pt}\ s{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}v{\isacharparenleft}{\kern0pt}s\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}iter{\isacharunderscore}{\kern0pt}max\ {\isacharparenleft}{\kern0pt}vec{\isacharunderscore}{\kern0pt}lambda\ v{\isacharparenright}{\kern0pt}\ s{\isacharparenright}{\kern0pt}{\isacharcomma}{\kern0pt}\ d{\isacharparenleft}{\kern0pt}s\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}iter{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max\ {\isacharparenleft}{\kern0pt}vec{\isacharunderscore}{\kern0pt}lambda\ v{\isacharparenright}{\kern0pt}\ s{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}{\isachardollar}{\kern0pt}{\isacharparenright}{\kern0pt}\ v{\isacharcomma}{\kern0pt}\ d{\isacharparenright}{\kern0pt}\ xs\ in\ {\isacharparenleft}{\kern0pt}v{\isacharprime}{\kern0pt}\ s{\isacharprime}{\kern0pt}{\isacharcomma}{\kern0pt}\ d{\isacharprime}{\kern0pt}\ s{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{for}\ xs\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}induction\ xs\ arbitrary{\isacharcolon}{\kern0pt}\ v\ d{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{case}\isamarkupfalse%
-\ Nil\isanewline
-\ \ \ \ \isacommand{then}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}case\ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{case}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}Cons\ a\ xs{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}case\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ {\isadigit{1}}\ Cons{\isachardot}{\kern0pt}IH{\isacharbrackleft}{\kern0pt}of\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}vec{\isacharunderscore}{\kern0pt}lambda\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}{\isachardollar}{\kern0pt}{\isacharparenright}{\kern0pt}\ v{\isacharparenright}{\kern0pt}{\isacharparenleft}{\kern0pt}a\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}iter{\isacharunderscore}{\kern0pt}max\ v\ a{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}{\isacharcomma}{\kern0pt}\ unfolded\ {\isadigit{2}}{\isacharbrackleft}{\kern0pt}symmetric{\isacharbrackright}{\kern0pt}{\isacharbrackright}{\kern0pt}\ del{\isacharcolon}{\kern0pt}\ fun{\isacharunderscore}{\kern0pt}upd{\isacharunderscore}{\kern0pt}apply{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}am{\isacharunderscore}{\kern0pt}code{\isacharprime}{\kern0pt}{\isacharunderscore}{\kern0pt}def\ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ code{\isacharunderscore}{\kern0pt}GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}am{\isacharunderscore}{\kern0pt}arr{\isacharunderscore}{\kern0pt}opt{\isacharbrackleft}{\kern0pt}code{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}opt{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}gs{\isacharprime}{\kern0pt}{\isacharprime}{\kern0pt}\ {\isacharparenleft}{\kern0pt}vec{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}bfun\ v{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ vec{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}fun\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}snd\ {\isacharparenleft}{\kern0pt}foldl\ {\isacharparenleft}{\kern0pt}{\isasymlambda}{\isacharparenleft}{\kern0pt}v{\isacharcomma}{\kern0pt}\ d{\isacharparenright}{\kern0pt}\ s{\isachardot}{\kern0pt}\isanewline
-\ \ let\ {\isacharparenleft}{\kern0pt}am{\isacharcomma}{\kern0pt}\ m{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ least{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}enum\ {\isacharparenleft}{\kern0pt}{\isasymlambda}a{\isachardot}{\kern0pt}\ r\ {\isacharparenleft}{\kern0pt}s{\isacharcomma}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isacharplus}{\kern0pt}\ l\ {\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSum}s{\isacharprime}{\kern0pt}\ {\isasymin}\ UNIV{\isachardot}{\kern0pt}\ pmf\ {\isacharparenleft}{\kern0pt}K\ {\isacharparenleft}{\kern0pt}s{\isacharcomma}{\kern0pt}a{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ s{\isacharprime}{\kern0pt}\ {\isacharasterisk}{\kern0pt}\ v\ {\isachardollar}{\kern0pt}\ s{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}A\ s{\isacharparenright}{\kern0pt}\ in\isanewline
-\ \ {\isacharparenleft}{\kern0pt}vec{\isacharunderscore}{\kern0pt}upd\ v\ s\ m{\isacharcomma}{\kern0pt}\ vec{\isacharunderscore}{\kern0pt}upd\ d\ s\ am{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ \isanewline
-\ \ {\isacharparenleft}{\kern0pt}v{\isacharcomma}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymchi}\ s{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}least{\isacharunderscore}{\kern0pt}enum\ {\isacharparenleft}{\kern0pt}{\isasymlambda}a{\isachardot}{\kern0pt}\ a\ {\isasymin}\ A\ s{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}sorted{\isacharunderscore}{\kern0pt}list{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}set\ UNIV{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}opt{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}gs{\isacharprime}{\kern0pt}{\isacharprime}{\kern0pt}\ v{\isacharprime}{\kern0pt}\ {\isacharequal}{\kern0pt}\ Abs{\isacharunderscore}{\kern0pt}Fun\ {\isacharparenleft}{\kern0pt}opt{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}gs\ v{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{for}\ v{\isacharprime}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ opt{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}gs{\isacharprime}{\kern0pt}{\isacharprime}{\kern0pt}{\isachardot}{\kern0pt}abs{\isacharunderscore}{\kern0pt}eq{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{2}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}opt{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}gs\ {\isacharparenleft}{\kern0pt}vec{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}bfun\ v{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ opt{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}gs{\isacharprime}{\kern0pt}\ d\ v{\isachardoublequoteclose}\ \isakeyword{for}\ v\ d\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ Bfun{\isacharunderscore}{\kern0pt}inverse{\isacharunderscore}{\kern0pt}fin\ opt{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}gs{\isacharunderscore}{\kern0pt}eq{\isacharprime}{\kern0pt}\ vec{\isacharunderscore}{\kern0pt}lambda{\isacharunderscore}{\kern0pt}eta\ vec{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}bfun{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{3}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}opt{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}gs{\isacharprime}{\kern0pt}\ d\ v\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymlambda}s{\isachardot}{\kern0pt}\ snd\ {\isacharparenleft}{\kern0pt}GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}am{\isacharunderscore}{\kern0pt}code\ v\ d\ s{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{for}\ d\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}am{\isacharunderscore}{\kern0pt}code{\isacharunderscore}{\kern0pt}eq{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{4}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}am{\isacharunderscore}{\kern0pt}code\ v\ d\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymlambda}s{\isacharprime}{\kern0pt}{\isachardot}{\kern0pt}\ let\ {\isacharparenleft}{\kern0pt}v{\isacharprime}{\kern0pt}{\isacharcomma}{\kern0pt}\ d{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ foldl\ {\isacharparenleft}{\kern0pt}{\isasymlambda}{\isacharparenleft}{\kern0pt}v{\isacharcomma}{\kern0pt}\ d{\isacharparenright}{\kern0pt}\ s{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}v{\isacharparenleft}{\kern0pt}s\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}iter{\isacharunderscore}{\kern0pt}max\ {\isacharparenleft}{\kern0pt}vec{\isacharunderscore}{\kern0pt}lambda\ v{\isacharparenright}{\kern0pt}\ s{\isacharparenright}{\kern0pt}{\isacharcomma}{\kern0pt}\ d{\isacharparenleft}{\kern0pt}s\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}iter{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max\ {\isacharparenleft}{\kern0pt}vec{\isacharunderscore}{\kern0pt}lambda\ v{\isacharparenright}{\kern0pt}\ s{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}{\isachardollar}{\kern0pt}{\isacharparenright}{\kern0pt}\ v{\isacharcomma}{\kern0pt}\ d{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}sorted{\isacharunderscore}{\kern0pt}list{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}set\ UNIV{\isacharparenright}{\kern0pt}\ in\ {\isacharparenleft}{\kern0pt}v{\isacharprime}{\kern0pt}\ s{\isacharprime}{\kern0pt}{\isacharcomma}{\kern0pt}\ d{\isacharprime}{\kern0pt}\ s{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{for}\ d\ s\ v\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}am{\isacharunderscore}{\kern0pt}code{\isacharprime}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}am{\isacharunderscore}{\kern0pt}code{\isacharprime}{\kern0pt}{\isacharunderscore}{\kern0pt}prod\ \isacommand{by}\isamarkupfalse%
-\ presburger\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{5}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}foldl\ {\isacharparenleft}{\kern0pt}{\isasymlambda}{\isacharparenleft}{\kern0pt}v{\isacharcomma}{\kern0pt}\ d{\isacharparenright}{\kern0pt}\ s{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}v{\isacharparenleft}{\kern0pt}s\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}iter{\isacharunderscore}{\kern0pt}max\ {\isacharparenleft}{\kern0pt}vec{\isacharunderscore}{\kern0pt}lambda\ v{\isacharparenright}{\kern0pt}\ s{\isacharparenright}{\kern0pt}{\isacharcomma}{\kern0pt}\ d{\isacharparenleft}{\kern0pt}s\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}iter{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max\ {\isacharparenleft}{\kern0pt}vec{\isacharunderscore}{\kern0pt}lambda\ v{\isacharparenright}{\kern0pt}\ s{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}{\isachardollar}{\kern0pt}{\isacharparenright}{\kern0pt}\ v{\isacharcomma}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isachardollar}{\kern0pt}{\isacharparenright}{\kern0pt}\ d{\isacharparenright}{\kern0pt}\ xs\ {\isacharequal}{\kern0pt}\ \isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}let\ {\isacharparenleft}{\kern0pt}v{\isacharprime}{\kern0pt}{\isacharcomma}{\kern0pt}\ d{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ foldl\ {\isacharparenleft}{\kern0pt}{\isasymlambda}{\isacharparenleft}{\kern0pt}v{\isacharcomma}{\kern0pt}\ d{\isacharparenright}{\kern0pt}\ s{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}vec{\isacharunderscore}{\kern0pt}upd\ v\ s\ {\isacharparenleft}{\kern0pt}GS{\isacharunderscore}{\kern0pt}iter{\isacharunderscore}{\kern0pt}max\ v\ s{\isacharparenright}{\kern0pt}{\isacharcomma}{\kern0pt}\ vec{\isacharunderscore}{\kern0pt}upd\ d\ s\ {\isacharparenleft}{\kern0pt}GS{\isacharunderscore}{\kern0pt}iter{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max\ v\ s{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}v{\isacharcomma}{\kern0pt}\ d{\isacharparenright}{\kern0pt}\ xs\ in\ {\isacharparenleft}{\kern0pt}vec{\isacharunderscore}{\kern0pt}nth\ v{\isacharprime}{\kern0pt}{\isacharcomma}{\kern0pt}\ vec{\isacharunderscore}{\kern0pt}nth\ d{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{for}\ d\ v\ xs\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}induction\ xs\ arbitrary{\isacharcolon}{\kern0pt}\ d\ v{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{case}\isamarkupfalse%
-\ Nil\isanewline
-\ \ \ \ \isacommand{then}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}case\ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{case}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}Cons\ a\ xs{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}case\isanewline
-\ \ \ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ \ vec{\isacharunderscore}{\kern0pt}lambda{\isacharunderscore}{\kern0pt}inverse\ Let{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ Cons{\isacharbrackleft}{\kern0pt}symmetric{\isacharcomma}{\kern0pt}\ unfolded\ Let{\isacharunderscore}{\kern0pt}def{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ vec{\isacharunderscore}{\kern0pt}lambda{\isacharunderscore}{\kern0pt}inverse\ vec{\isacharunderscore}{\kern0pt}upd{\isacharunderscore}{\kern0pt}def\ Let{\isacharunderscore}{\kern0pt}def\ eq{\isacharunderscore}{\kern0pt}commute\ cong{\isacharcolon}{\kern0pt}\ if{\isacharunderscore}{\kern0pt}cong{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{6}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}opt{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}gs{\isacharprime}{\kern0pt}{\isacharprime}{\kern0pt}\ {\isacharparenleft}{\kern0pt}vec{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}bfun\ v{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ vec{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}fun\ {\isacharparenleft}{\kern0pt}snd\ {\isacharparenleft}{\kern0pt}foldl\ {\isacharparenleft}{\kern0pt}{\isasymlambda}{\isacharparenleft}{\kern0pt}v{\isacharcomma}{\kern0pt}\ d{\isacharparenright}{\kern0pt}\ s{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}vec{\isacharunderscore}{\kern0pt}upd\ v\ s\ {\isacharparenleft}{\kern0pt}GS{\isacharunderscore}{\kern0pt}iter{\isacharunderscore}{\kern0pt}max\ v\ s{\isacharparenright}{\kern0pt}{\isacharcomma}{\kern0pt}\ vec{\isacharunderscore}{\kern0pt}upd\ d\ s\ {\isacharparenleft}{\kern0pt}GS{\isacharunderscore}{\kern0pt}iter{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max\ v\ s{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}v{\isacharcomma}{\kern0pt}\ d{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}sorted{\isacharunderscore}{\kern0pt}list{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}set\ UNIV{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{for}\ d\isanewline
-\ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ {\isadigit{1}}\isanewline
-\ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ {\isadigit{2}}{\isacharbrackleft}{\kern0pt}of\ {\isacharunderscore}{\kern0pt}\ {\isachardoublequoteopen}Rep{\isacharunderscore}{\kern0pt}Fun\ {\isacharparenleft}{\kern0pt}vec{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}fun\ d{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ {\isadigit{3}}\isanewline
-\ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ {\isadigit{4}}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isadigit{5}}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ Let{\isacharunderscore}{\kern0pt}def\ case{\isacharunderscore}{\kern0pt}prod{\isacharunderscore}{\kern0pt}beta\ vec{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}fun{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ Let{\isacharunderscore}{\kern0pt}def\ case{\isacharunderscore}{\kern0pt}prod{\isacharunderscore}{\kern0pt}beta\isanewline
-\ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ least{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}enum{\isacharunderscore}{\kern0pt}correct{\isadigit{1}}{\isacharbrackleft}{\kern0pt}OF\ A{\isacharunderscore}{\kern0pt}ne{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ least{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}enum{\isacharunderscore}{\kern0pt}correct{\isadigit{2}}{\isacharbrackleft}{\kern0pt}OF\ A{\isacharunderscore}{\kern0pt}ne{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ least{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}enum{\isacharunderscore}{\kern0pt}correct{\isadigit{2}}{\isacharbrackleft}{\kern0pt}OF\ A{\isacharunderscore}{\kern0pt}ne{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isadigit{6}}\ fin{\isacharunderscore}{\kern0pt}actions\ A{\isacharunderscore}{\kern0pt}ne\ \isanewline
-\ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ GS{\isacharunderscore}{\kern0pt}iter{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}def\ GS{\isacharunderscore}{\kern0pt}iter{\isacharunderscore}{\kern0pt}def\ GS{\isacharunderscore}{\kern0pt}iter{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}def\ \isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ cSup{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}Max\ split{\isacharunderscore}{\kern0pt}beta{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Modified Policy Iteration%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{sublocale}\isamarkupfalse%
-\ MDP{\isacharunderscore}{\kern0pt}MPI\ A\ K\ r\ l\ {\isachardoublequoteopen}{\isasymlambda}X{\isachardot}{\kern0pt}\ Least\ {\isacharparenleft}{\kern0pt}{\isasymlambda}x{\isachardot}{\kern0pt}\ x\ {\isasymin}\ X{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ MDP{\isacharunderscore}{\kern0pt}act{\isacharunderscore}{\kern0pt}axioms\ MDP{\isacharunderscore}{\kern0pt}reward{\isacharunderscore}{\kern0pt}axioms\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ unfold{\isacharunderscore}{\kern0pt}locales\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ {\isachardoublequoteopen}d{\isadigit{0}}\ s\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}LEAST\ a{\isachardot}{\kern0pt}\ a\ {\isasymin}\ A\ s{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isanewline
-\isacommand{lift{\isacharunderscore}{\kern0pt}definition}\isamarkupfalse%
-\ d{\isadigit{0}}{\isacharprime}{\kern0pt}\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}s{\isacharcomma}{\kern0pt}\ {\isacharprime}{\kern0pt}a{\isacharparenright}{\kern0pt}\ Fun{\isachardoublequoteclose}\ \isakeyword{is}\ d{\isadigit{0}}%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{{\isachardot}{\kern0pt}}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ d{\isadigit{0}}{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}is{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\ d{\isadigit{0}}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ A{\isacharunderscore}{\kern0pt}ne\ \isacommand{unfolding}\isamarkupfalse%
-\ d{\isadigit{0}}{\isacharunderscore}{\kern0pt}def\ is{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ simp%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ v{\isadigit{0}}{\isacharunderscore}{\kern0pt}code{\isacharbrackleft}{\kern0pt}code{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}v{\isadigit{0}}{\isacharunderscore}{\kern0pt}mpi\isactrlsub b\ {\isacharequal}{\kern0pt}\ vec{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}bfun\ {\isacharparenleft}{\kern0pt}{\isasymchi}\ s{\isachardot}{\kern0pt}\ r{\isacharunderscore}{\kern0pt}min\ {\isacharslash}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}\ {\isacharminus}{\kern0pt}\ l{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ v{\isadigit{0}}{\isacharunderscore}{\kern0pt}mpi\isactrlsub b{\isacharunderscore}{\kern0pt}def\ v{\isadigit{0}}{\isacharunderscore}{\kern0pt}mpi{\isacharunderscore}{\kern0pt}def\ vec{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}bfun{\isacharunderscore}{\kern0pt}def\ {\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ d{\isadigit{0}}{\isacharprime}{\kern0pt}{\isacharunderscore}{\kern0pt}code{\isacharbrackleft}{\kern0pt}code{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}d{\isadigit{0}}{\isacharprime}{\kern0pt}\ {\isacharequal}{\kern0pt}\ vec{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}fun\ {\isacharparenleft}{\kern0pt}{\isasymchi}\ s{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}LEAST\ a{\isachardot}{\kern0pt}\ a\ {\isasymin}\ A\ s{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ d{\isadigit{0}}{\isacharprime}{\kern0pt}{\isachardot}{\kern0pt}rep{\isacharunderscore}{\kern0pt}eq\ d{\isadigit{0}}{\isacharunderscore}{\kern0pt}def\ Rep{\isacharunderscore}{\kern0pt}Fun{\isacharunderscore}{\kern0pt}inject{\isacharbrackleft}{\kern0pt}symmetric{\isacharbrackright}{\kern0pt}\ vec{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}fun{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ step{\isacharunderscore}{\kern0pt}value{\isacharunderscore}{\kern0pt}code{\isacharbrackleft}{\kern0pt}code{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}L{\isacharunderscore}{\kern0pt}pow\ v\ d\ m\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}L{\isacharunderscore}{\kern0pt}det\ d\ {\isacharcircum}{\kern0pt}{\isacharcircum}{\kern0pt}\ Suc\ m{\isacharparenright}{\kern0pt}\ v{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ L{\isacharunderscore}{\kern0pt}pow{\isacharunderscore}{\kern0pt}def\ L{\isacharunderscore}{\kern0pt}det{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{partial{\isacharunderscore}{\kern0pt}function}\isamarkupfalse%
-\ \ {\isacharparenleft}{\kern0pt}tailrec{\isacharparenright}{\kern0pt}\ mpi{\isacharunderscore}{\kern0pt}partial\ \isakeyword{where}\ {\isacharbrackleft}{\kern0pt}code{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}mpi{\isacharunderscore}{\kern0pt}partial\ eps\ d\ v\ m\ {\isacharequal}{\kern0pt}\ \isanewline
-\ \ {\isacharparenleft}{\kern0pt}let\ d{\isacharprime}{\kern0pt}\ {\isacharequal}{\kern0pt}\ policy{\isacharunderscore}{\kern0pt}improvement\ d\ v\ in\ {\isacharparenleft}{\kern0pt}\isanewline
-\ \ \ \ if\ {\isadigit{2}}\ {\isacharasterisk}{\kern0pt}\ l\ {\isacharasterisk}{\kern0pt}\ dist\ v\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ v{\isacharparenright}{\kern0pt}\ {\isacharless}{\kern0pt}\ \ eps\ {\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}\ {\isacharminus}{\kern0pt}\ l{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ then\ {\isacharparenleft}{\kern0pt}d{\isacharprime}{\kern0pt}{\isacharcomma}{\kern0pt}\ v{\isacharparenright}{\kern0pt}\ \isanewline
-\ \ \ \ else\ mpi{\isacharunderscore}{\kern0pt}partial\ eps\ d{\isacharprime}{\kern0pt}\ {\isacharparenleft}{\kern0pt}L{\isacharunderscore}{\kern0pt}pow\ v\ d{\isacharprime}{\kern0pt}\ {\isacharparenleft}{\kern0pt}m\ {\isadigit{0}}\ v{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymlambda}n{\isachardot}{\kern0pt}\ m\ {\isacharparenleft}{\kern0pt}Suc\ n{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ mpi{\isacharunderscore}{\kern0pt}partial{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}algo{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}eps\ {\isachargreater}{\kern0pt}\ {\isadigit{0}}{\isachardoublequoteclose}\ {\isachardoublequoteopen}d\ {\isasymin}\ D\isactrlsub D{\isachardoublequoteclose}\ {\isachardoublequoteopen}v\ {\isasymle}\ {\isasymL}\isactrlsub b\ v{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}mpi{\isacharunderscore}{\kern0pt}partial\ eps\ d\ v\ m\ {\isacharequal}{\kern0pt}\ mpi{\isacharunderscore}{\kern0pt}algo\ eps\ d\ v\ m{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}mpi{\isacharunderscore}{\kern0pt}algo{\isacharunderscore}{\kern0pt}dom\ eps\ {\isacharparenleft}{\kern0pt}d{\isacharcomma}{\kern0pt}v{\isacharcomma}{\kern0pt}m{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assms\ termination{\isacharunderscore}{\kern0pt}mpi{\isacharunderscore}{\kern0pt}algo\ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}induction\ rule{\isacharcolon}{\kern0pt}\ mpi{\isacharunderscore}{\kern0pt}algo{\isachardot}{\kern0pt}pinduct{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ Let{\isacharunderscore}{\kern0pt}def\ mpi{\isacharunderscore}{\kern0pt}algo{\isachardot}{\kern0pt}psimps\ mpi{\isacharunderscore}{\kern0pt}partial{\isachardot}{\kern0pt}simps{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lift{\isacharunderscore}{\kern0pt}definition}\isamarkupfalse%
-\ mpi{\isacharunderscore}{\kern0pt}partial{\isacharprime}{\kern0pt}\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}real\ {\isasymRightarrow}\ {\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}s{\isacharcomma}{\kern0pt}\ {\isacharprime}{\kern0pt}a{\isacharparenright}{\kern0pt}\ Fun\ {\isasymRightarrow}\ {\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}s\ {\isasymRightarrow}\isactrlsub b\ real{\isacharparenright}{\kern0pt}\ {\isasymRightarrow}\ {\isacharparenleft}{\kern0pt}nat\ {\isasymRightarrow}\ {\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}s\ {\isasymRightarrow}\isactrlsub b\ real{\isacharparenright}{\kern0pt}\ {\isasymRightarrow}\ nat{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ {\isasymRightarrow}\ {\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}s{\isacharcomma}{\kern0pt}\ {\isacharprime}{\kern0pt}a{\isacharparenright}{\kern0pt}\ Fun\ {\isasymtimes}\ {\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}s\ {\isasymRightarrow}\isactrlsub b\ real{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{is}\ mpi{\isacharunderscore}{\kern0pt}partial%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{{\isachardot}{\kern0pt}}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ mpi{\isacharunderscore}{\kern0pt}partial{\isacharprime}{\kern0pt}{\isacharunderscore}{\kern0pt}code{\isacharbrackleft}{\kern0pt}code{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}mpi{\isacharunderscore}{\kern0pt}partial{\isacharprime}{\kern0pt}\ eps\ d\ v\ m\ {\isacharequal}{\kern0pt}\isanewline
-\ \ {\isacharparenleft}{\kern0pt}let\ d{\isacharprime}{\kern0pt}\ {\isacharequal}{\kern0pt}\ policy{\isacharunderscore}{\kern0pt}improvement{\isacharprime}{\kern0pt}\ d\ v\ in\ {\isacharparenleft}{\kern0pt}\isanewline
-\ \ \ \ if\ {\isadigit{2}}\ {\isacharasterisk}{\kern0pt}\ l\ {\isacharasterisk}{\kern0pt}\ dist\ v\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ v{\isacharparenright}{\kern0pt}\ {\isacharless}{\kern0pt}\ \ eps\ {\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}\ {\isacharminus}{\kern0pt}\ l{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ then\ {\isacharparenleft}{\kern0pt}d{\isacharprime}{\kern0pt}{\isacharcomma}{\kern0pt}\ v{\isacharparenright}{\kern0pt}\ \isanewline
-\ \ \ \ else\ mpi{\isacharunderscore}{\kern0pt}partial{\isacharprime}{\kern0pt}\ eps\ d{\isacharprime}{\kern0pt}\ {\isacharparenleft}{\kern0pt}L{\isacharunderscore}{\kern0pt}pow\ v\ {\isacharparenleft}{\kern0pt}Rep{\isacharunderscore}{\kern0pt}Fun\ d{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}m\ {\isadigit{0}}\ v{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymlambda}n{\isachardot}{\kern0pt}\ m\ {\isacharparenleft}{\kern0pt}Suc\ n{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ mpi{\isacharunderscore}{\kern0pt}partial{\isacharprime}{\kern0pt}{\isacharunderscore}{\kern0pt}def\ mpi{\isacharunderscore}{\kern0pt}partial{\isachardot}{\kern0pt}simps\ Let{\isacharunderscore}{\kern0pt}def\ policy{\isacharunderscore}{\kern0pt}improvement{\isacharprime}{\kern0pt}{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ r{\isacharunderscore}{\kern0pt}min{\isacharunderscore}{\kern0pt}code{\isacharbrackleft}{\kern0pt}code{\isacharunderscore}{\kern0pt}unfold{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}r{\isacharunderscore}{\kern0pt}min\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}MIN\ s{\isachardot}{\kern0pt}\ MIN\ a{\isachardot}{\kern0pt}\ r{\isacharparenleft}{\kern0pt}s{\isacharcomma}{\kern0pt}a{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ cInf{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}Min{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ mpi{\isacharunderscore}{\kern0pt}user{\isacharunderscore}{\kern0pt}code{\isacharbrackleft}{\kern0pt}code{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}mpi{\isacharunderscore}{\kern0pt}user\ eps\ m\ {\isacharequal}{\kern0pt}\ \isanewline
-\ \ {\isacharparenleft}{\kern0pt}if\ eps\ {\isasymle}\ {\isadigit{0}}\ then\ undefined\ else\ \isanewline
-\ \ \ \ let\ {\isacharparenleft}{\kern0pt}d{\isacharcomma}{\kern0pt}\ v{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ mpi{\isacharunderscore}{\kern0pt}partial{\isacharprime}{\kern0pt}\ eps\ d{\isadigit{0}}{\isacharprime}{\kern0pt}\ v{\isadigit{0}}{\isacharunderscore}{\kern0pt}mpi\isactrlsub b\ m\ in\ {\isacharparenleft}{\kern0pt}Rep{\isacharunderscore}{\kern0pt}Fun\ d{\isacharcomma}{\kern0pt}\ v{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ mpi{\isacharunderscore}{\kern0pt}user{\isacharunderscore}{\kern0pt}def\ case{\isacharunderscore}{\kern0pt}prod{\isacharunderscore}{\kern0pt}beta\ mpi{\isacharunderscore}{\kern0pt}partial{\isacharprime}{\kern0pt}{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \isacommand{using}\isamarkupfalse%
-\ mpi{\isacharunderscore}{\kern0pt}partial{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}algo\ A{\isacharunderscore}{\kern0pt}ne\ v{\isadigit{0}}{\isacharunderscore}{\kern0pt}mpi\isactrlsub b{\isacharunderscore}{\kern0pt}le{\isacharunderscore}{\kern0pt}{\isasymL}\isactrlsub b\ d{\isadigit{0}}{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ d{\isadigit{0}}{\isacharprime}{\kern0pt}{\isachardot}{\kern0pt}rep{\isacharunderscore}{\kern0pt}eq\ d{\isadigit{0}}{\isacharunderscore}{\kern0pt}def{\isacharbrackleft}{\kern0pt}symmetric{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isacommand{end}\isamarkupfalse%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Auxiliary Equations%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{lemma}\isamarkupfalse%
-\ {\isacharbrackleft}{\kern0pt}code{\isacharunderscore}{\kern0pt}unfold{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}dist\ {\isacharparenleft}{\kern0pt}f{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}{\isacharprime}{\kern0pt}a{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}finite\ {\isasymRightarrow}\isactrlsub b\ {\isacharprime}{\kern0pt}b{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}metric{\isacharunderscore}{\kern0pt}space{\isacharparenright}{\kern0pt}\ g\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}MAX\ a{\isachardot}{\kern0pt}\ dist\ {\isacharparenleft}{\kern0pt}apply{\isacharunderscore}{\kern0pt}bfun\ f\ a{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}g\ a{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ dist{\isacharunderscore}{\kern0pt}bfun{\isacharunderscore}{\kern0pt}def\ cSup{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}Max{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ member{\isacharunderscore}{\kern0pt}code{\isacharbrackleft}{\kern0pt}code\ del{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ List{\isachardot}{\kern0pt}coset\ xs\ {\isasymlongleftrightarrow}\ {\isasymnot}\ List{\isachardot}{\kern0pt}member\ xs\ x{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ in{\isacharunderscore}{\kern0pt}set{\isacharunderscore}{\kern0pt}member{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ {\isacharbrackleft}{\kern0pt}code{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}iarray{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}vec\ v\ {\isacharplus}{\kern0pt}\ iarray{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}vec\ u\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}Matrix{\isacharunderscore}{\kern0pt}To{\isacharunderscore}{\kern0pt}IArray{\isachardot}{\kern0pt}iarray{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}vec\ {\isacharparenleft}{\kern0pt}Rep{\isacharunderscore}{\kern0pt}iarray{\isacharunderscore}{\kern0pt}type\ v\ {\isacharplus}{\kern0pt}\ Rep{\isacharunderscore}{\kern0pt}iarray{\isacharunderscore}{\kern0pt}type\ u{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharcomma}{\kern0pt}\ lifting{\isacharparenright}{\kern0pt}\ Matrix{\isacharunderscore}{\kern0pt}To{\isacharunderscore}{\kern0pt}IArray{\isachardot}{\kern0pt}vec{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}iarray{\isacharunderscore}{\kern0pt}def\ iarray{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}vec{\isacharunderscore}{\kern0pt}vec{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}iarray\ vec{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}iarray{\isachardot}{\kern0pt}rep{\isacharunderscore}{\kern0pt}eq\ vec{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}iarray{\isacharunderscore}{\kern0pt}inv\ vec{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}iarray{\isacharunderscore}{\kern0pt}plus{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ {\isacharbrackleft}{\kern0pt}code{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}iarray{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}vec\ v\ {\isacharminus}{\kern0pt}\ iarray{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}vec\ u\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}Matrix{\isacharunderscore}{\kern0pt}To{\isacharunderscore}{\kern0pt}IArray{\isachardot}{\kern0pt}iarray{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}vec\ {\isacharparenleft}{\kern0pt}Rep{\isacharunderscore}{\kern0pt}iarray{\isacharunderscore}{\kern0pt}type\ v\ {\isacharminus}{\kern0pt}\ Rep{\isacharunderscore}{\kern0pt}iarray{\isacharunderscore}{\kern0pt}type\ u{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ minus{\isacharunderscore}{\kern0pt}iarray{\isacharunderscore}{\kern0pt}def\ Matrix{\isacharunderscore}{\kern0pt}To{\isacharunderscore}{\kern0pt}IArray{\isachardot}{\kern0pt}iarray{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}vec{\isacharunderscore}{\kern0pt}def\ iarray{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}vec{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ vec{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}iff\ to{\isacharunderscore}{\kern0pt}nat{\isacharunderscore}{\kern0pt}less{\isacharunderscore}{\kern0pt}card{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ matrix{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}iarray{\isacharunderscore}{\kern0pt}minus{\isacharbrackleft}{\kern0pt}code{\isacharunderscore}{\kern0pt}unfold{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}matrix{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}iarray\ {\isacharparenleft}{\kern0pt}A\ {\isacharminus}{\kern0pt}\ B{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ matrix{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}iarray\ A\ {\isacharminus}{\kern0pt}\ matrix{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}iarray\ B{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ matrix{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}iarray{\isacharunderscore}{\kern0pt}def\ o{\isacharunderscore}{\kern0pt}def\ minus{\isacharunderscore}{\kern0pt}iarray{\isacharunderscore}{\kern0pt}def\ Matrix{\isacharunderscore}{\kern0pt}To{\isacharunderscore}{\kern0pt}IArray{\isachardot}{\kern0pt}vec{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}iarray{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ simp%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{declare}\isamarkupfalse%
-\ matrix{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}iarray{\isacharunderscore}{\kern0pt}fst{\isacharunderscore}{\kern0pt}Gauss{\isacharunderscore}{\kern0pt}Jordan{\isacharunderscore}{\kern0pt}PA{\isacharbrackleft}{\kern0pt}code{\isacharunderscore}{\kern0pt}unfold{\isacharbrackright}{\kern0pt}\ \isanewline
-%
-\isadelimtheory
-\isanewline
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{end}\isamarkupfalse%
-%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\end{isabellebody}%
-\endinput
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-%:%1973=516%:%
-%:%1976=517%:%
-%:%1980=517%:%
-%:%1981=517%:%
-%:%1982=518%:%
-%:%1983=518%:%
-%:%1988=518%:%
-%:%1991=519%:%
-%:%1992=520%:%
-%:%1993=520%:%
-%:%1996=521%:%
-%:%2000=521%:%
-%:%2001=521%:%
-%:%2002=522%:%
-%:%2003=522%:%
-%:%2008=522%:%
-%:%2011=523%:%
-%:%2012=524%:%
-%:%2013=524%:%
-%:%2016=525%:%
-%:%2021=526%:%
diff --git a/thys/MDP-Algorithms/output/outline/Code_Gridworld.tex b/thys/MDP-Algorithms/output/outline/Code_Gridworld.tex
deleted file mode 100644
--- a/thys/MDP-Algorithms/output/outline/Code_Gridworld.tex
+++ /dev/null
@@ -1,493 +0,0 @@
-%
-\begin{isabellebody}%
-\setisabellecontext{Code{\isacharunderscore}{\kern0pt}Gridworld}%
-%
-\isadelimtheory
-\isanewline
-\isanewline
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{theory}\isamarkupfalse%
-\ Code{\isacharunderscore}{\kern0pt}Gridworld\isanewline
-\ \ \isakeyword{imports}\ \isanewline
-\ \ \ \ Code{\isacharunderscore}{\kern0pt}Mod\isanewline
-\isakeyword{begin}%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsection{Gridworld Example%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{lemma}\isamarkupfalse%
-\ {\isacharbrackleft}{\kern0pt}code\ abstype{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}embed{\isacharunderscore}{\kern0pt}pmf\ {\isacharparenleft}{\kern0pt}pmf\ P{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ P{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharcomma}{\kern0pt}\ lifting{\isacharparenright}{\kern0pt}\ td{\isacharunderscore}{\kern0pt}pmf{\isacharunderscore}{\kern0pt}embed{\isacharunderscore}{\kern0pt}pmf\ type{\isacharunderscore}{\kern0pt}definition{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemmas}\isamarkupfalse%
-\ {\isacharbrackleft}{\kern0pt}code{\isacharunderscore}{\kern0pt}abbrev\ del{\isacharbrackright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ pmf{\isacharunderscore}{\kern0pt}integral{\isacharunderscore}{\kern0pt}code{\isacharunderscore}{\kern0pt}unfold\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ {\isacharbrackleft}{\kern0pt}code{\isacharunderscore}{\kern0pt}unfold{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\ \isanewline
-\ \ {\isachardoublequoteopen}measure{\isacharunderscore}{\kern0pt}pmf{\isachardot}{\kern0pt}expectation\ P\ {\isacharparenleft}{\kern0pt}f\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isacharprime}{\kern0pt}a\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ enum\ {\isasymRightarrow}\ real{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSum}x\ {\isasymin}\ UNIV{\isachardot}{\kern0pt}\ pmf\ P\ x\ {\isacharasterisk}{\kern0pt}\ f\ x{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharcomma}{\kern0pt}\ lifting{\isacharparenright}{\kern0pt}\ UNIV{\isacharunderscore}{\kern0pt}I\ finite{\isacharunderscore}{\kern0pt}class{\isachardot}{\kern0pt}finite{\isacharunderscore}{\kern0pt}UNIV\ integral{\isacharunderscore}{\kern0pt}measure{\isacharunderscore}{\kern0pt}pmf\ \isanewline
-\ \ \ \ \ \ real{\isacharunderscore}{\kern0pt}scaleR{\isacharunderscore}{\kern0pt}def\ sum{\isachardot}{\kern0pt}cong{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ {\isacharbrackleft}{\kern0pt}code{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}pmf\ {\isacharparenleft}{\kern0pt}return{\isacharunderscore}{\kern0pt}pmf\ x{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymlambda}y{\isachardot}{\kern0pt}\ indicat{\isacharunderscore}{\kern0pt}real\ {\isacharbraceleft}{\kern0pt}y{\isacharbraceright}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isachardoublequoteclose}\ \isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ {\isacharbrackleft}{\kern0pt}code{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\ \isanewline
-\ \ {\isachardoublequoteopen}pmf\ {\isacharparenleft}{\kern0pt}bind{\isacharunderscore}{\kern0pt}pmf\ N\ f{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymlambda}i\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isacharprime}{\kern0pt}a{\isachardot}{\kern0pt}\ measure{\isacharunderscore}{\kern0pt}pmf{\isachardot}{\kern0pt}expectation\ N\ {\isacharparenleft}{\kern0pt}{\isasymlambda}{\isacharparenleft}{\kern0pt}x\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isacharprime}{\kern0pt}b\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}enum{\isacharparenright}{\kern0pt}{\isachardot}{\kern0pt}\ pmf\ {\isacharparenleft}{\kern0pt}f\ x{\isacharparenright}{\kern0pt}\ i{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ Probability{\isacharunderscore}{\kern0pt}Mass{\isacharunderscore}{\kern0pt}Function{\isachardot}{\kern0pt}pmf{\isacharunderscore}{\kern0pt}bind\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ fast%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{type{\isacharunderscore}{\kern0pt}synonym}\isamarkupfalse%
-\ state{\isacharunderscore}{\kern0pt}robot\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}{\isadigit{1}}{\isadigit{3}}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ {\isachardoublequoteopen}from{\isacharunderscore}{\kern0pt}state\ x\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}Rep\ x\ div\ {\isadigit{4}}{\isacharcomma}{\kern0pt}\ Rep\ x\ mod\ {\isadigit{4}}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ {\isachardoublequoteopen}to{\isacharunderscore}{\kern0pt}state\ x\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}Abs\ {\isacharparenleft}{\kern0pt}fst\ x\ {\isacharasterisk}{\kern0pt}\ {\isadigit{4}}\ {\isacharplus}{\kern0pt}\ snd\ x{\isacharparenright}{\kern0pt}\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ state{\isacharunderscore}{\kern0pt}robot{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isanewline
-\isacommand{type{\isacharunderscore}{\kern0pt}synonym}\isamarkupfalse%
-\ action{\isacharunderscore}{\kern0pt}robot\ {\isacharequal}{\kern0pt}\ {\isadigit{4}}\isanewline
-\isanewline
-\isacommand{fun}\isamarkupfalse%
-\ A{\isacharunderscore}{\kern0pt}robot\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}state{\isacharunderscore}{\kern0pt}robot\ {\isasymRightarrow}\ action{\isacharunderscore}{\kern0pt}robot\ set{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}A{\isacharunderscore}{\kern0pt}robot\ pos\ {\isacharequal}{\kern0pt}\ UNIV{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{abbreviation}\isamarkupfalse%
-\ {\isachardoublequoteopen}noise\ {\isasymequiv}\ {\isacharparenleft}{\kern0pt}{\isadigit{0}}{\isachardot}{\kern0pt}{\isadigit{2}}\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ real{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{lift{\isacharunderscore}{\kern0pt}definition}\isamarkupfalse%
-\ add{\isacharunderscore}{\kern0pt}noise\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}action{\isacharunderscore}{\kern0pt}robot\ {\isasymRightarrow}\ action{\isacharunderscore}{\kern0pt}robot\ pmf{\isachardoublequoteclose}\ \isakeyword{is}\ {\isachardoublequoteopen}{\isasymlambda}det\ rnd{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}\isanewline
-\ \ if\ det\ {\isacharequal}{\kern0pt}\ rnd\ then\ {\isadigit{1}}\ {\isacharminus}{\kern0pt}\ noise\ else\ if\ det\ {\isacharequal}{\kern0pt}\ rnd\ {\isacharminus}{\kern0pt}\ {\isadigit{1}}\ {\isasymor}\ det\ {\isacharequal}{\kern0pt}\ rnd\ {\isacharplus}{\kern0pt}\ {\isadigit{1}}\ then\ noise\ {\isacharslash}{\kern0pt}\ {\isadigit{2}}\ else\ {\isadigit{0}}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{subgoal}\isamarkupfalse%
-\ \isakeyword{for}\ n\isanewline
-\ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ nn{\isacharunderscore}{\kern0pt}integral{\isacharunderscore}{\kern0pt}count{\isacharunderscore}{\kern0pt}space{\isacharunderscore}{\kern0pt}finite{\isacharbrackleft}{\kern0pt}OF\ finite{\isacharbrackright}{\kern0pt}\ UNIV{\isacharunderscore}{\kern0pt}{\isadigit{4}}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ exhaust{\isacharunderscore}{\kern0pt}{\isadigit{4}}{\isacharbrackleft}{\kern0pt}of\ n{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ fastforce\isanewline
-\ \ \isacommand{done}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{fun}\isamarkupfalse%
-\ r{\isacharunderscore}{\kern0pt}robot\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}state{\isacharunderscore}{\kern0pt}robot\ {\isasymtimes}\ action{\isacharunderscore}{\kern0pt}robot{\isacharparenright}{\kern0pt}\ {\isasymRightarrow}\ real{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}r{\isacharunderscore}{\kern0pt}robot\ {\isacharparenleft}{\kern0pt}s{\isacharcomma}{\kern0pt}a{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}\isanewline
-\ \ if\ from{\isacharunderscore}{\kern0pt}state\ s\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharcomma}{\kern0pt}{\isadigit{3}}{\isacharparenright}{\kern0pt}\ then\ {\isadigit{1}}\ else\isanewline
-\ \ if\ from{\isacharunderscore}{\kern0pt}state\ s\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharcomma}{\kern0pt}{\isadigit{3}}{\isacharparenright}{\kern0pt}\ then\ {\isacharminus}{\kern0pt}{\isadigit{1}}\ else\ \isanewline
-\ \ if\ from{\isacharunderscore}{\kern0pt}state\ s\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isadigit{3}}{\isacharcomma}{\kern0pt}{\isadigit{0}}{\isacharparenright}{\kern0pt}\ then\ {\isadigit{0}}\ else\isanewline
-\ \ {\isadigit{0}}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{fun}\isamarkupfalse%
-\ K{\isacharunderscore}{\kern0pt}robot\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}state{\isacharunderscore}{\kern0pt}robot\ {\isasymtimes}\ action{\isacharunderscore}{\kern0pt}robot{\isacharparenright}{\kern0pt}\ {\isasymRightarrow}\ state{\isacharunderscore}{\kern0pt}robot\ pmf{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}K{\isacharunderscore}{\kern0pt}robot\ {\isacharparenleft}{\kern0pt}loc{\isacharcomma}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ \isanewline
-\ \ do\ {\isacharbraceleft}{\kern0pt}\isanewline
-\ \ a\ {\isasymleftarrow}\ add{\isacharunderscore}{\kern0pt}noise\ a{\isacharsemicolon}{\kern0pt}\isanewline
-\ \ let\ {\isacharparenleft}{\kern0pt}y{\isacharcomma}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ from{\isacharunderscore}{\kern0pt}state\ loc{\isacharsemicolon}{\kern0pt}\isanewline
-\ \ let\ {\isacharparenleft}{\kern0pt}y{\isacharprime}{\kern0pt}{\isacharcomma}{\kern0pt}\ x{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ {\isacharparenleft}{\kern0pt}if\ a\ {\isacharequal}{\kern0pt}\ {\isadigit{0}}\ then\ {\isacharparenleft}{\kern0pt}y\ {\isacharplus}{\kern0pt}\ {\isadigit{1}}\ {\isacharcomma}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ else\ if\ a\ {\isacharequal}{\kern0pt}\ {\isadigit{1}}\ then\ {\isacharparenleft}{\kern0pt}y{\isacharcomma}{\kern0pt}\ x{\isacharplus}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ else\ if\ a\ {\isacharequal}{\kern0pt}\ {\isadigit{2}}\ then\ {\isacharparenleft}{\kern0pt}y{\isacharminus}{\kern0pt}{\isadigit{1}}{\isacharcomma}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ else\ if\ a\ {\isacharequal}{\kern0pt}\ {\isadigit{3}}\ then\ {\isacharparenleft}{\kern0pt}y{\isacharcomma}{\kern0pt}\ x{\isacharminus}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ else\ undefined{\isacharparenright}{\kern0pt}{\isacharsemicolon}{\kern0pt}\isanewline
-\ \ \ return{\isacharunderscore}{\kern0pt}pmf\ {\isacharparenleft}{\kern0pt}\isanewline
-\ \ \ \ \ \ if\ {\isacharparenleft}{\kern0pt}y{\isacharcomma}{\kern0pt}x{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharcomma}{\kern0pt}{\isadigit{3}}{\isacharparenright}{\kern0pt}\ {\isasymor}\ {\isacharparenleft}{\kern0pt}y{\isacharcomma}{\kern0pt}x{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharcomma}{\kern0pt}{\isadigit{3}}{\isacharparenright}{\kern0pt}\ {\isasymor}\ {\isacharparenleft}{\kern0pt}y{\isacharcomma}{\kern0pt}x{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isadigit{3}}{\isacharcomma}{\kern0pt}{\isadigit{0}}{\isacharparenright}{\kern0pt}\ \isanewline
-\ \ \ \ \ \ \ \ then\ to{\isacharunderscore}{\kern0pt}state\ {\isacharparenleft}{\kern0pt}{\isadigit{3}}{\isacharcomma}{\kern0pt}{\isadigit{0}}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ else\ if\ y{\isacharprime}{\kern0pt}\ {\isacharless}{\kern0pt}\ {\isadigit{0}}\ {\isasymor}\ y{\isacharprime}{\kern0pt}\ {\isachargreater}{\kern0pt}\ {\isadigit{2}}\ {\isasymor}\ x{\isacharprime}{\kern0pt}\ {\isacharless}{\kern0pt}\ {\isadigit{0}}\ {\isasymor}\ x{\isacharprime}{\kern0pt}\ {\isachargreater}{\kern0pt}\ {\isadigit{3}}\ {\isasymor}\ {\isacharparenleft}{\kern0pt}y{\isacharprime}{\kern0pt}{\isacharcomma}{\kern0pt}x{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharcomma}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ then\ to{\isacharunderscore}{\kern0pt}state\ {\isacharparenleft}{\kern0pt}y{\isacharcomma}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ else\ to{\isacharunderscore}{\kern0pt}state\ {\isacharparenleft}{\kern0pt}y{\isacharprime}{\kern0pt}{\isacharcomma}{\kern0pt}\ x{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ {\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ {\isachardoublequoteopen}l{\isacharunderscore}{\kern0pt}robot\ {\isacharequal}{\kern0pt}\ {\isadigit{0}}{\isachardot}{\kern0pt}{\isadigit{9}}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ {\isachardoublequoteopen}vi{\isacharunderscore}{\kern0pt}code\ A{\isacharunderscore}{\kern0pt}robot\ r{\isacharunderscore}{\kern0pt}robot\ l{\isacharunderscore}{\kern0pt}robot{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ standard\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ l{\isacharunderscore}{\kern0pt}robot{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{abbreviation}\isamarkupfalse%
-\ {\isachardoublequoteopen}to{\isacharunderscore}{\kern0pt}gridworld\ f\ {\isasymequiv}\ f\ K{\isacharunderscore}{\kern0pt}robot\ r{\isacharunderscore}{\kern0pt}robot\ l{\isacharunderscore}{\kern0pt}robot{\isachardoublequoteclose}\isanewline
-\isacommand{abbreviation}\isamarkupfalse%
-\ {\isachardoublequoteopen}to{\isacharunderscore}{\kern0pt}gridworld{\isacharprime}{\kern0pt}\ f\ {\isasymequiv}\ f\ K{\isacharunderscore}{\kern0pt}robot\ A{\isacharunderscore}{\kern0pt}robot\ r{\isacharunderscore}{\kern0pt}robot\ l{\isacharunderscore}{\kern0pt}robot{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{abbreviation}\isamarkupfalse%
-\ {\isachardoublequoteopen}gridworld{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}eval{\isacharprime}{\kern0pt}\ {\isasymequiv}\ to{\isacharunderscore}{\kern0pt}gridworld\ mod{\isacharunderscore}{\kern0pt}MDP{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}eval{\isacharprime}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isacommand{abbreviation}\isamarkupfalse%
-\ {\isachardoublequoteopen}gridworld{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}step{\isacharprime}{\kern0pt}\ {\isasymequiv}\ to{\isacharunderscore}{\kern0pt}gridworld{\isacharprime}{\kern0pt}\ mod{\isacharunderscore}{\kern0pt}MDP{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}iteration{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}step{\isacharprime}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isacommand{abbreviation}\isamarkupfalse%
-\ {\isachardoublequoteopen}gridworld{\isacharunderscore}{\kern0pt}mpi{\isacharunderscore}{\kern0pt}user\ {\isasymequiv}\ to{\isacharunderscore}{\kern0pt}gridworld{\isacharprime}{\kern0pt}\ mod{\isacharunderscore}{\kern0pt}MDP{\isacharunderscore}{\kern0pt}mpi{\isacharunderscore}{\kern0pt}user{\isachardoublequoteclose}\isanewline
-\isacommand{abbreviation}\isamarkupfalse%
-\ {\isachardoublequoteopen}gridworld{\isacharunderscore}{\kern0pt}opt{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}gs\ {\isasymequiv}\ to{\isacharunderscore}{\kern0pt}gridworld{\isacharprime}{\kern0pt}\ mod{\isacharunderscore}{\kern0pt}MDP{\isacharunderscore}{\kern0pt}opt{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}gs{\isachardoublequoteclose}\isanewline
-\isacommand{abbreviation}\isamarkupfalse%
-\ {\isachardoublequoteopen}gridworld{\isacharunderscore}{\kern0pt}{\isasymL}\isactrlsub b\ {\isasymequiv}\ to{\isacharunderscore}{\kern0pt}gridworld{\isacharprime}{\kern0pt}\ mod{\isacharunderscore}{\kern0pt}MDP{\isacharunderscore}{\kern0pt}{\isasymL}\isactrlsub b{\isachardoublequoteclose}\isanewline
-\isacommand{abbreviation}\isamarkupfalse%
-\ {\isachardoublequoteopen}gridworld{\isacharunderscore}{\kern0pt}find{\isacharunderscore}{\kern0pt}policy{\isacharprime}{\kern0pt}\ {\isasymequiv}\ to{\isacharunderscore}{\kern0pt}gridworld{\isacharprime}{\kern0pt}\ mod{\isacharunderscore}{\kern0pt}MDP{\isacharunderscore}{\kern0pt}find{\isacharunderscore}{\kern0pt}policy{\isacharprime}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isacommand{abbreviation}\isamarkupfalse%
-\ {\isachardoublequoteopen}gridworld{\isacharunderscore}{\kern0pt}GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}fun\isactrlsub b\ {\isasymequiv}\ to{\isacharunderscore}{\kern0pt}gridworld{\isacharprime}{\kern0pt}\ mod{\isacharunderscore}{\kern0pt}MDP{\isacharunderscore}{\kern0pt}GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}fun\isactrlsub b{\isachardoublequoteclose}\isanewline
-\isacommand{abbreviation}\isamarkupfalse%
-\ {\isachardoublequoteopen}gridworld{\isacharunderscore}{\kern0pt}vi{\isacharunderscore}{\kern0pt}policy{\isacharprime}{\kern0pt}\ {\isasymequiv}\ to{\isacharunderscore}{\kern0pt}gridworld{\isacharprime}{\kern0pt}\ mod{\isacharunderscore}{\kern0pt}MDP{\isacharunderscore}{\kern0pt}vi{\isacharunderscore}{\kern0pt}policy{\isacharprime}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isacommand{abbreviation}\isamarkupfalse%
-\ {\isachardoublequoteopen}gridworld{\isacharunderscore}{\kern0pt}vi{\isacharunderscore}{\kern0pt}gs{\isacharunderscore}{\kern0pt}policy\ {\isasymequiv}\ to{\isacharunderscore}{\kern0pt}gridworld{\isacharprime}{\kern0pt}\ mod{\isacharunderscore}{\kern0pt}MDP{\isacharunderscore}{\kern0pt}vi{\isacharunderscore}{\kern0pt}gs{\isacharunderscore}{\kern0pt}policy{\isachardoublequoteclose}\isanewline
-\isacommand{abbreviation}\isamarkupfalse%
-\ {\isachardoublequoteopen}gridworld{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}iteration\ {\isasymequiv}\ to{\isacharunderscore}{\kern0pt}gridworld{\isacharprime}{\kern0pt}\ mod{\isacharunderscore}{\kern0pt}MDP{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}iteration{\isachardoublequoteclose}\isanewline
-\isanewline
-\isanewline
-\isacommand{fun}\isamarkupfalse%
-\ pi{\isacharunderscore}{\kern0pt}robot{\isacharunderscore}{\kern0pt}n\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}pi{\isacharunderscore}{\kern0pt}robot{\isacharunderscore}{\kern0pt}n\ {\isadigit{0}}\ d\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}d{\isacharcomma}{\kern0pt}\ gridworld{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}eval{\isacharprime}{\kern0pt}\ d{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ {\isacharbar}{\kern0pt}\ \isanewline
-\ \ {\isachardoublequoteopen}pi{\isacharunderscore}{\kern0pt}robot{\isacharunderscore}{\kern0pt}n\ {\isacharparenleft}{\kern0pt}Suc\ n{\isacharparenright}{\kern0pt}\ d\ {\isacharequal}{\kern0pt}\ pi{\isacharunderscore}{\kern0pt}robot{\isacharunderscore}{\kern0pt}n\ n\ {\isacharparenleft}{\kern0pt}gridworld{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}step{\isacharprime}{\kern0pt}\ d{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ {\isachardoublequoteopen}mpi{\isacharunderscore}{\kern0pt}robot\ eps\ {\isacharequal}{\kern0pt}\ gridworld{\isacharunderscore}{\kern0pt}mpi{\isacharunderscore}{\kern0pt}user\ eps\ {\isacharparenleft}{\kern0pt}{\isasymlambda}{\isacharunderscore}{\kern0pt}{\isachardot}{\kern0pt}\ {\isadigit{3}}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{fun}\isamarkupfalse%
-\ gs{\isacharunderscore}{\kern0pt}robot{\isacharunderscore}{\kern0pt}n\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}gs{\isacharunderscore}{\kern0pt}robot{\isacharunderscore}{\kern0pt}n\ {\isacharparenleft}{\kern0pt}{\isadigit{0}}\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ nat{\isacharparenright}{\kern0pt}\ v\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}gridworld{\isacharunderscore}{\kern0pt}opt{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}gs\ v{\isacharcomma}{\kern0pt}\ v{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ {\isacharbar}{\kern0pt}\isanewline
-\ \ {\isachardoublequoteopen}gs{\isacharunderscore}{\kern0pt}robot{\isacharunderscore}{\kern0pt}n\ {\isacharparenleft}{\kern0pt}Suc\ n\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ nat{\isacharparenright}{\kern0pt}\ v\ {\isacharequal}{\kern0pt}\ gs{\isacharunderscore}{\kern0pt}robot{\isacharunderscore}{\kern0pt}n\ n\ {\isacharparenleft}{\kern0pt}gridworld{\isacharunderscore}{\kern0pt}GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}fun\isactrlsub b\ v{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{fun}\isamarkupfalse%
-\ vi{\isacharunderscore}{\kern0pt}robot{\isacharunderscore}{\kern0pt}n\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}vi{\isacharunderscore}{\kern0pt}robot{\isacharunderscore}{\kern0pt}n\ {\isacharparenleft}{\kern0pt}{\isadigit{0}}\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ nat{\isacharparenright}{\kern0pt}\ v\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}gridworld{\isacharunderscore}{\kern0pt}find{\isacharunderscore}{\kern0pt}policy{\isacharprime}{\kern0pt}\ v{\isacharcomma}{\kern0pt}\ v{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ {\isacharbar}{\kern0pt}\isanewline
-\ \ {\isachardoublequoteopen}vi{\isacharunderscore}{\kern0pt}robot{\isacharunderscore}{\kern0pt}n\ {\isacharparenleft}{\kern0pt}Suc\ n\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ nat{\isacharparenright}{\kern0pt}\ v\ {\isacharequal}{\kern0pt}\ vi{\isacharunderscore}{\kern0pt}robot{\isacharunderscore}{\kern0pt}n\ n\ {\isacharparenleft}{\kern0pt}gridworld{\isacharunderscore}{\kern0pt}{\isasymL}\isactrlsub b\ v{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ {\isachardoublequoteopen}mpi{\isacharunderscore}{\kern0pt}result\ eps\ {\isacharequal}{\kern0pt}\ \isanewline
-\ \ {\isacharparenleft}{\kern0pt}let\ {\isacharparenleft}{\kern0pt}d{\isacharcomma}{\kern0pt}\ v{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ mpi{\isacharunderscore}{\kern0pt}robot\ eps\ in\ {\isacharparenleft}{\kern0pt}d{\isacharcomma}{\kern0pt}v{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ {\isachardoublequoteopen}gs{\isacharunderscore}{\kern0pt}result\ n\ {\isacharequal}{\kern0pt}\ \isanewline
-\ \ {\isacharparenleft}{\kern0pt}let\ {\isacharparenleft}{\kern0pt}d{\isacharcomma}{\kern0pt}v{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ gs{\isacharunderscore}{\kern0pt}robot{\isacharunderscore}{\kern0pt}n\ n\ {\isadigit{0}}\ in\ {\isacharparenleft}{\kern0pt}d{\isacharcomma}{\kern0pt}v{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ {\isachardoublequoteopen}vi{\isacharunderscore}{\kern0pt}result{\isacharunderscore}{\kern0pt}n\ n\ {\isacharequal}{\kern0pt}\ \isanewline
-\ \ {\isacharparenleft}{\kern0pt}let\ {\isacharparenleft}{\kern0pt}d{\isacharcomma}{\kern0pt}\ v{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ vi{\isacharunderscore}{\kern0pt}robot{\isacharunderscore}{\kern0pt}n\ n\ {\isadigit{0}}\ in\ {\isacharparenleft}{\kern0pt}d{\isacharcomma}{\kern0pt}v{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ {\isachardoublequoteopen}pi{\isacharunderscore}{\kern0pt}result{\isacharunderscore}{\kern0pt}n\ n\ {\isacharequal}{\kern0pt}\ \isanewline
-\ \ {\isacharparenleft}{\kern0pt}let\ {\isacharparenleft}{\kern0pt}d{\isacharcomma}{\kern0pt}\ v{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ pi{\isacharunderscore}{\kern0pt}robot{\isacharunderscore}{\kern0pt}n\ n\ {\isacharparenleft}{\kern0pt}vec{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}fun\ {\isadigit{0}}{\isacharparenright}{\kern0pt}\ in\ {\isacharparenleft}{\kern0pt}d{\isacharcomma}{\kern0pt}v{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ {\isachardoublequoteopen}fun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}list\ f\ {\isacharequal}{\kern0pt}\ map\ f\ {\isacharparenleft}{\kern0pt}sorted{\isacharunderscore}{\kern0pt}list{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}set\ UNIV{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ {\isachardoublequoteopen}benchmark{\isacharunderscore}{\kern0pt}gs\ {\isacharequal}{\kern0pt}\ fun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}list\ {\isacharparenleft}{\kern0pt}gridworld{\isacharunderscore}{\kern0pt}vi{\isacharunderscore}{\kern0pt}policy{\isacharprime}{\kern0pt}\ {\isadigit{0}}{\isachardot}{\kern0pt}{\isadigit{1}}\ {\isadigit{0}}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ {\isachardoublequoteopen}benchmark{\isacharunderscore}{\kern0pt}vi\ {\isacharequal}{\kern0pt}\ fun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}list\ {\isacharparenleft}{\kern0pt}gridworld{\isacharunderscore}{\kern0pt}vi{\isacharunderscore}{\kern0pt}gs{\isacharunderscore}{\kern0pt}policy\ {\isadigit{0}}{\isachardot}{\kern0pt}{\isadigit{1}}\ {\isadigit{0}}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ {\isachardoublequoteopen}benchmark{\isacharunderscore}{\kern0pt}mpi\ {\isacharequal}{\kern0pt}\ fun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}list\ {\isacharparenleft}{\kern0pt}fst\ {\isacharparenleft}{\kern0pt}gridworld{\isacharunderscore}{\kern0pt}mpi{\isacharunderscore}{\kern0pt}user\ {\isadigit{0}}{\isachardot}{\kern0pt}{\isadigit{1}}\ {\isacharparenleft}{\kern0pt}{\isasymlambda}{\isacharunderscore}{\kern0pt}\ {\isacharunderscore}{\kern0pt}{\isachardot}{\kern0pt}\ {\isadigit{3}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ {\isachardoublequoteopen}benchmark{\isacharunderscore}{\kern0pt}pi\ {\isacharequal}{\kern0pt}\ fun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}list\ {\isacharparenleft}{\kern0pt}gridworld{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}iteration\ {\isadigit{0}}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isanewline
-\isanewline
-\isacommand{export{\isacharunderscore}{\kern0pt}code}\isamarkupfalse%
-\ benchmark{\isacharunderscore}{\kern0pt}gs\ benchmark{\isacharunderscore}{\kern0pt}vi\ benchmark{\isacharunderscore}{\kern0pt}mpi\ benchmark{\isacharunderscore}{\kern0pt}pi\ \isakeyword{in}\ Haskell\ \isakeyword{module{\isacharunderscore}{\kern0pt}name}\ DP\isanewline
-\isacommand{export{\isacharunderscore}{\kern0pt}code}\isamarkupfalse%
-\ benchmark{\isacharunderscore}{\kern0pt}gs\ benchmark{\isacharunderscore}{\kern0pt}vi\ benchmark{\isacharunderscore}{\kern0pt}mpi\ benchmark{\isacharunderscore}{\kern0pt}pi\ \isakeyword{in}\ SML\ \isakeyword{module{\isacharunderscore}{\kern0pt}name}\ DP\isanewline
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{end}\isamarkupfalse%
-%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\end{isabellebody}%
-\endinput
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diff --git a/thys/MDP-Algorithms/output/outline/Code_Inventory.tex b/thys/MDP-Algorithms/output/outline/Code_Inventory.tex
deleted file mode 100644
--- a/thys/MDP-Algorithms/output/outline/Code_Inventory.tex
+++ /dev/null
@@ -1,877 +0,0 @@
-%
-\begin{isabellebody}%
-\setisabellecontext{Code{\isacharunderscore}{\kern0pt}Inventory}%
-%
-\isadelimtheory
-\isanewline
-\isanewline
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{theory}\isamarkupfalse%
-\ Code{\isacharunderscore}{\kern0pt}Inventory\isanewline
-\ \ \isakeyword{imports}\ \isanewline
-\ \ \ \ Code{\isacharunderscore}{\kern0pt}Mod\isanewline
-\ \ \isanewline
-\ \ \ \ Code{\isacharunderscore}{\kern0pt}Real{\isacharunderscore}{\kern0pt}Approx{\isacharunderscore}{\kern0pt}By{\isacharunderscore}{\kern0pt}Float{\isacharunderscore}{\kern0pt}Fix\isanewline
-\isakeyword{begin}%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsection{Inventory Management Example%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{lemma}\isamarkupfalse%
-\ {\isacharbrackleft}{\kern0pt}code\ abstype{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}embed{\isacharunderscore}{\kern0pt}pmf\ {\isacharparenleft}{\kern0pt}pmf\ P{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ P{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharcomma}{\kern0pt}\ lifting{\isacharparenright}{\kern0pt}\ td{\isacharunderscore}{\kern0pt}pmf{\isacharunderscore}{\kern0pt}embed{\isacharunderscore}{\kern0pt}pmf\ type{\isacharunderscore}{\kern0pt}definition{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemmas}\isamarkupfalse%
-\ {\isacharbrackleft}{\kern0pt}code{\isacharunderscore}{\kern0pt}abbrev\ del{\isacharbrackright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ pmf{\isacharunderscore}{\kern0pt}integral{\isacharunderscore}{\kern0pt}code{\isacharunderscore}{\kern0pt}unfold\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ {\isacharbrackleft}{\kern0pt}code{\isacharunderscore}{\kern0pt}unfold{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\ \isanewline
-\ \ {\isachardoublequoteopen}measure{\isacharunderscore}{\kern0pt}pmf{\isachardot}{\kern0pt}expectation\ P\ {\isacharparenleft}{\kern0pt}f\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isacharprime}{\kern0pt}a\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ enum\ {\isasymRightarrow}\ real{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSum}x\ {\isasymin}\ UNIV{\isachardot}{\kern0pt}\ pmf\ P\ x\ {\isacharasterisk}{\kern0pt}\ f\ x{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharcomma}{\kern0pt}\ lifting{\isacharparenright}{\kern0pt}\ UNIV{\isacharunderscore}{\kern0pt}I\ finite{\isacharunderscore}{\kern0pt}class{\isachardot}{\kern0pt}finite{\isacharunderscore}{\kern0pt}UNIV\ integral{\isacharunderscore}{\kern0pt}measure{\isacharunderscore}{\kern0pt}pmf\ \isanewline
-\ \ \ \ \ \ real{\isacharunderscore}{\kern0pt}scaleR{\isacharunderscore}{\kern0pt}def\ sum{\isachardot}{\kern0pt}cong{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ {\isacharbrackleft}{\kern0pt}code{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}pmf\ {\isacharparenleft}{\kern0pt}return{\isacharunderscore}{\kern0pt}pmf\ x{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymlambda}y{\isachardot}{\kern0pt}\ indicat{\isacharunderscore}{\kern0pt}real\ {\isacharbraceleft}{\kern0pt}y{\isacharbraceright}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isachardoublequoteclose}\ \isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ {\isacharbrackleft}{\kern0pt}code{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\ \isanewline
-\ \ {\isachardoublequoteopen}pmf\ {\isacharparenleft}{\kern0pt}bind{\isacharunderscore}{\kern0pt}pmf\ N\ f{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymlambda}i\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isacharprime}{\kern0pt}a{\isachardot}{\kern0pt}\ measure{\isacharunderscore}{\kern0pt}pmf{\isachardot}{\kern0pt}expectation\ N\ {\isacharparenleft}{\kern0pt}{\isasymlambda}{\isacharparenleft}{\kern0pt}x\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isacharprime}{\kern0pt}b\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}enum{\isacharparenright}{\kern0pt}{\isachardot}{\kern0pt}\ pmf\ {\isacharparenleft}{\kern0pt}f\ x{\isacharparenright}{\kern0pt}\ i{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ Probability{\isacharunderscore}{\kern0pt}Mass{\isacharunderscore}{\kern0pt}Function{\isachardot}{\kern0pt}pmf{\isacharunderscore}{\kern0pt}bind\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ fast%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ pmf{\isacharunderscore}{\kern0pt}finite{\isacharunderscore}{\kern0pt}le{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite\ {\isacharparenleft}{\kern0pt}X\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}a{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}finite{\isacharparenright}{\kern0pt}\ set{\isacharparenright}{\kern0pt}\ {\isasymLongrightarrow}\ sum\ {\isacharparenleft}{\kern0pt}pmf\ p{\isacharparenright}{\kern0pt}\ X\ {\isasymle}\ {\isadigit{1}}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}subst\ sum{\isacharunderscore}{\kern0pt}pmf{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}{\isadigit{1}}{\isacharbrackleft}{\kern0pt}symmetric{\isacharcomma}{\kern0pt}\ of\ UNIV\ p{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}auto\ intro{\isacharcolon}{\kern0pt}\ sum{\isacharunderscore}{\kern0pt}mono{\isadigit{2}}{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ mod{\isacharunderscore}{\kern0pt}less{\isacharunderscore}{\kern0pt}diff{\isacharcolon}{\kern0pt}\ \isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}{\isadigit{0}}\ {\isacharless}{\kern0pt}\ {\isacharparenleft}{\kern0pt}x{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}{\isacharprime}{\kern0pt}s{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}{\isacharbraceleft}{\kern0pt}mod{\isacharunderscore}{\kern0pt}type{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ {\isachardoublequoteopen}x\ {\isasymle}\ y{\isachardoublequoteclose}\ \isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}y\ {\isacharminus}{\kern0pt}\ x\ {\isacharless}{\kern0pt}\ y{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isadigit{0}}\ {\isasymle}\ Rep\ y\ {\isacharminus}{\kern0pt}\ Rep\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assms\ mono{\isacharunderscore}{\kern0pt}Rep\ \isacommand{unfolding}\isamarkupfalse%
-\ mono{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}Rep\ y\ {\isacharminus}{\kern0pt}\ Rep\ x\ {\isacharequal}{\kern0pt}\ Rep\ {\isacharparenleft}{\kern0pt}y\ {\isacharminus}{\kern0pt}\ x{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ mod{\isacharunderscore}{\kern0pt}type{\isacharunderscore}{\kern0pt}class{\isachardot}{\kern0pt}diff{\isacharunderscore}{\kern0pt}def\ Rep{\isacharunderscore}{\kern0pt}Abs{\isacharunderscore}{\kern0pt}mod\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ Rep{\isacharunderscore}{\kern0pt}ge{\isacharunderscore}{\kern0pt}{\isadigit{0}}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ mod{\isacharunderscore}{\kern0pt}pos{\isacharunderscore}{\kern0pt}pos{\isacharunderscore}{\kern0pt}trivial{\isacharbrackleft}{\kern0pt}OF\ {\isacartoucheopen}{\isadigit{0}}\ {\isasymle}\ Rep\ y\ {\isacharminus}{\kern0pt}\ Rep\ x{\isacartoucheclose}\ order{\isachardot}{\kern0pt}strict{\isacharunderscore}{\kern0pt}trans{\isadigit{1}}{\isacharbrackleft}{\kern0pt}OF\ {\isacharunderscore}{\kern0pt}\ Rep{\isacharunderscore}{\kern0pt}less{\isacharunderscore}{\kern0pt}n{\isacharcomma}{\kern0pt}\ of\ {\isacharunderscore}{\kern0pt}\ y{\isacharbrackright}{\kern0pt}{\isacharcomma}{\kern0pt}\ symmetric{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}Rep\ y\ {\isacharminus}{\kern0pt}\ Rep\ x\ {\isacharless}{\kern0pt}\ Rep\ y{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assms{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}\ strict{\isacharunderscore}{\kern0pt}mono{\isacharunderscore}{\kern0pt}Rep\ Rep{\isacharunderscore}{\kern0pt}ge{\isacharunderscore}{\kern0pt}{\isadigit{0}}\ le{\isacharunderscore}{\kern0pt}less\ not{\isacharunderscore}{\kern0pt}less\ \isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}fastforce\ simp{\isacharcolon}{\kern0pt}\ strict{\isacharunderscore}{\kern0pt}mono{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}Rep\ {\isacharparenleft}{\kern0pt}y\ {\isacharminus}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isacharless}{\kern0pt}\ Rep\ y{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ {\isadigit{1}}\ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ not{\isacharunderscore}{\kern0pt}less{\isacharunderscore}{\kern0pt}iff{\isacharunderscore}{\kern0pt}gr{\isacharunderscore}{\kern0pt}or{\isacharunderscore}{\kern0pt}eq\ strict{\isacharunderscore}{\kern0pt}mono{\isacharunderscore}{\kern0pt}Rep\ strict{\isacharunderscore}{\kern0pt}mono{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{locale}\isamarkupfalse%
-\ inventory\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \isakeyword{fixes}\ fixed{\isacharunderscore}{\kern0pt}cost\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ real\isanewline
-\ \ \ \ \isakeyword{and}\ var{\isacharunderscore}{\kern0pt}cost\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}s{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}{\isacharbraceleft}{\kern0pt}mod{\isacharunderscore}{\kern0pt}type{\isacharcomma}{\kern0pt}\ finite{\isacharbraceright}{\kern0pt}\ {\isasymRightarrow}\ real{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isakeyword{and}\ inv{\isacharunderscore}{\kern0pt}cost\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}s\ {\isasymRightarrow}\ real{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isakeyword{and}\ demand{\isacharunderscore}{\kern0pt}prob\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}s\ pmf{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isakeyword{and}\ revenue\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}s\ {\isasymRightarrow}\ real{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isakeyword{and}\ discount\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}real{\isachardoublequoteclose}\isanewline
-\isakeyword{begin}\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ {\isachardoublequoteopen}order{\isacharunderscore}{\kern0pt}cost\ u\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}if\ u\ {\isacharequal}{\kern0pt}\ {\isadigit{0}}\ then\ {\isadigit{0}}\ else\ fixed{\isacharunderscore}{\kern0pt}cost\ {\isacharplus}{\kern0pt}\ var{\isacharunderscore}{\kern0pt}cost\ u{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isanewline
-\isacommand{definition}\isamarkupfalse%
-\ {\isachardoublequoteopen}prob{\isacharunderscore}{\kern0pt}ge{\isacharunderscore}{\kern0pt}inv\ u\ {\isacharequal}{\kern0pt}\ {\isadigit{1}}\ {\isacharminus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSum}j{\isacharless}{\kern0pt}u{\isachardot}{\kern0pt}\ pmf\ demand{\isacharunderscore}{\kern0pt}prob\ j{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ {\isachardoublequoteopen}exp{\isacharunderscore}{\kern0pt}rev\ u\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSum}j{\isacharless}{\kern0pt}u{\isachardot}{\kern0pt}\ revenue\ j\ {\isacharasterisk}{\kern0pt}\ pmf\ demand{\isacharunderscore}{\kern0pt}prob\ j{\isacharparenright}{\kern0pt}\ {\isacharplus}{\kern0pt}\ revenue\ u\ {\isacharasterisk}{\kern0pt}\ prob{\isacharunderscore}{\kern0pt}ge{\isacharunderscore}{\kern0pt}inv\ u{\isachardoublequoteclose}\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ {\isachardoublequoteopen}reward\ sa\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}case\ sa\ of\ {\isacharparenleft}{\kern0pt}s{\isacharcomma}{\kern0pt}a{\isacharparenright}{\kern0pt}\ {\isasymRightarrow}\ exp{\isacharunderscore}{\kern0pt}rev\ {\isacharparenleft}{\kern0pt}s\ {\isacharplus}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isacharminus}{\kern0pt}\ order{\isacharunderscore}{\kern0pt}cost\ a\ {\isacharminus}{\kern0pt}\ inv{\isacharunderscore}{\kern0pt}cost\ {\isacharparenleft}{\kern0pt}s\ {\isacharplus}{\kern0pt}\ a{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isacommand{lift{\isacharunderscore}{\kern0pt}definition}\isamarkupfalse%
-\ transition\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}s\ {\isasymtimes}\ {\isacharprime}{\kern0pt}s{\isacharparenright}{\kern0pt}\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}s\ pmf{\isachardoublequoteclose}\ \isakeyword{is}\ {\isachardoublequoteopen}{\isasymlambda}{\isacharparenleft}{\kern0pt}s{\isacharcomma}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ s{\isacharprime}{\kern0pt}{\isachardot}{\kern0pt}\ \isanewline
-\ \ {\isacharparenleft}{\kern0pt}if\ CARD{\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}s{\isacharparenright}{\kern0pt}\ {\isasymle}\ Rep\ s\ {\isacharplus}{\kern0pt}\ Rep\ a\ \isanewline
-\ \ then\ {\isacharparenleft}{\kern0pt}if\ s{\isacharprime}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isadigit{0}}\ then\ {\isadigit{1}}\ else\ {\isadigit{0}}{\isacharparenright}{\kern0pt}\isanewline
-\ \ else\ {\isacharparenleft}{\kern0pt}if\ s\ {\isacharplus}{\kern0pt}\ a\ {\isacharless}{\kern0pt}\ s{\isacharprime}{\kern0pt}\ then\ {\isadigit{0}}\ else\isanewline
-\ \ \ if\ s{\isacharprime}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isadigit{0}}\ then\ prob{\isacharunderscore}{\kern0pt}ge{\isacharunderscore}{\kern0pt}inv\ {\isacharparenleft}{\kern0pt}s{\isacharplus}{\kern0pt}a{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ else\ pmf\ demand{\isacharunderscore}{\kern0pt}prob\ {\isacharparenleft}{\kern0pt}s\ {\isacharplus}{\kern0pt}\ a\ {\isacharminus}{\kern0pt}\ s{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ {\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}safe{\isacharcomma}{\kern0pt}\ goal{\isacharunderscore}{\kern0pt}cases{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{case}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}\ a\ b\ x{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{then}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}case\ \isacommand{unfolding}\isamarkupfalse%
-\ prob{\isacharunderscore}{\kern0pt}ge{\isacharunderscore}{\kern0pt}inv{\isacharunderscore}{\kern0pt}def\ \isacommand{using}\isamarkupfalse%
-\ pmf{\isacharunderscore}{\kern0pt}finite{\isacharunderscore}{\kern0pt}le\ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{case}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}{\isadigit{2}}\ a\ b{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{then}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}case\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}cases\ {\isachardoublequoteopen}int\ CARD{\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}s{\isacharparenright}{\kern0pt}\ {\isasymle}\ Rep\ a\ {\isacharplus}{\kern0pt}\ Rep\ b{\isachardoublequoteclose}{\isacharparenright}{\kern0pt}\ \ \isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{case}\isamarkupfalse%
-\ False\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isasymintegral}\isactrlsup {\isacharplus}{\kern0pt}\ x{\isachardot}{\kern0pt}\ ennreal\ {\isacharparenleft}{\kern0pt}if\ int\ CARD{\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}s{\isacharparenright}{\kern0pt}\ {\isasymle}\ Rep\ a\ {\isacharplus}{\kern0pt}\ Rep\ b\ then\ if\ x\ {\isacharequal}{\kern0pt}\ {\isadigit{0}}\ then\ {\isadigit{1}}\ else\ {\isadigit{0}}\ else\ if\ a\ {\isacharplus}{\kern0pt}\ b\ {\isacharless}{\kern0pt}\ x\ then\ {\isadigit{0}}\ else\ if\ x\ {\isacharequal}{\kern0pt}\ {\isadigit{0}}\ then\ prob{\isacharunderscore}{\kern0pt}ge{\isacharunderscore}{\kern0pt}inv\ {\isacharparenleft}{\kern0pt}a\ {\isacharplus}{\kern0pt}\ b{\isacharparenright}{\kern0pt}\ else\ pmf\ demand{\isacharunderscore}{\kern0pt}prob\ {\isacharparenleft}{\kern0pt}a\ {\isacharplus}{\kern0pt}\ b\ {\isacharminus}{\kern0pt}\ x{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasympartial}count{\isacharunderscore}{\kern0pt}space\ UNIV{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymSum}x\ {\isasymin}\ UNIV{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}if\ a\ {\isacharplus}{\kern0pt}\ b\ {\isacharless}{\kern0pt}\ x\ then\ {\isadigit{0}}\ else\ if\ x\ {\isacharequal}{\kern0pt}\ {\isadigit{0}}\ then\ prob{\isacharunderscore}{\kern0pt}ge{\isacharunderscore}{\kern0pt}inv\ {\isacharparenleft}{\kern0pt}a\ {\isacharplus}{\kern0pt}\ b{\isacharparenright}{\kern0pt}\ else\ pmf\ demand{\isacharunderscore}{\kern0pt}prob\ {\isacharparenleft}{\kern0pt}a\ {\isacharplus}{\kern0pt}\ b\ {\isacharminus}{\kern0pt}\ x{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ pmf{\isacharunderscore}{\kern0pt}nonneg\ prob{\isacharunderscore}{\kern0pt}ge{\isacharunderscore}{\kern0pt}inv{\isacharunderscore}{\kern0pt}def\ pmf{\isacharunderscore}{\kern0pt}finite{\isacharunderscore}{\kern0pt}le\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ nn{\isacharunderscore}{\kern0pt}integral{\isacharunderscore}{\kern0pt}count{\isacharunderscore}{\kern0pt}space{\isacharunderscore}{\kern0pt}finite\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ sum{\isacharunderscore}{\kern0pt}ennreal{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isacharequal}{\kern0pt}{\isacharparenleft}{\kern0pt}{\isasymSum}x\ {\isasymin}\ UNIV{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}if\ x\ {\isacharequal}{\kern0pt}\ {\isadigit{0}}\ then\ prob{\isacharunderscore}{\kern0pt}ge{\isacharunderscore}{\kern0pt}inv\ {\isacharparenleft}{\kern0pt}a\ {\isacharplus}{\kern0pt}\ b{\isacharparenright}{\kern0pt}\ else\ if\ a\ {\isacharplus}{\kern0pt}\ b\ {\isacharless}{\kern0pt}\ x\ then\ {\isadigit{0}}\ else\ pmf\ demand{\isacharunderscore}{\kern0pt}prob\ {\isacharparenleft}{\kern0pt}a\ {\isacharplus}{\kern0pt}\ b\ {\isacharminus}{\kern0pt}\ x{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ sum{\isachardot}{\kern0pt}cong\ ennreal{\isacharunderscore}{\kern0pt}cong\ simp\ add{\isacharcolon}{\kern0pt}\ leD\ least{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}type{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isacharequal}{\kern0pt}\ prob{\isacharunderscore}{\kern0pt}ge{\isacharunderscore}{\kern0pt}inv\ {\isacharparenleft}{\kern0pt}a\ {\isacharplus}{\kern0pt}\ b{\isacharparenright}{\kern0pt}\ {\isacharplus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSum}x\ {\isasymin}\ UNIV{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}{\isadigit{0}}{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}if\ a\ {\isacharplus}{\kern0pt}\ b\ {\isacharless}{\kern0pt}\ x\ then\ {\isadigit{0}}\ else\ pmf\ demand{\isacharunderscore}{\kern0pt}prob\ {\isacharparenleft}{\kern0pt}a\ {\isacharplus}{\kern0pt}\ b\ {\isacharminus}{\kern0pt}\ x{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ sum{\isachardot}{\kern0pt}remove{\isacharbrackleft}{\kern0pt}of\ UNIV\ {\isadigit{0}}{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isacharequal}{\kern0pt}\ prob{\isacharunderscore}{\kern0pt}ge{\isacharunderscore}{\kern0pt}inv\ {\isacharparenleft}{\kern0pt}a\ {\isacharplus}{\kern0pt}\ b{\isacharparenright}{\kern0pt}\ {\isacharplus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSum}x{\isasymin}{\isacharbraceleft}{\kern0pt}{\isadigit{0}}{\isacharless}{\kern0pt}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}if\ a\ {\isacharplus}{\kern0pt}\ b\ {\isacharless}{\kern0pt}\ x\ then\ {\isadigit{0}}\ else\ pmf\ demand{\isacharunderscore}{\kern0pt}prob\ {\isacharparenleft}{\kern0pt}a\ {\isacharplus}{\kern0pt}\ b\ {\isacharminus}{\kern0pt}\ x{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp\ add{\isacharcolon}{\kern0pt}\ greaterThan{\isacharunderscore}{\kern0pt}def\ le{\isacharunderscore}{\kern0pt}neq{\isacharunderscore}{\kern0pt}trans\ least{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}type\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ cong{\isacharbrackleft}{\kern0pt}of\ {\isachardoublequoteopen}sum\ {\isacharunderscore}{\kern0pt}{\isachardoublequoteclose}{\isacharbrackright}{\kern0pt}\ ennreal{\isacharunderscore}{\kern0pt}cong{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isacharequal}{\kern0pt}\ prob{\isacharunderscore}{\kern0pt}ge{\isacharunderscore}{\kern0pt}inv\ {\isacharparenleft}{\kern0pt}a\ {\isacharplus}{\kern0pt}\ b{\isacharparenright}{\kern0pt}\ {\isacharplus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSum}x{\isasymin}{\isacharbraceleft}{\kern0pt}{\isadigit{0}}{\isacharless}{\kern0pt}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}a{\isacharplus}{\kern0pt}b{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}pmf\ demand{\isacharunderscore}{\kern0pt}prob\ {\isacharparenleft}{\kern0pt}a\ {\isacharplus}{\kern0pt}\ b\ {\isacharminus}{\kern0pt}\ x{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ atMost{\isacharunderscore}{\kern0pt}def\ greaterThan{\isacharunderscore}{\kern0pt}def\ greaterThanAtMost{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ Collect{\isacharunderscore}{\kern0pt}neg{\isacharunderscore}{\kern0pt}eq{\isacharbrackleft}{\kern0pt}symmetric{\isacharbrackright}{\kern0pt}\ not{\isacharunderscore}{\kern0pt}less\ sum{\isachardot}{\kern0pt}If{\isacharunderscore}{\kern0pt}cases{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isacharequal}{\kern0pt}\ \ {\isadigit{1}}\ {\isacharminus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSum}j{\isacharless}{\kern0pt}{\isacharparenleft}{\kern0pt}a\ {\isacharplus}{\kern0pt}\ b{\isacharparenright}{\kern0pt}{\isachardot}{\kern0pt}\ pmf\ demand{\isacharunderscore}{\kern0pt}prob\ j{\isacharparenright}{\kern0pt}\ {\isacharplus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSum}x{\isasymin}{\isacharbraceleft}{\kern0pt}{\isadigit{0}}{\isacharless}{\kern0pt}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}a{\isacharplus}{\kern0pt}b{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ pmf\ demand{\isacharunderscore}{\kern0pt}prob\ {\isacharparenleft}{\kern0pt}a\ {\isacharplus}{\kern0pt}\ b\ {\isacharminus}{\kern0pt}\ x{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ prob{\isacharunderscore}{\kern0pt}ge{\isacharunderscore}{\kern0pt}inv{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isacharequal}{\kern0pt}\ \ {\isadigit{1}}\ {\isacharminus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSum}j{\isacharless}{\kern0pt}{\isacharparenleft}{\kern0pt}a\ {\isacharplus}{\kern0pt}\ b{\isacharparenright}{\kern0pt}{\isachardot}{\kern0pt}\ pmf\ demand{\isacharunderscore}{\kern0pt}prob\ j{\isacharparenright}{\kern0pt}\ {\isacharplus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSum}x{\isasymin}{\isacharbraceleft}{\kern0pt}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharless}{\kern0pt}a{\isacharplus}{\kern0pt}b{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}pmf\ demand{\isacharunderscore}{\kern0pt}prob\ x{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isasymSum}x{\isasymin}{\isacharbraceleft}{\kern0pt}{\isadigit{0}}{\isacharless}{\kern0pt}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}a{\isacharplus}{\kern0pt}b{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ pmf\ demand{\isacharunderscore}{\kern0pt}prob\ {\isacharparenleft}{\kern0pt}a\ {\isacharplus}{\kern0pt}\ b\ {\isacharminus}{\kern0pt}\ x{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSum}x{\isasymin}{\isacharbraceleft}{\kern0pt}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharless}{\kern0pt}a{\isacharplus}{\kern0pt}b{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}pmf\ demand{\isacharunderscore}{\kern0pt}prob\ x{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}intro\ sum{\isachardot}{\kern0pt}reindex{\isacharunderscore}{\kern0pt}bij{\isacharunderscore}{\kern0pt}betw\ bij{\isacharunderscore}{\kern0pt}betw{\isacharunderscore}{\kern0pt}imageI{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}inj{\isacharunderscore}{\kern0pt}on\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}{\isacharminus}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}a\ {\isacharplus}{\kern0pt}\ b{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isadigit{0}}{\isacharless}{\kern0pt}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}a\ {\isacharplus}{\kern0pt}\ b{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ inj{\isacharunderscore}{\kern0pt}on{\isacharunderscore}{\kern0pt}def\ \isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ add{\isachardot}{\kern0pt}left{\isacharunderscore}{\kern0pt}cancel\ diff{\isacharunderscore}{\kern0pt}add{\isacharunderscore}{\kern0pt}cancel{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}x\ {\isacharplus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}a\ {\isacharplus}{\kern0pt}\ b{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ a\ {\isacharplus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}b\ {\isacharplus}{\kern0pt}\ x{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{for}\ x\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ add{\isachardot}{\kern0pt}assoc\ add{\isachardot}{\kern0pt}commute\ add{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}nat{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}x\ {\isacharless}{\kern0pt}\ a\ {\isacharplus}{\kern0pt}\ b\ {\isasymLongrightarrow}\ {\isadigit{0}}\ {\isacharless}{\kern0pt}\ a\ {\isacharplus}{\kern0pt}\ b\ {\isacharminus}{\kern0pt}\ x{\isachardoublequoteclose}\ \isakeyword{for}\ x\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ add{\isachardot}{\kern0pt}left{\isacharunderscore}{\kern0pt}neutral\ diff{\isacharunderscore}{\kern0pt}add{\isacharunderscore}{\kern0pt}cancel\ least{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}type\ less{\isacharunderscore}{\kern0pt}le{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}x\ {\isacharless}{\kern0pt}\ a\ {\isacharplus}{\kern0pt}\ b\ {\isasymLongrightarrow}\ a\ {\isacharplus}{\kern0pt}\ b\ {\isacharminus}{\kern0pt}\ x\ {\isasymle}\ a\ {\isacharplus}{\kern0pt}\ b{\isachardoublequoteclose}\ \isakeyword{for}\ x\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ diff{\isacharunderscore}{\kern0pt}{\isadigit{0}}{\isacharunderscore}{\kern0pt}right\ least{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}type\ less{\isacharunderscore}{\kern0pt}le\ mod{\isacharunderscore}{\kern0pt}less{\isacharunderscore}{\kern0pt}diff\ not{\isacharunderscore}{\kern0pt}less{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}x\ {\isacharless}{\kern0pt}\ a\ {\isacharplus}{\kern0pt}\ b\ {\isasymLongrightarrow}\ {\isasymexists}xa{\isachardot}{\kern0pt}\ x\ {\isacharequal}{\kern0pt}\ a\ {\isacharplus}{\kern0pt}\ b\ {\isacharminus}{\kern0pt}\ xa\ {\isasymand}\ {\isadigit{0}}\ {\isacharless}{\kern0pt}\ xa\ {\isasymand}\ xa\ {\isasymle}\ a\ {\isacharplus}{\kern0pt}\ b{\isachardoublequoteclose}\ \isakeyword{for}\ x\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ algebra{\isacharunderscore}{\kern0pt}simps\ intro{\isacharcolon}{\kern0pt}\ exI{\isacharbrackleft}{\kern0pt}of\ {\isacharunderscore}{\kern0pt}\ {\isachardoublequoteopen}a\ {\isacharplus}{\kern0pt}\ b\ {\isacharminus}{\kern0pt}\ x{\isachardoublequoteclose}{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isacharminus}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}a\ {\isacharplus}{\kern0pt}\ b{\isacharparenright}{\kern0pt}\ {\isacharbackquote}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isadigit{0}}{\isacharless}{\kern0pt}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}a\ {\isacharplus}{\kern0pt}\ b{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharless}{\kern0pt}a\ {\isacharplus}{\kern0pt}\ b{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\ \isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}fastforce\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ mod{\isacharunderscore}{\kern0pt}less{\isacharunderscore}{\kern0pt}diff{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\ \isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isacharequal}{\kern0pt}\ \ {\isadigit{1}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \isacommand{finally}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isacommand{{\isachardot}{\kern0pt}}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ nn{\isacharunderscore}{\kern0pt}integral{\isacharunderscore}{\kern0pt}count{\isacharunderscore}{\kern0pt}space{\isacharunderscore}{\kern0pt}finite{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ {\isachardoublequoteopen}A{\isacharunderscore}{\kern0pt}inv\ {\isacharparenleft}{\kern0pt}s{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}{\isacharprime}{\kern0pt}s{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}{\isacharprime}{\kern0pt}s{\isachardot}{\kern0pt}\ Rep\ s\ {\isacharplus}{\kern0pt}\ Rep\ a\ {\isacharless}{\kern0pt}\ CARD{\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}s{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{end}\isamarkupfalse%
-\isanewline
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ {\isachardoublequoteopen}var{\isacharunderscore}{\kern0pt}cost{\isacharunderscore}{\kern0pt}lin\ {\isacharparenleft}{\kern0pt}c{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}real{\isacharparenright}{\kern0pt}\ n\ {\isacharequal}{\kern0pt}\ c\ {\isacharasterisk}{\kern0pt}\ Rep\ n{\isachardoublequoteclose}\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ {\isachardoublequoteopen}inv{\isacharunderscore}{\kern0pt}cost{\isacharunderscore}{\kern0pt}lin\ {\isacharparenleft}{\kern0pt}c{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}real{\isacharparenright}{\kern0pt}\ n\ {\isacharequal}{\kern0pt}\ c\ {\isacharasterisk}{\kern0pt}\ Rep\ n{\isachardoublequoteclose}\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ {\isachardoublequoteopen}revenue{\isacharunderscore}{\kern0pt}lin\ {\isacharparenleft}{\kern0pt}c{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}real{\isacharparenright}{\kern0pt}\ n\ {\isacharequal}{\kern0pt}\ c\ {\isacharasterisk}{\kern0pt}\ Rep\ n{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{lift{\isacharunderscore}{\kern0pt}definition}\isamarkupfalse%
-\ demand{\isacharunderscore}{\kern0pt}unif\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}a{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}finite{\isacharparenright}{\kern0pt}\ pmf{\isachardoublequoteclose}\ \isakeyword{is}\ {\isachardoublequoteopen}{\isasymlambda}{\isacharunderscore}{\kern0pt}{\isachardot}{\kern0pt}\ {\isadigit{1}}\ {\isacharslash}{\kern0pt}\ card\ {\isacharparenleft}{\kern0pt}UNIV{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}{\isacharprime}{\kern0pt}a\ set{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ ennreal{\isacharunderscore}{\kern0pt}divide{\isacharunderscore}{\kern0pt}times\ divide{\isacharunderscore}{\kern0pt}ennreal{\isacharbrackleft}{\kern0pt}symmetric{\isacharbrackright}{\kern0pt}\ ennreal{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}nat{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}real{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}nat{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lift{\isacharunderscore}{\kern0pt}definition}\isamarkupfalse%
-\ demand{\isacharunderscore}{\kern0pt}three\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isadigit{3}}\ pmf{\isachardoublequoteclose}\ \isakeyword{is}\ {\isachardoublequoteopen}{\isasymlambda}i{\isachardot}{\kern0pt}\ if\ i\ {\isacharequal}{\kern0pt}\ {\isadigit{1}}\ then\ {\isadigit{1}}{\isacharslash}{\kern0pt}{\isadigit{4}}\ else\ if\ i\ {\isacharequal}{\kern0pt}\ {\isadigit{2}}\ then\ {\isadigit{1}}{\isacharslash}{\kern0pt}{\isadigit{2}}\ else\ {\isadigit{1}}{\isacharslash}{\kern0pt}{\isadigit{4}}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isacharasterisk}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}UNIV\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isadigit{3}}\ set{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isadigit{0}}{\isacharcomma}{\kern0pt}{\isadigit{1}}{\isacharcomma}{\kern0pt}{\isadigit{2}}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ exhaust{\isacharunderscore}{\kern0pt}{\isadigit{3}}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ fastforce\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{apply}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ nn{\isacharunderscore}{\kern0pt}integral{\isacharunderscore}{\kern0pt}count{\isacharunderscore}{\kern0pt}space{\isacharunderscore}{\kern0pt}finite{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ {\isacharasterisk}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{abbreviation}\isamarkupfalse%
-\ {\isachardoublequoteopen}fixed{\isacharunderscore}{\kern0pt}cost\ {\isasymequiv}\ {\isadigit{4}}{\isachardoublequoteclose}\isanewline
-\isacommand{abbreviation}\isamarkupfalse%
-\ {\isachardoublequoteopen}var{\isacharunderscore}{\kern0pt}cost\ {\isasymequiv}\ var{\isacharunderscore}{\kern0pt}cost{\isacharunderscore}{\kern0pt}lin\ {\isadigit{2}}{\isachardoublequoteclose}\isanewline
-\isacommand{abbreviation}\isamarkupfalse%
-\ {\isachardoublequoteopen}inv{\isacharunderscore}{\kern0pt}cost\ {\isasymequiv}\ inv{\isacharunderscore}{\kern0pt}cost{\isacharunderscore}{\kern0pt}lin\ {\isadigit{1}}{\isachardoublequoteclose}\isanewline
-\isacommand{abbreviation}\isamarkupfalse%
-\ {\isachardoublequoteopen}revenue\ {\isasymequiv}\ revenue{\isacharunderscore}{\kern0pt}lin\ {\isadigit{8}}{\isachardoublequoteclose}\isanewline
-\isacommand{abbreviation}\isamarkupfalse%
-\ {\isachardoublequoteopen}discount\ {\isasymequiv}\ {\isadigit{0}}{\isachardot}{\kern0pt}{\isadigit{9}}{\isadigit{9}}{\isachardoublequoteclose}\isanewline
-\isacommand{type{\isacharunderscore}{\kern0pt}synonym}\isamarkupfalse%
-\ capacity\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}{\isadigit{3}}{\isadigit{0}}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ card{\isacharunderscore}{\kern0pt}ge{\isacharunderscore}{\kern0pt}{\isadigit{2}}{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}ne{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}CARD{\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}a{\isacharparenright}{\kern0pt}\ {\isasymge}\ {\isadigit{2}}\ {\isasymLongrightarrow}\ {\isasymexists}{\isacharparenleft}{\kern0pt}x{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}{\isacharprime}{\kern0pt}a{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}finite{\isacharparenright}{\kern0pt}\ y{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}{\isacharprime}{\kern0pt}a{\isachardot}{\kern0pt}\ x\ {\isasymnoteq}\ y{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}meson\ card{\isacharunderscore}{\kern0pt}{\isadigit{2}}{\isacharunderscore}{\kern0pt}iff{\isacharprime}{\kern0pt}\ ex{\isacharunderscore}{\kern0pt}card{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\ \ \isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{global{\isacharunderscore}{\kern0pt}interpretation}\isamarkupfalse%
-\ inventory{\isacharunderscore}{\kern0pt}ex{\isacharcolon}{\kern0pt}\ inventory\ fixed{\isacharunderscore}{\kern0pt}cost\ {\isachardoublequoteopen}var{\isacharunderscore}{\kern0pt}cost{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ capacity\ {\isasymRightarrow}\ real{\isachardoublequoteclose}\ inv{\isacharunderscore}{\kern0pt}cost\ demand{\isacharunderscore}{\kern0pt}unif\ revenue\ discount\isanewline
-\ \ \isakeyword{defines}\ A{\isacharunderscore}{\kern0pt}inv\ {\isacharequal}{\kern0pt}\ inventory{\isacharunderscore}{\kern0pt}ex{\isachardot}{\kern0pt}A{\isacharunderscore}{\kern0pt}inv\isanewline
-\ \ \ \ \isakeyword{and}\ transition\ {\isacharequal}{\kern0pt}\ inventory{\isacharunderscore}{\kern0pt}ex{\isachardot}{\kern0pt}transition\isanewline
-\ \ \ \ \isakeyword{and}\ reward\ {\isacharequal}{\kern0pt}\ inventory{\isacharunderscore}{\kern0pt}ex{\isachardot}{\kern0pt}reward\isanewline
-\ \ \ \ \isakeyword{and}\ prob{\isacharunderscore}{\kern0pt}ge{\isacharunderscore}{\kern0pt}inv\ {\isacharequal}{\kern0pt}\ inventory{\isacharunderscore}{\kern0pt}ex{\isachardot}{\kern0pt}prob{\isacharunderscore}{\kern0pt}ge{\isacharunderscore}{\kern0pt}inv\isanewline
-\ \ \ \ \isakeyword{and}\ order{\isacharunderscore}{\kern0pt}cost\ {\isacharequal}{\kern0pt}\ inventory{\isacharunderscore}{\kern0pt}ex{\isachardot}{\kern0pt}order{\isacharunderscore}{\kern0pt}cost\isanewline
-\ \ \ \ \isakeyword{and}\ exp{\isacharunderscore}{\kern0pt}rev\ {\isacharequal}{\kern0pt}\ inventory{\isacharunderscore}{\kern0pt}ex{\isachardot}{\kern0pt}exp{\isacharunderscore}{\kern0pt}rev%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{{\isachardot}{\kern0pt}}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{abbreviation}\isamarkupfalse%
-\ {\isachardoublequoteopen}K\ {\isasymequiv}\ inventory{\isacharunderscore}{\kern0pt}ex{\isachardot}{\kern0pt}transition{\isachardoublequoteclose}\isanewline
-\isacommand{abbreviation}\isamarkupfalse%
-\ {\isachardoublequoteopen}A\ {\isasymequiv}\ inventory{\isacharunderscore}{\kern0pt}ex{\isachardot}{\kern0pt}A{\isacharunderscore}{\kern0pt}inv{\isachardoublequoteclose}\isanewline
-\isacommand{abbreviation}\isamarkupfalse%
-\ {\isachardoublequoteopen}r\ {\isasymequiv}\ inventory{\isacharunderscore}{\kern0pt}ex{\isachardot}{\kern0pt}reward{\isachardoublequoteclose}\isanewline
-\isacommand{abbreviation}\isamarkupfalse%
-\ {\isachardoublequoteopen}l\ {\isasymequiv}\ {\isadigit{0}}{\isachardot}{\kern0pt}{\isadigit{9}}{\isadigit{5}}{\isachardoublequoteclose}\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ {\isachardoublequoteopen}eps\ {\isacharequal}{\kern0pt}\ {\isadigit{0}}{\isachardot}{\kern0pt}{\isadigit{1}}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ {\isachardoublequoteopen}fun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}list\ f\ {\isacharequal}{\kern0pt}\ map\ f\ {\isacharparenleft}{\kern0pt}sorted{\isacharunderscore}{\kern0pt}list{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}set\ UNIV{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ {\isachardoublequoteopen}benchmark{\isacharunderscore}{\kern0pt}gs\ {\isacharparenleft}{\kern0pt}{\isacharunderscore}{\kern0pt}\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ unit{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ map\ Rep\ {\isacharparenleft}{\kern0pt}fun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}list\ {\isacharparenleft}{\kern0pt}vi{\isacharunderscore}{\kern0pt}policy{\isacharprime}{\kern0pt}\ K\ A\ r\ l\ eps\ {\isadigit{0}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ {\isachardoublequoteopen}benchmark{\isacharunderscore}{\kern0pt}vi\ {\isacharparenleft}{\kern0pt}{\isacharunderscore}{\kern0pt}\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ unit{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ map\ Rep\ {\isacharparenleft}{\kern0pt}fun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}list\ {\isacharparenleft}{\kern0pt}vi{\isacharunderscore}{\kern0pt}gs{\isacharunderscore}{\kern0pt}policy\ K\ A\ r\ l\ eps\ {\isadigit{0}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ {\isachardoublequoteopen}benchmark{\isacharunderscore}{\kern0pt}mpi\ {\isacharparenleft}{\kern0pt}{\isacharunderscore}{\kern0pt}\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ unit\ {\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ map\ Rep\ {\isacharparenleft}{\kern0pt}fun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}list\ {\isacharparenleft}{\kern0pt}fst\ {\isacharparenleft}{\kern0pt}mpi{\isacharunderscore}{\kern0pt}user\ K\ A\ r\ l\ eps\ {\isacharparenleft}{\kern0pt}{\isasymlambda}{\isacharunderscore}{\kern0pt}\ {\isacharunderscore}{\kern0pt}{\isachardot}{\kern0pt}\ {\isadigit{3}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ {\isachardoublequoteopen}benchmark{\isacharunderscore}{\kern0pt}pi\ {\isacharparenleft}{\kern0pt}{\isacharunderscore}{\kern0pt}\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ unit{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ map\ Rep\ {\isacharparenleft}{\kern0pt}fun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}list\ {\isacharparenleft}{\kern0pt}policy{\isacharunderscore}{\kern0pt}iteration\ K\ A\ r\ l\ {\isadigit{0}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{fun}\isamarkupfalse%
-\ vs{\isacharunderscore}{\kern0pt}n\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}vs{\isacharunderscore}{\kern0pt}n\ {\isadigit{0}}\ v\ {\isacharequal}{\kern0pt}\ v{\isachardoublequoteclose}\isanewline
-{\isacharbar}{\kern0pt}\ {\isachardoublequoteopen}vs{\isacharunderscore}{\kern0pt}n\ {\isacharparenleft}{\kern0pt}Suc\ n{\isacharparenright}{\kern0pt}\ v\ {\isacharequal}{\kern0pt}\ vs{\isacharunderscore}{\kern0pt}n\ n\ {\isacharparenleft}{\kern0pt}mod{\isacharunderscore}{\kern0pt}MDP{\isacharunderscore}{\kern0pt}{\isasymL}\isactrlsub b\ K\ A\ r\ l\ v{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ {\isachardoublequoteopen}vs{\isacharunderscore}{\kern0pt}n{\isacharprime}{\kern0pt}\ n\ {\isacharequal}{\kern0pt}\ vs{\isacharunderscore}{\kern0pt}n\ n\ {\isadigit{0}}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ {\isachardoublequoteopen}benchmark{\isacharunderscore}{\kern0pt}vi{\isacharunderscore}{\kern0pt}n\ n\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}fun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}list\ {\isacharparenleft}{\kern0pt}vs{\isacharunderscore}{\kern0pt}n\ n\ {\isadigit{0}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ {\isachardoublequoteopen}benchmark{\isacharunderscore}{\kern0pt}vi{\isacharunderscore}{\kern0pt}nopol\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}fun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}list\ {\isacharparenleft}{\kern0pt}mod{\isacharunderscore}{\kern0pt}MDP{\isacharunderscore}{\kern0pt}value{\isacharunderscore}{\kern0pt}iteration\ K\ A\ r\ l\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharslash}{\kern0pt}{\isadigit{1}}{\isadigit{0}}{\isacharparenright}{\kern0pt}\ {\isadigit{0}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isanewline
-\isanewline
-\isanewline
-\isacommand{export{\isacharunderscore}{\kern0pt}code}\isamarkupfalse%
-\ dist\ vs{\isacharunderscore}{\kern0pt}n{\isacharprime}{\kern0pt}\ benchmark{\isacharunderscore}{\kern0pt}vi{\isacharunderscore}{\kern0pt}nopol\ benchmark{\isacharunderscore}{\kern0pt}vi{\isacharunderscore}{\kern0pt}n\ nat{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}integer\ integer{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}int\ benchmark{\isacharunderscore}{\kern0pt}gs\ benchmark{\isacharunderscore}{\kern0pt}vi\ benchmark{\isacharunderscore}{\kern0pt}mpi\ benchmark{\isacharunderscore}{\kern0pt}pi\isanewline
-\ \ \isakeyword{in}\ Haskell\ \isakeyword{module{\isacharunderscore}{\kern0pt}name}\ DP\isanewline
-\isanewline
-\isacommand{export{\isacharunderscore}{\kern0pt}code}\isamarkupfalse%
-\ integer{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}int\ benchmark{\isacharunderscore}{\kern0pt}gs\ benchmark{\isacharunderscore}{\kern0pt}vi\ benchmark{\isacharunderscore}{\kern0pt}mpi\ benchmark{\isacharunderscore}{\kern0pt}pi\ \isakeyword{in}\ SML\ \isakeyword{module{\isacharunderscore}{\kern0pt}name}\ DP\isanewline
-%
-\isadelimtheory
-\isanewline
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{end}\isamarkupfalse%
-%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\end{isabellebody}%
-\endinput
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diff --git a/thys/MDP-Algorithms/output/outline/Code_Mod.tex b/thys/MDP-Algorithms/output/outline/Code_Mod.tex
deleted file mode 100644
--- a/thys/MDP-Algorithms/output/outline/Code_Mod.tex
+++ /dev/null
@@ -1,303 +0,0 @@
-%
-\begin{isabellebody}%
-\setisabellecontext{Code{\isacharunderscore}{\kern0pt}Mod}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{theory}\isamarkupfalse%
-\ Code{\isacharunderscore}{\kern0pt}Mod\isanewline
-\ \ \isakeyword{imports}\ Code{\isacharunderscore}{\kern0pt}DP\isanewline
-\isakeyword{begin}%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsection{Code Generation for Concrete Finite MDPs%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{locale}\isamarkupfalse%
-\ mod{\isacharunderscore}{\kern0pt}MDP\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \isakeyword{fixes}\ transition\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}s{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}{\isacharbraceleft}{\kern0pt}enum{\isacharcomma}{\kern0pt}\ mod{\isacharunderscore}{\kern0pt}type{\isacharbraceright}{\kern0pt}\ {\isasymtimes}\ {\isacharprime}{\kern0pt}a{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}{\isacharbraceleft}{\kern0pt}enum{\isacharcomma}{\kern0pt}\ mod{\isacharunderscore}{\kern0pt}type{\isacharbraceright}{\kern0pt}\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}s\ pmf{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isakeyword{and}\ A\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}s\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isakeyword{and}\ reward\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}s\ {\isasymtimes}\ {\isacharprime}{\kern0pt}a\ {\isasymRightarrow}\ real{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isakeyword{and}\ discount\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}real{\isachardoublequoteclose}\isanewline
-\isakeyword{begin}\isanewline
-\isanewline
-\isacommand{sublocale}\isamarkupfalse%
-\ mdp{\isacharcolon}{\kern0pt}\ vi{\isacharunderscore}{\kern0pt}code\ \isanewline
-\ \ {\isachardoublequoteopen}{\isasymlambda}s{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}if\ Set{\isachardot}{\kern0pt}is{\isacharunderscore}{\kern0pt}empty\ {\isacharparenleft}{\kern0pt}A\ s{\isacharparenright}{\kern0pt}\ then\ UNIV\ else\ A\ s{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isanewline
-\ \ transition\ \isanewline
-\ \ reward\isanewline
-\ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}if\ {\isadigit{1}}\ {\isasymle}\ discount\ {\isasymor}\ discount\ {\isacharless}{\kern0pt}\ {\isadigit{0}}\ then\ {\isadigit{0}}\ else\ discount{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{defines}\ {\isasymL}\isactrlsub b\ {\isacharequal}{\kern0pt}\ mdp{\isachardot}{\kern0pt}{\isasymL}\isactrlsub b\isanewline
-\ \ \ \ \isakeyword{and}\ L{\isacharunderscore}{\kern0pt}det\ {\isacharequal}{\kern0pt}\ mdp{\isachardot}{\kern0pt}L{\isacharunderscore}{\kern0pt}det\isanewline
-\ \ \ \ \isakeyword{and}\ value{\isacharunderscore}{\kern0pt}iteration\ {\isacharequal}{\kern0pt}\ mdp{\isachardot}{\kern0pt}value{\isacharunderscore}{\kern0pt}iteration\isanewline
-\ \ \ \ \isakeyword{and}\ vi{\isacharunderscore}{\kern0pt}policy{\isacharprime}{\kern0pt}\ {\isacharequal}{\kern0pt}\ mdp{\isachardot}{\kern0pt}vi{\isacharunderscore}{\kern0pt}policy{\isacharprime}{\kern0pt}\isanewline
-\ \ \ \ \isakeyword{and}\ find{\isacharunderscore}{\kern0pt}policy{\isacharprime}{\kern0pt}\ {\isacharequal}{\kern0pt}\ mdp{\isachardot}{\kern0pt}find{\isacharunderscore}{\kern0pt}policy{\isacharprime}{\kern0pt}\isanewline
-\ \ \ \ \isakeyword{and}\ find{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}impl\ {\isacharequal}{\kern0pt}\ mdp{\isachardot}{\kern0pt}find{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}impl\isanewline
-\ \ \ \ \isakeyword{and}\ is{\isacharunderscore}{\kern0pt}opt{\isacharunderscore}{\kern0pt}act\ {\isacharequal}{\kern0pt}\ mdp{\isachardot}{\kern0pt}is{\isacharunderscore}{\kern0pt}opt{\isacharunderscore}{\kern0pt}act\isanewline
-\ \ \ \ \isakeyword{and}\ value{\isacharunderscore}{\kern0pt}iteration{\isacharunderscore}{\kern0pt}partial\ {\isacharequal}{\kern0pt}\ mdp{\isachardot}{\kern0pt}value{\isacharunderscore}{\kern0pt}iteration{\isacharunderscore}{\kern0pt}partial\isanewline
-\ \ \ \ \isakeyword{and}\ policy{\isacharunderscore}{\kern0pt}iteration\ {\isacharequal}{\kern0pt}\ mdp{\isachardot}{\kern0pt}policy{\isacharunderscore}{\kern0pt}iteration\isanewline
-\ \ \ \ \isakeyword{and}\ is{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\ {\isacharequal}{\kern0pt}\ mdp{\isachardot}{\kern0pt}is{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\isanewline
-\ \ \ \ \isakeyword{and}\ policy{\isacharunderscore}{\kern0pt}step\ {\isacharequal}{\kern0pt}\ mdp{\isachardot}{\kern0pt}policy{\isacharunderscore}{\kern0pt}step\isanewline
-\ \ \ \ \isakeyword{and}\ policy{\isacharunderscore}{\kern0pt}improvement\ {\isacharequal}{\kern0pt}\ mdp{\isachardot}{\kern0pt}policy{\isacharunderscore}{\kern0pt}improvement\isanewline
-\ \ \ \ \isakeyword{and}\ policy{\isacharunderscore}{\kern0pt}eval\ {\isacharequal}{\kern0pt}\ mdp{\isachardot}{\kern0pt}policy{\isacharunderscore}{\kern0pt}eval\isanewline
-\ \ \ \ \isakeyword{and}\ mk{\isacharunderscore}{\kern0pt}markovian\ {\isacharequal}{\kern0pt}\ mdp{\isachardot}{\kern0pt}mk{\isacharunderscore}{\kern0pt}markovian\isanewline
-\ \ \ \ \isakeyword{and}\ policy{\isacharunderscore}{\kern0pt}eval{\isacharprime}{\kern0pt}\ {\isacharequal}{\kern0pt}\ mdp{\isachardot}{\kern0pt}policy{\isacharunderscore}{\kern0pt}eval{\isacharprime}{\kern0pt}\isanewline
-\ \ \ \ \isakeyword{and}\ policy{\isacharunderscore}{\kern0pt}iteration{\isacharunderscore}{\kern0pt}partial{\isacharprime}{\kern0pt}\ {\isacharequal}{\kern0pt}\ mdp{\isachardot}{\kern0pt}policy{\isacharunderscore}{\kern0pt}iteration{\isacharunderscore}{\kern0pt}partial{\isacharprime}{\kern0pt}\isanewline
-\ \ \ \ \isakeyword{and}\ policy{\isacharunderscore}{\kern0pt}iteration{\isacharprime}{\kern0pt}\ {\isacharequal}{\kern0pt}\ mdp{\isachardot}{\kern0pt}policy{\isacharunderscore}{\kern0pt}iteration{\isacharprime}{\kern0pt}\isanewline
-\ \ \ \ \isakeyword{and}\ policy{\isacharunderscore}{\kern0pt}iteration{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}step{\isacharprime}{\kern0pt}\ {\isacharequal}{\kern0pt}\ mdp{\isachardot}{\kern0pt}policy{\isacharunderscore}{\kern0pt}step{\isacharprime}{\kern0pt}\isanewline
-\ \ \ \ \isakeyword{and}\ policy{\isacharunderscore}{\kern0pt}iteration{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}eval{\isacharprime}{\kern0pt}\ {\isacharequal}{\kern0pt}\ mdp{\isachardot}{\kern0pt}policy{\isacharunderscore}{\kern0pt}eval{\isacharprime}{\kern0pt}\isanewline
-\ \ \ \ \isakeyword{and}\ policy{\isacharunderscore}{\kern0pt}iteration{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}improvement{\isacharprime}{\kern0pt}\ {\isacharequal}{\kern0pt}\ mdp{\isachardot}{\kern0pt}policy{\isacharunderscore}{\kern0pt}improvement{\isacharprime}{\kern0pt}\isanewline
-\ \ \ \ \isakeyword{and}\ gs{\isacharunderscore}{\kern0pt}iteration\ {\isacharequal}{\kern0pt}\ mdp{\isachardot}{\kern0pt}gs{\isacharunderscore}{\kern0pt}iteration\isanewline
-\ \ \ \ \isakeyword{and}\ gs{\isacharunderscore}{\kern0pt}iteration{\isacharunderscore}{\kern0pt}partial\ {\isacharequal}{\kern0pt}\ mdp{\isachardot}{\kern0pt}gs{\isacharunderscore}{\kern0pt}iteration{\isacharunderscore}{\kern0pt}partial\isanewline
-\ \ \ \ \isakeyword{and}\ vi{\isacharunderscore}{\kern0pt}gs{\isacharunderscore}{\kern0pt}policy\ {\isacharequal}{\kern0pt}\ mdp{\isachardot}{\kern0pt}vi{\isacharunderscore}{\kern0pt}gs{\isacharunderscore}{\kern0pt}policy\isanewline
-\ \ \ \ \isakeyword{and}\ opt{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}gs\ {\isacharequal}{\kern0pt}\ mdp{\isachardot}{\kern0pt}opt{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}gs\isanewline
-\ \ \ \ \isakeyword{and}\ opt{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}gs{\isacharprime}{\kern0pt}{\isacharprime}{\kern0pt}\ {\isacharequal}{\kern0pt}\ mdp{\isachardot}{\kern0pt}opt{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}gs{\isacharprime}{\kern0pt}{\isacharprime}{\kern0pt}\isanewline
-\ \ \ \ \isakeyword{and}\ P{\isacharunderscore}{\kern0pt}mat\ {\isacharequal}{\kern0pt}\ mdp{\isachardot}{\kern0pt}P{\isacharunderscore}{\kern0pt}mat\isanewline
-\ \ \ \ \isakeyword{and}\ r{\isacharunderscore}{\kern0pt}vec{\isacharprime}{\kern0pt}\ {\isacharequal}{\kern0pt}\ mdp{\isachardot}{\kern0pt}r{\isacharunderscore}{\kern0pt}vec{\isacharprime}{\kern0pt}\isanewline
-\ \ \ \ \isakeyword{and}\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}fun\isactrlsub b\ {\isacharequal}{\kern0pt}\ mdp{\isachardot}{\kern0pt}GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}fun\isactrlsub b\isanewline
-\ \ \ \ \isakeyword{and}\ GS{\isacharunderscore}{\kern0pt}iter{\isacharunderscore}{\kern0pt}max\ {\isacharequal}{\kern0pt}\ mdp{\isachardot}{\kern0pt}GS{\isacharunderscore}{\kern0pt}iter{\isacharunderscore}{\kern0pt}max\isanewline
-\ \ \ \ \isakeyword{and}\ GS{\isacharunderscore}{\kern0pt}iter\ {\isacharequal}{\kern0pt}\ mdp{\isachardot}{\kern0pt}GS{\isacharunderscore}{\kern0pt}iter\isanewline
-\ \ \ \ \isakeyword{and}\ mpi{\isacharunderscore}{\kern0pt}user\ {\isacharequal}{\kern0pt}\ mdp{\isachardot}{\kern0pt}mpi{\isacharunderscore}{\kern0pt}user\isanewline
-\ \ \ \ \isakeyword{and}\ v{\isadigit{0}}{\isacharunderscore}{\kern0pt}mpi\isactrlsub b\ {\isacharequal}{\kern0pt}\ mdp{\isachardot}{\kern0pt}v{\isadigit{0}}{\isacharunderscore}{\kern0pt}mpi\isactrlsub b\isanewline
-\ \ \ \ \isakeyword{and}\ mpi{\isacharunderscore}{\kern0pt}partial{\isacharprime}{\kern0pt}\ {\isacharequal}{\kern0pt}\ mdp{\isachardot}{\kern0pt}mpi{\isacharunderscore}{\kern0pt}partial{\isacharprime}{\kern0pt}\isanewline
-\ \ \ \ \isakeyword{and}\ L{\isacharunderscore}{\kern0pt}pow\ {\isacharequal}{\kern0pt}\ mdp{\isachardot}{\kern0pt}L{\isacharunderscore}{\kern0pt}pow\isanewline
-\ \ \ \ \isakeyword{and}\ v{\isadigit{0}}{\isacharunderscore}{\kern0pt}mpi\ {\isacharequal}{\kern0pt}\ mdp{\isachardot}{\kern0pt}v{\isadigit{0}}{\isacharunderscore}{\kern0pt}mpi\isanewline
-\ \ \ \ \isakeyword{and}\ r{\isacharunderscore}{\kern0pt}min\ {\isacharequal}{\kern0pt}\ mdp{\isachardot}{\kern0pt}r{\isacharunderscore}{\kern0pt}min\isanewline
-\ \ \ \ \isakeyword{and}\ d{\isadigit{0}}\ {\isacharequal}{\kern0pt}\ mdp{\isachardot}{\kern0pt}d{\isadigit{0}}\isanewline
-\ \ \ \ \isakeyword{and}\ d{\isadigit{0}}{\isacharprime}{\kern0pt}\ {\isacharequal}{\kern0pt}\ mdp{\isachardot}{\kern0pt}d{\isadigit{0}}{\isacharprime}{\kern0pt}\isanewline
-\ \ \ \ \isakeyword{and}\ {\isasymnu}\isactrlsub b\ {\isacharequal}{\kern0pt}\ mdp{\isachardot}{\kern0pt}{\isasymnu}\isactrlsub b\isanewline
-\ \ \ \ \isakeyword{and}\ vi{\isacharunderscore}{\kern0pt}test\ {\isacharequal}{\kern0pt}\ mdp{\isachardot}{\kern0pt}vi{\isacharunderscore}{\kern0pt}test\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ unfold{\isacharunderscore}{\kern0pt}locales\ {\isacharparenleft}{\kern0pt}auto\ simp\ add{\isacharcolon}{\kern0pt}\ Set{\isachardot}{\kern0pt}is{\isacharunderscore}{\kern0pt}empty{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isacommand{end}\isamarkupfalse%
-\isanewline
-\isanewline
-\isacommand{global{\isacharunderscore}{\kern0pt}interpretation}\isamarkupfalse%
-\ mod{\isacharunderscore}{\kern0pt}MDP\ transition\ A\ reward\ discount\ \isanewline
-\ \ \isakeyword{for}\ transition\ A\ reward\ discount\ \isanewline
-\ \ \isakeyword{defines}\ mod{\isacharunderscore}{\kern0pt}MDP{\isacharunderscore}{\kern0pt}{\isasymL}\isactrlsub b\ {\isacharequal}{\kern0pt}\ mdp{\isachardot}{\kern0pt}{\isasymL}\isactrlsub b\isanewline
-\ \ \ \ \isakeyword{and}\ mod{\isacharunderscore}{\kern0pt}MDP{\isacharunderscore}{\kern0pt}{\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}L{\isacharunderscore}{\kern0pt}det\ {\isacharequal}{\kern0pt}\ mdp{\isachardot}{\kern0pt}L{\isacharunderscore}{\kern0pt}det\isanewline
-\ \ \ \ \isakeyword{and}\ mod{\isacharunderscore}{\kern0pt}MDP{\isacharunderscore}{\kern0pt}value{\isacharunderscore}{\kern0pt}iteration\ {\isacharequal}{\kern0pt}\ mdp{\isachardot}{\kern0pt}value{\isacharunderscore}{\kern0pt}iteration\isanewline
-\ \ \ \ \isakeyword{and}\ mod{\isacharunderscore}{\kern0pt}MDP{\isacharunderscore}{\kern0pt}vi{\isacharunderscore}{\kern0pt}policy{\isacharprime}{\kern0pt}\ {\isacharequal}{\kern0pt}\ mdp{\isachardot}{\kern0pt}vi{\isacharunderscore}{\kern0pt}policy{\isacharprime}{\kern0pt}\isanewline
-\ \ \ \ \isakeyword{and}\ mod{\isacharunderscore}{\kern0pt}MDP{\isacharunderscore}{\kern0pt}find{\isacharunderscore}{\kern0pt}policy{\isacharprime}{\kern0pt}\ {\isacharequal}{\kern0pt}\ mdp{\isachardot}{\kern0pt}find{\isacharunderscore}{\kern0pt}policy{\isacharprime}{\kern0pt}\isanewline
-\ \ \ \ \isakeyword{and}\ mod{\isacharunderscore}{\kern0pt}MDP{\isacharunderscore}{\kern0pt}find{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}impl\ {\isacharequal}{\kern0pt}\ mdp{\isachardot}{\kern0pt}find{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}impl\isanewline
-\ \ \ \ \isakeyword{and}\ mod{\isacharunderscore}{\kern0pt}MDP{\isacharunderscore}{\kern0pt}is{\isacharunderscore}{\kern0pt}opt{\isacharunderscore}{\kern0pt}act\ {\isacharequal}{\kern0pt}\ mdp{\isachardot}{\kern0pt}is{\isacharunderscore}{\kern0pt}opt{\isacharunderscore}{\kern0pt}act\isanewline
-\ \ \ \ \isakeyword{and}\ mod{\isacharunderscore}{\kern0pt}MDP{\isacharunderscore}{\kern0pt}value{\isacharunderscore}{\kern0pt}iteration{\isacharunderscore}{\kern0pt}partial\ {\isacharequal}{\kern0pt}\ mdp{\isachardot}{\kern0pt}value{\isacharunderscore}{\kern0pt}iteration{\isacharunderscore}{\kern0pt}partial\isanewline
-\ \ \ \ \isakeyword{and}\ mod{\isacharunderscore}{\kern0pt}MDP{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}iteration\ {\isacharequal}{\kern0pt}\ mdp{\isachardot}{\kern0pt}policy{\isacharunderscore}{\kern0pt}iteration\isanewline
-\ \ \ \ \isakeyword{and}\ mod{\isacharunderscore}{\kern0pt}MDP{\isacharunderscore}{\kern0pt}is{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\ {\isacharequal}{\kern0pt}\ mdp{\isachardot}{\kern0pt}is{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\isanewline
-\ \ \ \ \isakeyword{and}\ mod{\isacharunderscore}{\kern0pt}MDP{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}step\ {\isacharequal}{\kern0pt}\ mdp{\isachardot}{\kern0pt}policy{\isacharunderscore}{\kern0pt}step\isanewline
-\ \ \ \ \isakeyword{and}\ mod{\isacharunderscore}{\kern0pt}MDP{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}improvement\ {\isacharequal}{\kern0pt}\ mdp{\isachardot}{\kern0pt}policy{\isacharunderscore}{\kern0pt}improvement\isanewline
-\ \ \ \ \isakeyword{and}\ mod{\isacharunderscore}{\kern0pt}MDP{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}eval\ {\isacharequal}{\kern0pt}\ mdp{\isachardot}{\kern0pt}policy{\isacharunderscore}{\kern0pt}eval\isanewline
-\ \ \ \ \isakeyword{and}\ mod{\isacharunderscore}{\kern0pt}MDP{\isacharunderscore}{\kern0pt}mk{\isacharunderscore}{\kern0pt}markovian\ {\isacharequal}{\kern0pt}\ mdp{\isachardot}{\kern0pt}mk{\isacharunderscore}{\kern0pt}markovian\isanewline
-\ \ \ \ \isakeyword{and}\ mod{\isacharunderscore}{\kern0pt}MDP{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}eval{\isacharprime}{\kern0pt}\ {\isacharequal}{\kern0pt}\ mdp{\isachardot}{\kern0pt}policy{\isacharunderscore}{\kern0pt}eval{\isacharprime}{\kern0pt}\isanewline
-\ \ \ \ \isakeyword{and}\ mod{\isacharunderscore}{\kern0pt}MDP{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}iteration{\isacharunderscore}{\kern0pt}partial{\isacharprime}{\kern0pt}\ {\isacharequal}{\kern0pt}\ mdp{\isachardot}{\kern0pt}policy{\isacharunderscore}{\kern0pt}iteration{\isacharunderscore}{\kern0pt}partial{\isacharprime}{\kern0pt}\isanewline
-\ \ \ \ \isakeyword{and}\ mod{\isacharunderscore}{\kern0pt}MDP{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}iteration{\isacharprime}{\kern0pt}\ {\isacharequal}{\kern0pt}\ mdp{\isachardot}{\kern0pt}policy{\isacharunderscore}{\kern0pt}iteration{\isacharprime}{\kern0pt}\isanewline
-\ \ \ \ \isakeyword{and}\ mod{\isacharunderscore}{\kern0pt}MDP{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}iteration{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}step{\isacharprime}{\kern0pt}\ {\isacharequal}{\kern0pt}\ mdp{\isachardot}{\kern0pt}policy{\isacharunderscore}{\kern0pt}step{\isacharprime}{\kern0pt}\isanewline
-\ \ \ \ \isakeyword{and}\ mod{\isacharunderscore}{\kern0pt}MDP{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}iteration{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}eval{\isacharprime}{\kern0pt}\ {\isacharequal}{\kern0pt}\ mdp{\isachardot}{\kern0pt}policy{\isacharunderscore}{\kern0pt}eval{\isacharprime}{\kern0pt}\isanewline
-\ \ \ \ \isakeyword{and}\ mod{\isacharunderscore}{\kern0pt}MDP{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}iteration{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}improvement{\isacharprime}{\kern0pt}\ {\isacharequal}{\kern0pt}\ mdp{\isachardot}{\kern0pt}policy{\isacharunderscore}{\kern0pt}improvement{\isacharprime}{\kern0pt}\isanewline
-\ \ \ \ \isakeyword{and}\ mod{\isacharunderscore}{\kern0pt}MDP{\isacharunderscore}{\kern0pt}gs{\isacharunderscore}{\kern0pt}iteration\ {\isacharequal}{\kern0pt}\ mdp{\isachardot}{\kern0pt}gs{\isacharunderscore}{\kern0pt}iteration\isanewline
-\ \ \ \ \isakeyword{and}\ mod{\isacharunderscore}{\kern0pt}MDP{\isacharunderscore}{\kern0pt}gs{\isacharunderscore}{\kern0pt}iteration{\isacharunderscore}{\kern0pt}partial\ {\isacharequal}{\kern0pt}\ mdp{\isachardot}{\kern0pt}gs{\isacharunderscore}{\kern0pt}iteration{\isacharunderscore}{\kern0pt}partial\isanewline
-\ \ \ \ \isakeyword{and}\ mod{\isacharunderscore}{\kern0pt}MDP{\isacharunderscore}{\kern0pt}vi{\isacharunderscore}{\kern0pt}gs{\isacharunderscore}{\kern0pt}policy\ {\isacharequal}{\kern0pt}\ mdp{\isachardot}{\kern0pt}vi{\isacharunderscore}{\kern0pt}gs{\isacharunderscore}{\kern0pt}policy\isanewline
-\ \ \ \ \isakeyword{and}\ mod{\isacharunderscore}{\kern0pt}MDP{\isacharunderscore}{\kern0pt}opt{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}gs\ {\isacharequal}{\kern0pt}\ mdp{\isachardot}{\kern0pt}opt{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}gs\isanewline
-\ \ \ \ \isakeyword{and}\ mod{\isacharunderscore}{\kern0pt}MDP{\isacharunderscore}{\kern0pt}opt{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}gs{\isacharprime}{\kern0pt}{\isacharprime}{\kern0pt}\ {\isacharequal}{\kern0pt}\ mdp{\isachardot}{\kern0pt}opt{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}gs{\isacharprime}{\kern0pt}{\isacharprime}{\kern0pt}\isanewline
-\ \ \ \ \isakeyword{and}\ mod{\isacharunderscore}{\kern0pt}MDP{\isacharunderscore}{\kern0pt}P{\isacharunderscore}{\kern0pt}mat\ {\isacharequal}{\kern0pt}\ mdp{\isachardot}{\kern0pt}P{\isacharunderscore}{\kern0pt}mat\isanewline
-\ \ \ \ \isakeyword{and}\ mod{\isacharunderscore}{\kern0pt}MDP{\isacharunderscore}{\kern0pt}r{\isacharunderscore}{\kern0pt}vec{\isacharprime}{\kern0pt}\ {\isacharequal}{\kern0pt}\ mdp{\isachardot}{\kern0pt}r{\isacharunderscore}{\kern0pt}vec{\isacharprime}{\kern0pt}\isanewline
-\ \ \ \ \isakeyword{and}\ mod{\isacharunderscore}{\kern0pt}MDP{\isacharunderscore}{\kern0pt}GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}fun\isactrlsub b\ {\isacharequal}{\kern0pt}\ mdp{\isachardot}{\kern0pt}GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}fun\isactrlsub b\isanewline
-\ \ \ \ \isakeyword{and}\ mod{\isacharunderscore}{\kern0pt}MDP{\isacharunderscore}{\kern0pt}GS{\isacharunderscore}{\kern0pt}iter{\isacharunderscore}{\kern0pt}max\ {\isacharequal}{\kern0pt}\ mdp{\isachardot}{\kern0pt}GS{\isacharunderscore}{\kern0pt}iter{\isacharunderscore}{\kern0pt}max\isanewline
-\ \ \ \ \isakeyword{and}\ mod{\isacharunderscore}{\kern0pt}MDP{\isacharunderscore}{\kern0pt}GS{\isacharunderscore}{\kern0pt}iter\ {\isacharequal}{\kern0pt}\ mdp{\isachardot}{\kern0pt}GS{\isacharunderscore}{\kern0pt}iter\isanewline
-\ \ \ \ \isakeyword{and}\ mod{\isacharunderscore}{\kern0pt}MDP{\isacharunderscore}{\kern0pt}mpi{\isacharunderscore}{\kern0pt}user\ {\isacharequal}{\kern0pt}\ mdp{\isachardot}{\kern0pt}mpi{\isacharunderscore}{\kern0pt}user\isanewline
-\ \ \ \ \isakeyword{and}\ mod{\isacharunderscore}{\kern0pt}MDP{\isacharunderscore}{\kern0pt}v{\isadigit{0}}{\isacharunderscore}{\kern0pt}mpi\isactrlsub b\ {\isacharequal}{\kern0pt}\ mdp{\isachardot}{\kern0pt}v{\isadigit{0}}{\isacharunderscore}{\kern0pt}mpi\isactrlsub b\isanewline
-\ \ \ \ \isakeyword{and}\ mod{\isacharunderscore}{\kern0pt}MDP{\isacharunderscore}{\kern0pt}mpi{\isacharunderscore}{\kern0pt}partial{\isacharprime}{\kern0pt}\ {\isacharequal}{\kern0pt}\ mdp{\isachardot}{\kern0pt}mpi{\isacharunderscore}{\kern0pt}partial{\isacharprime}{\kern0pt}\isanewline
-\ \ \ \ \isakeyword{and}\ mod{\isacharunderscore}{\kern0pt}MDP{\isacharunderscore}{\kern0pt}L{\isacharunderscore}{\kern0pt}pow\ {\isacharequal}{\kern0pt}\ mdp{\isachardot}{\kern0pt}L{\isacharunderscore}{\kern0pt}pow\isanewline
-\ \ \ \ \isakeyword{and}\ mod{\isacharunderscore}{\kern0pt}MDP{\isacharunderscore}{\kern0pt}v{\isadigit{0}}{\isacharunderscore}{\kern0pt}mpi\ {\isacharequal}{\kern0pt}\ mdp{\isachardot}{\kern0pt}v{\isadigit{0}}{\isacharunderscore}{\kern0pt}mpi\isanewline
-\ \ \ \ \isakeyword{and}\ mod{\isacharunderscore}{\kern0pt}MDP{\isacharunderscore}{\kern0pt}r{\isacharunderscore}{\kern0pt}min\ {\isacharequal}{\kern0pt}\ mdp{\isachardot}{\kern0pt}r{\isacharunderscore}{\kern0pt}min\isanewline
-\ \ \ \ \isakeyword{and}\ mod{\isacharunderscore}{\kern0pt}MDP{\isacharunderscore}{\kern0pt}d{\isadigit{0}}\ {\isacharequal}{\kern0pt}\ mdp{\isachardot}{\kern0pt}d{\isadigit{0}}\isanewline
-\ \ \ \ \isakeyword{and}\ mod{\isacharunderscore}{\kern0pt}MDP{\isacharunderscore}{\kern0pt}d{\isadigit{0}}{\isacharprime}{\kern0pt}\ {\isacharequal}{\kern0pt}\ mdp{\isachardot}{\kern0pt}d{\isadigit{0}}{\isacharprime}{\kern0pt}\isanewline
-\ \ \ \ \isakeyword{and}\ mod{\isacharunderscore}{\kern0pt}MDP{\isacharunderscore}{\kern0pt}{\isasymnu}\isactrlsub b\ {\isacharequal}{\kern0pt}\ mdp{\isachardot}{\kern0pt}{\isasymnu}\isactrlsub b\isanewline
-\ \ \ \ \isakeyword{and}\ mod{\isacharunderscore}{\kern0pt}MDP{\isacharunderscore}{\kern0pt}vi{\isacharunderscore}{\kern0pt}test\ {\isacharequal}{\kern0pt}\ mdp{\isachardot}{\kern0pt}vi{\isacharunderscore}{\kern0pt}test\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{{\isachardot}{\kern0pt}}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-%
-\isadelimtheory
-\isanewline
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{end}\isamarkupfalse%
-%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\end{isabellebody}%
-\endinput
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diff --git a/thys/MDP-Algorithms/output/outline/Code_Real_Approx_By_Float_Fix.tex b/thys/MDP-Algorithms/output/outline/Code_Real_Approx_By_Float_Fix.tex
deleted file mode 100644
--- a/thys/MDP-Algorithms/output/outline/Code_Real_Approx_By_Float_Fix.tex
+++ /dev/null
@@ -1,59 +0,0 @@
-%
-\begin{isabellebody}%
-\setisabellecontext{Code{\isacharunderscore}{\kern0pt}Real{\isacharunderscore}{\kern0pt}Approx{\isacharunderscore}{\kern0pt}By{\isacharunderscore}{\kern0pt}Float{\isacharunderscore}{\kern0pt}Fix}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{theory}\isamarkupfalse%
-\ Code{\isacharunderscore}{\kern0pt}Real{\isacharunderscore}{\kern0pt}Approx{\isacharunderscore}{\kern0pt}By{\isacharunderscore}{\kern0pt}Float{\isacharunderscore}{\kern0pt}Fix\isanewline
-\ \ \isakeyword{imports}\isanewline
-\ \ {\isachardoublequoteopen}HOL{\isacharminus}{\kern0pt}Library{\isachardot}{\kern0pt}Code{\isacharunderscore}{\kern0pt}Real{\isacharunderscore}{\kern0pt}Approx{\isacharunderscore}{\kern0pt}By{\isacharunderscore}{\kern0pt}Float{\isachardoublequoteclose}\isanewline
-\ \ {\isachardoublequoteopen}Gauss{\isacharunderscore}{\kern0pt}Jordan{\isachardot}{\kern0pt}Code{\isacharunderscore}{\kern0pt}Real{\isacharunderscore}{\kern0pt}Approx{\isacharunderscore}{\kern0pt}By{\isacharunderscore}{\kern0pt}Float{\isacharunderscore}{\kern0pt}Haskell{\isachardoublequoteclose}\isanewline
-\isakeyword{begin}%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
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-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
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-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{end}\isamarkupfalse%
-%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\end{isabellebody}%
-\endinput
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diff --git a/thys/MDP-Algorithms/output/outline/Examples.tex b/thys/MDP-Algorithms/output/outline/Examples.tex
deleted file mode 100644
--- a/thys/MDP-Algorithms/output/outline/Examples.tex
+++ /dev/null
@@ -1,38 +0,0 @@
-%
-\begin{isabellebody}%
-\setisabellecontext{Examples}%
-%
-\isadelimtheory
-\isanewline
-\isanewline
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{theory}\isamarkupfalse%
-\ Examples\isanewline
-\ \ \isakeyword{imports}\isanewline
-\ \ \ \ Code{\isacharunderscore}{\kern0pt}Inventory\isanewline
-\ \ \ \ Code{\isacharunderscore}{\kern0pt}Gridworld\isanewline
-\isakeyword{begin}\isanewline
-\isacommand{end}\isamarkupfalse%
-%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
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-\end{isabellebody}%
-\endinput
-%:%file=Examples.tex%:%
-%:%6=1%:%
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diff --git a/thys/MDP-Algorithms/output/outline/Matrix_Util.tex b/thys/MDP-Algorithms/output/outline/Matrix_Util.tex
deleted file mode 100644
--- a/thys/MDP-Algorithms/output/outline/Matrix_Util.tex
+++ /dev/null
@@ -1,1103 +0,0 @@
-%
-\begin{isabellebody}%
-\setisabellecontext{Matrix{\isacharunderscore}{\kern0pt}Util}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{theory}\isamarkupfalse%
-\ Matrix{\isacharunderscore}{\kern0pt}Util\isanewline
-\ \ \isakeyword{imports}\ {\isachardoublequoteopen}HOL{\isacharminus}{\kern0pt}Analysis{\isachardot}{\kern0pt}Analysis{\isachardoublequoteclose}\isanewline
-\isakeyword{begin}%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsection{Matrices%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{proposition}\isamarkupfalse%
-\ scalar{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}assoc{\isacharprime}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{fixes}\ C\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}b{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}real{\isacharunderscore}{\kern0pt}algebra{\isacharunderscore}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}{\isacharcircum}{\kern0pt}{\isacharprime}{\kern0pt}m{\isacharcircum}{\kern0pt}{\isacharprime}{\kern0pt}n{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}k\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isacharparenleft}{\kern0pt}C\ {\isacharasterisk}{\kern0pt}{\isacharasterisk}{\kern0pt}\ D{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ C\ {\isacharasterisk}{\kern0pt}{\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}k\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ D{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ matrix{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}mult{\isacharunderscore}{\kern0pt}def\ sum{\isacharunderscore}{\kern0pt}distrib{\isacharunderscore}{\kern0pt}left\ mult{\isacharunderscore}{\kern0pt}ac\ vec{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}iff\ scaleR{\isacharunderscore}{\kern0pt}sum{\isacharunderscore}{\kern0pt}right{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Nonnegative Matrices%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{lemma}\isamarkupfalse%
-\ nonneg{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}nonneg\ {\isacharbrackleft}{\kern0pt}dest{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isadigit{0}}\ {\isasymle}\ m\ {\isasymLongrightarrow}\ {\isadigit{0}}\ {\isasymle}\ m\ {\isachardollar}{\kern0pt}\ i\ {\isachardollar}{\kern0pt}\ j{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ Finite{\isacharunderscore}{\kern0pt}Cartesian{\isacharunderscore}{\kern0pt}Product{\isachardot}{\kern0pt}less{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}vec{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ matrix{\isacharunderscore}{\kern0pt}mult{\isacharunderscore}{\kern0pt}mono{\isacharcolon}{\kern0pt}\ \isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}{\isadigit{0}}\ {\isasymle}\ E{\isachardoublequoteclose}\ {\isachardoublequoteopen}{\isadigit{0}}\ {\isasymle}\ C{\isachardoublequoteclose}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}E\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ real{\isacharcircum}{\kern0pt}{\isacharprime}{\kern0pt}c{\isacharcircum}{\kern0pt}{\isacharprime}{\kern0pt}c{\isacharparenright}{\kern0pt}\ {\isasymle}\ B{\isachardoublequoteclose}\ {\isachardoublequoteopen}C\ {\isasymle}\ D{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}E\ {\isacharasterisk}{\kern0pt}{\isacharasterisk}{\kern0pt}\ C\ {\isasymle}\ B\ {\isacharasterisk}{\kern0pt}{\isacharasterisk}{\kern0pt}\ D{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ order{\isachardot}{\kern0pt}trans{\isacharbrackleft}{\kern0pt}OF\ assms{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}\ assms{\isacharparenleft}{\kern0pt}{\isadigit{3}}{\isacharparenright}{\kern0pt}{\isacharbrackright}{\kern0pt}\ assms\isanewline
-\ \ \isacommand{unfolding}\isamarkupfalse%
-\ Finite{\isacharunderscore}{\kern0pt}Cartesian{\isacharunderscore}{\kern0pt}Product{\isachardot}{\kern0pt}less{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}vec{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ sum{\isacharunderscore}{\kern0pt}mono\ mult{\isacharunderscore}{\kern0pt}mono\ simp{\isacharcolon}{\kern0pt}\ matrix{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}mult{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ nonneg{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}mult{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isadigit{0}}\ {\isasymle}\ {\isacharparenleft}{\kern0pt}C\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}b{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}{\isacharbraceleft}{\kern0pt}field{\isacharcomma}{\kern0pt}\ ordered{\isacharunderscore}{\kern0pt}ring{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharcircum}{\kern0pt}{\isacharunderscore}{\kern0pt}{\isacharcircum}{\kern0pt}{\isacharunderscore}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymLongrightarrow}\ {\isadigit{0}}\ {\isasymle}\ D\ {\isasymLongrightarrow}\ {\isadigit{0}}\ {\isasymle}\ C\ {\isacharasterisk}{\kern0pt}{\isacharasterisk}{\kern0pt}\ D{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ Finite{\isacharunderscore}{\kern0pt}Cartesian{\isacharunderscore}{\kern0pt}Product{\isachardot}{\kern0pt}less{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}vec{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ matrix{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}mult{\isacharunderscore}{\kern0pt}def\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ sum{\isacharunderscore}{\kern0pt}nonneg{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ zero{\isacharunderscore}{\kern0pt}le{\isacharunderscore}{\kern0pt}mat{\isacharunderscore}{\kern0pt}iff\ {\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isadigit{0}}\ {\isasymle}\ mat\ {\isacharparenleft}{\kern0pt}x\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isacharprime}{\kern0pt}c\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}zero{\isacharcomma}{\kern0pt}\ order{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymlongleftrightarrow}\ {\isadigit{0}}\ {\isasymle}\ x{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ Finite{\isacharunderscore}{\kern0pt}Cartesian{\isacharunderscore}{\kern0pt}Product{\isachardot}{\kern0pt}less{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}vec{\isacharunderscore}{\kern0pt}def\ mat{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ nonneg{\isacharunderscore}{\kern0pt}mat{\isacharunderscore}{\kern0pt}ge{\isacharunderscore}{\kern0pt}zero{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isadigit{0}}\ {\isasymle}\ Q\ {\isasymLongrightarrow}\ {\isadigit{0}}\ {\isasymle}\ v\ {\isasymLongrightarrow}\ {\isadigit{0}}\ {\isasymle}\ Q\ {\isacharasterisk}{\kern0pt}v\ {\isacharparenleft}{\kern0pt}v\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ real{\isacharcircum}{\kern0pt}{\isacharprime}{\kern0pt}c{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ Finite{\isacharunderscore}{\kern0pt}Cartesian{\isacharunderscore}{\kern0pt}Product{\isachardot}{\kern0pt}less{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}vec{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ sum{\isacharunderscore}{\kern0pt}nonneg\ simp{\isacharcolon}{\kern0pt}\ matrix{\isacharunderscore}{\kern0pt}vector{\isacharunderscore}{\kern0pt}mult{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ nonneg{\isacharunderscore}{\kern0pt}mat{\isacharunderscore}{\kern0pt}mono{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isadigit{0}}\ {\isasymle}\ Q\ {\isasymLongrightarrow}\ u\ {\isasymle}\ v\ {\isasymLongrightarrow}\ Q\ {\isacharasterisk}{\kern0pt}v\ u\ {\isasymle}\ Q\ {\isacharasterisk}{\kern0pt}v\ {\isacharparenleft}{\kern0pt}v\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ real{\isacharcircum}{\kern0pt}{\isacharprime}{\kern0pt}c{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ nonneg{\isacharunderscore}{\kern0pt}mat{\isacharunderscore}{\kern0pt}ge{\isacharunderscore}{\kern0pt}zero{\isacharbrackleft}{\kern0pt}of\ Q\ {\isachardoublequoteopen}v\ {\isacharminus}{\kern0pt}\ u{\isachardoublequoteclose}{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ vec{\isachardot}{\kern0pt}diff{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ nonneg{\isacharunderscore}{\kern0pt}mult{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}nonneg{\isacharunderscore}{\kern0pt}mat{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}{\isasymAnd}v{\isachardot}{\kern0pt}\ v\ {\isasymge}\ {\isadigit{0}}\ {\isasymLongrightarrow}\ X\ {\isacharasterisk}{\kern0pt}v\ v\ {\isasymge}\ {\isadigit{0}}{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}X\ {\isasymge}\ {\isacharparenleft}{\kern0pt}{\isadigit{0}}\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ real\ {\isacharcircum}{\kern0pt}\ {\isacharunderscore}{\kern0pt}\ {\isacharcircum}{\kern0pt}{\isacharunderscore}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{{\isacharbraceleft}{\kern0pt}}\isamarkupfalse%
-\ \isacommand{assume}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymnot}\ {\isacharparenleft}{\kern0pt}{\isadigit{0}}\ {\isasymle}\ X{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{then}\isamarkupfalse%
-\ \isacommand{obtain}\isamarkupfalse%
-\ i\ j\ \isakeyword{where}\ neg{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}X\ {\isachardollar}{\kern0pt}\ i\ {\isachardollar}{\kern0pt}\ j\ {\isacharless}{\kern0pt}\ {\isadigit{0}}{\isachardoublequoteclose}\ \isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ less{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}vec{\isacharunderscore}{\kern0pt}def\ not{\isacharunderscore}{\kern0pt}le\ zero{\isacharunderscore}{\kern0pt}index{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}v\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}{\isasymchi}\ k{\isachardot}{\kern0pt}\ if\ j\ {\isacharequal}{\kern0pt}\ k\ then\ {\isadigit{1}}{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}real\ else\ {\isadigit{0}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}X\ {\isacharasterisk}{\kern0pt}v\ {\isacharquery}{\kern0pt}v{\isacharparenright}{\kern0pt}\ {\isachardollar}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ {\isadigit{0}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ neg\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ matrix{\isacharunderscore}{\kern0pt}vector{\isacharunderscore}{\kern0pt}mult{\isacharunderscore}{\kern0pt}def\ if{\isacharunderscore}{\kern0pt}distrib\ cong{\isacharcolon}{\kern0pt}\ if{\isacharunderscore}{\kern0pt}cong{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharquery}{\kern0pt}v\ {\isasymge}\ {\isadigit{0}}\ {\isasymand}\ {\isasymnot}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}X\ {\isacharasterisk}{\kern0pt}v\ {\isacharquery}{\kern0pt}v{\isacharparenright}{\kern0pt}\ {\isasymge}\ {\isadigit{0}}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ less{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}vec{\isacharunderscore}{\kern0pt}def\ not{\isacharunderscore}{\kern0pt}le{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymexists}v{\isachardot}{\kern0pt}\ v\ {\isasymge}\ {\isadigit{0}}\ {\isasymand}\ {\isasymnot}\ X\ {\isacharasterisk}{\kern0pt}v\ v\ {\isasymge}\ {\isadigit{0}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{{\isacharbraceright}{\kern0pt}}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assms\ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ nonneg{\isacharunderscore}{\kern0pt}mat{\isacharunderscore}{\kern0pt}iff{\isacharcolon}{\kern0pt}\isanewline
-\ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}X\ {\isasymge}\ {\isacharparenleft}{\kern0pt}{\isadigit{0}}\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ real\ {\isacharcircum}{\kern0pt}\ {\isacharunderscore}{\kern0pt}\ {\isacharcircum}{\kern0pt}{\isacharunderscore}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymlongleftrightarrow}\ {\isacharparenleft}{\kern0pt}{\isasymforall}v{\isachardot}{\kern0pt}\ v\ {\isasymge}\ {\isadigit{0}}\ {\isasymlongrightarrow}\ X\ {\isacharasterisk}{\kern0pt}v\ v\ {\isasymge}\ {\isadigit{0}}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ nonneg{\isacharunderscore}{\kern0pt}mat{\isacharunderscore}{\kern0pt}ge{\isacharunderscore}{\kern0pt}zero\ nonneg{\isacharunderscore}{\kern0pt}mult{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}nonneg{\isacharunderscore}{\kern0pt}mat\ \isacommand{by}\isamarkupfalse%
-\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ mat{\isacharunderscore}{\kern0pt}le{\isacharunderscore}{\kern0pt}iff{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}X\ {\isasymle}\ Y{\isacharparenright}{\kern0pt}\ {\isasymlongleftrightarrow}\ {\isacharparenleft}{\kern0pt}{\isasymforall}x{\isasymge}{\isadigit{0}}{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}X{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}real{\isacharcircum}{\kern0pt}{\isacharunderscore}{\kern0pt}{\isacharcircum}{\kern0pt}{\isacharunderscore}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharasterisk}{\kern0pt}v\ x\ {\isasymle}\ Y\ {\isacharasterisk}{\kern0pt}v\ x{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ diff{\isacharunderscore}{\kern0pt}ge{\isacharunderscore}{\kern0pt}{\isadigit{0}}{\isacharunderscore}{\kern0pt}iff{\isacharunderscore}{\kern0pt}ge\ matrix{\isacharunderscore}{\kern0pt}vector{\isacharunderscore}{\kern0pt}mult{\isacharunderscore}{\kern0pt}diff{\isacharunderscore}{\kern0pt}rdistrib\ nonneg{\isacharunderscore}{\kern0pt}mat{\isacharunderscore}{\kern0pt}iff{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Matrix Powers%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{primrec}\isamarkupfalse%
-\ matpow\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}semiring{\isacharunderscore}{\kern0pt}{\isadigit{1}}{\isacharcircum}{\kern0pt}{\isacharprime}{\kern0pt}n{\isacharcircum}{\kern0pt}{\isacharprime}{\kern0pt}n\ {\isasymRightarrow}\ nat\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a{\isacharcircum}{\kern0pt}{\isacharprime}{\kern0pt}n{\isacharcircum}{\kern0pt}{\isacharprime}{\kern0pt}n{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ matpow{\isacharunderscore}{\kern0pt}{\isadigit{0}}{\isacharcolon}{\kern0pt}\ \ \ {\isachardoublequoteopen}matpow\ A\ {\isadigit{0}}\ {\isacharequal}{\kern0pt}\ mat\ {\isadigit{1}}{\isachardoublequoteclose}\ {\isacharbar}{\kern0pt}\isanewline
-\ \ matpow{\isacharunderscore}{\kern0pt}Suc{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}matpow\ A\ {\isacharparenleft}{\kern0pt}Suc\ n{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}matpow\ A\ n{\isacharparenright}{\kern0pt}\ {\isacharasterisk}{\kern0pt}{\isacharasterisk}{\kern0pt}\ A{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ nonneg{\isacharunderscore}{\kern0pt}matpow{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isadigit{0}}\ {\isasymle}\ X\ {\isasymLongrightarrow}\ {\isadigit{0}}\ {\isasymle}\ matpow\ {\isacharparenleft}{\kern0pt}X\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ real\ {\isacharcircum}{\kern0pt}\ {\isacharunderscore}{\kern0pt}\ {\isacharcircum}{\kern0pt}\ {\isacharunderscore}{\kern0pt}{\isacharparenright}{\kern0pt}\ i{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}induction\ i{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ nonneg{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}mult{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ matpow{\isacharunderscore}{\kern0pt}mono{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isadigit{0}}\ {\isasymle}\ C\ {\isasymLongrightarrow}\ C\ {\isasymle}\ D\ {\isasymLongrightarrow}\ matpow\ {\isacharparenleft}{\kern0pt}C\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ real{\isacharcircum}{\kern0pt}{\isacharunderscore}{\kern0pt}{\isacharcircum}{\kern0pt}{\isacharunderscore}{\kern0pt}{\isacharparenright}{\kern0pt}\ n\ {\isasymle}\ matpow\ D\ n{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}induction\ n{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}auto\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ matrix{\isacharunderscore}{\kern0pt}mult{\isacharunderscore}{\kern0pt}mono\ nonneg{\isacharunderscore}{\kern0pt}matpow{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ matpow{\isacharunderscore}{\kern0pt}scaleR{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}matpow\ {\isacharparenleft}{\kern0pt}c\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isacharparenleft}{\kern0pt}X\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isacharprime}{\kern0pt}b\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ real{\isacharunderscore}{\kern0pt}algebra{\isacharunderscore}{\kern0pt}{\isadigit{1}}{\isacharcircum}{\kern0pt}{\isacharunderscore}{\kern0pt}{\isacharcircum}{\kern0pt}{\isacharunderscore}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ n\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}c{\isacharcircum}{\kern0pt}n{\isacharparenright}{\kern0pt}\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isacharparenleft}{\kern0pt}matpow\ X{\isacharparenright}{\kern0pt}\ n{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}induction\ n\ arbitrary{\isacharcolon}{\kern0pt}\ X\ c{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{case}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}Suc\ n{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}matpow\ {\isacharparenleft}{\kern0pt}c\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ X{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}Suc\ n{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}c{\isacharcircum}{\kern0pt}n{\isacharparenright}{\kern0pt}{\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isacharparenleft}{\kern0pt}matpow\ X{\isacharparenright}{\kern0pt}\ n\ {\isacharasterisk}{\kern0pt}{\isacharasterisk}{\kern0pt}\ c\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ X{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ Suc\ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isacharequal}{\kern0pt}\ c\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}c{\isacharcircum}{\kern0pt}n{\isacharparenright}{\kern0pt}\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isacharparenleft}{\kern0pt}matpow\ X{\isacharparenright}{\kern0pt}\ n\ {\isacharasterisk}{\kern0pt}{\isacharasterisk}{\kern0pt}\ X{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ scalar{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}assoc{\isacharprime}{\kern0pt}\ \isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ scalar{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}assoc{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{finally}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}case\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ scalar{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}assoc{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ matrix{\isacharunderscore}{\kern0pt}vector{\isacharunderscore}{\kern0pt}mult{\isacharunderscore}{\kern0pt}code{\isacharprime}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}X\ {\isacharasterisk}{\kern0pt}v\ x{\isacharparenright}{\kern0pt}\ {\isachardollar}{\kern0pt}\ i\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSum}j{\isasymin}UNIV{\isachardot}{\kern0pt}\ X\ {\isachardollar}{\kern0pt}\ i\ {\isachardollar}{\kern0pt}\ j\ {\isacharasterisk}{\kern0pt}\ x\ {\isachardollar}{\kern0pt}\ j{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ matrix{\isacharunderscore}{\kern0pt}vector{\isacharunderscore}{\kern0pt}mult{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ matrix{\isacharunderscore}{\kern0pt}vector{\isacharunderscore}{\kern0pt}mult{\isacharunderscore}{\kern0pt}mono{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isadigit{0}}{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}real{\isacharcircum}{\kern0pt}{\isacharunderscore}{\kern0pt}{\isacharcircum}{\kern0pt}{\isacharunderscore}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymle}\ X\ {\isasymLongrightarrow}\ {\isadigit{0}}\ {\isasymle}\ v\ {\isasymLongrightarrow}\ X\ {\isasymle}\ Y\ {\isasymLongrightarrow}\ X\ {\isacharasterisk}{\kern0pt}v\ v\ {\isasymle}\ Y\ {\isacharasterisk}{\kern0pt}v\ v{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ diff{\isacharunderscore}{\kern0pt}ge{\isacharunderscore}{\kern0pt}{\isadigit{0}}{\isacharunderscore}{\kern0pt}iff{\isacharunderscore}{\kern0pt}ge\ matrix{\isacharunderscore}{\kern0pt}vector{\isacharunderscore}{\kern0pt}mult{\isacharunderscore}{\kern0pt}diff{\isacharunderscore}{\kern0pt}rdistrib\ nonneg{\isacharunderscore}{\kern0pt}mat{\isacharunderscore}{\kern0pt}iff{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Triangular Matrices%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{definition}\isamarkupfalse%
-\ {\isachardoublequoteopen}lower{\isacharunderscore}{\kern0pt}triangular{\isacharunderscore}{\kern0pt}mat\ X\ {\isasymlongleftrightarrow}\ {\isacharparenleft}{\kern0pt}{\isasymforall}i\ j{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}i\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isacharprime}{\kern0pt}b{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}{\isacharbraceleft}{\kern0pt}finite{\isacharcomma}{\kern0pt}\ linorder{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharless}{\kern0pt}\ j\ {\isasymlongrightarrow}\ X\ {\isachardollar}{\kern0pt}\ i\ {\isachardollar}{\kern0pt}\ j\ {\isacharequal}{\kern0pt}\ {\isadigit{0}}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ {\isachardoublequoteopen}strict{\isacharunderscore}{\kern0pt}lower{\isacharunderscore}{\kern0pt}triangular{\isacharunderscore}{\kern0pt}mat\ X\ {\isasymlongleftrightarrow}\ {\isacharparenleft}{\kern0pt}{\isasymforall}i\ j{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}i\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isacharprime}{\kern0pt}b{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}{\isacharbraceleft}{\kern0pt}finite{\isacharcomma}{\kern0pt}\ linorder{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymle}\ j\ {\isasymlongrightarrow}\ X\ {\isachardollar}{\kern0pt}\ i\ {\isachardollar}{\kern0pt}\ j\ {\isacharequal}{\kern0pt}\ {\isadigit{0}}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ {\isachardoublequoteopen}upper{\isacharunderscore}{\kern0pt}triangular{\isacharunderscore}{\kern0pt}mat\ X\ {\isasymlongleftrightarrow}\ {\isacharparenleft}{\kern0pt}{\isasymforall}i\ j{\isachardot}{\kern0pt}\ j\ {\isacharless}{\kern0pt}\ i\ {\isasymlongrightarrow}\ X\ {\isachardollar}{\kern0pt}\ i\ {\isachardollar}{\kern0pt}\ j\ {\isacharequal}{\kern0pt}\ {\isadigit{0}}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ stlI{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}strict{\isacharunderscore}{\kern0pt}lower{\isacharunderscore}{\kern0pt}triangular{\isacharunderscore}{\kern0pt}mat\ X\ {\isasymLongrightarrow}\ lower{\isacharunderscore}{\kern0pt}triangular{\isacharunderscore}{\kern0pt}mat\ X{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ strict{\isacharunderscore}{\kern0pt}lower{\isacharunderscore}{\kern0pt}triangular{\isacharunderscore}{\kern0pt}mat{\isacharunderscore}{\kern0pt}def\ lower{\isacharunderscore}{\kern0pt}triangular{\isacharunderscore}{\kern0pt}mat{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ lower{\isacharunderscore}{\kern0pt}triangular{\isacharunderscore}{\kern0pt}mat{\isacharunderscore}{\kern0pt}mat{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}lower{\isacharunderscore}{\kern0pt}triangular{\isacharunderscore}{\kern0pt}mat\ {\isacharparenleft}{\kern0pt}mat\ x{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ lower{\isacharunderscore}{\kern0pt}triangular{\isacharunderscore}{\kern0pt}mat{\isacharunderscore}{\kern0pt}def\ mat{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ lower{\isacharunderscore}{\kern0pt}triangular{\isacharunderscore}{\kern0pt}mult{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}lower{\isacharunderscore}{\kern0pt}triangular{\isacharunderscore}{\kern0pt}mat\ X{\isachardoublequoteclose}\ {\isachardoublequoteopen}lower{\isacharunderscore}{\kern0pt}triangular{\isacharunderscore}{\kern0pt}mat\ Y{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}lower{\isacharunderscore}{\kern0pt}triangular{\isacharunderscore}{\kern0pt}mat\ {\isacharparenleft}{\kern0pt}X\ {\isacharasterisk}{\kern0pt}{\isacharasterisk}{\kern0pt}\ Y{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ assms\ \isanewline
-\ \ \isacommand{unfolding}\isamarkupfalse%
-\ matrix{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}mult{\isacharunderscore}{\kern0pt}def\ lower{\isacharunderscore}{\kern0pt}triangular{\isacharunderscore}{\kern0pt}mat{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ sum{\isachardot}{\kern0pt}neutral{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}metis\ mult{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}zero\ neqE\ less{\isacharunderscore}{\kern0pt}trans{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ lower{\isacharunderscore}{\kern0pt}triangular{\isacharunderscore}{\kern0pt}pow{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}lower{\isacharunderscore}{\kern0pt}triangular{\isacharunderscore}{\kern0pt}mat\ X{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}lower{\isacharunderscore}{\kern0pt}triangular{\isacharunderscore}{\kern0pt}mat\ {\isacharparenleft}{\kern0pt}matpow\ X\ i{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ assms\ lower{\isacharunderscore}{\kern0pt}triangular{\isacharunderscore}{\kern0pt}mult\ lower{\isacharunderscore}{\kern0pt}triangular{\isacharunderscore}{\kern0pt}mat{\isacharunderscore}{\kern0pt}mat\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}induction\ i{\isacharparenright}{\kern0pt}\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ lower{\isacharunderscore}{\kern0pt}triangular{\isacharunderscore}{\kern0pt}suminf{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}{\isasymAnd}i{\isachardot}{\kern0pt}\ lower{\isacharunderscore}{\kern0pt}triangular{\isacharunderscore}{\kern0pt}mat\ {\isacharparenleft}{\kern0pt}f\ i{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ {\isachardoublequoteopen}summable\ {\isacharparenleft}{\kern0pt}f\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ nat\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}b{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}real{\isacharunderscore}{\kern0pt}normed{\isacharunderscore}{\kern0pt}vector{\isacharcircum}{\kern0pt}{\isacharunderscore}{\kern0pt}{\isacharcircum}{\kern0pt}{\isacharunderscore}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}lower{\isacharunderscore}{\kern0pt}triangular{\isacharunderscore}{\kern0pt}mat\ {\isacharparenleft}{\kern0pt}{\isasymSum}i{\isachardot}{\kern0pt}\ f\ i{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ assms\isanewline
-\ \ \isacommand{unfolding}\isamarkupfalse%
-\ lower{\isacharunderscore}{\kern0pt}triangular{\isacharunderscore}{\kern0pt}mat{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}subst\ bounded{\isacharunderscore}{\kern0pt}linear{\isachardot}{\kern0pt}suminf{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}auto\ intro{\isacharcolon}{\kern0pt}\ bounded{\isacharunderscore}{\kern0pt}linear{\isacharunderscore}{\kern0pt}compose{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ lower{\isacharunderscore}{\kern0pt}triangular{\isacharunderscore}{\kern0pt}pow{\isacharunderscore}{\kern0pt}eq{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}lower{\isacharunderscore}{\kern0pt}triangular{\isacharunderscore}{\kern0pt}mat\ X{\isachardoublequoteclose}\ {\isachardoublequoteopen}lower{\isacharunderscore}{\kern0pt}triangular{\isacharunderscore}{\kern0pt}mat\ Y{\isachardoublequoteclose}\ {\isachardoublequoteopen}{\isasymAnd}s{\isacharprime}{\kern0pt}{\isachardot}{\kern0pt}\ s{\isacharprime}{\kern0pt}\ {\isasymle}\ s\ {\isasymLongrightarrow}\ row\ s{\isacharprime}{\kern0pt}\ X\ {\isacharequal}{\kern0pt}\ row\ s{\isacharprime}{\kern0pt}\ Y{\isachardoublequoteclose}\ {\isachardoublequoteopen}s{\isacharprime}{\kern0pt}\ {\isasymle}\ s{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}row\ s{\isacharprime}{\kern0pt}\ {\isacharparenleft}{\kern0pt}matpow\ X\ i{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ row\ s{\isacharprime}{\kern0pt}\ {\isacharparenleft}{\kern0pt}matpow\ Y\ i{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ assms\isanewline
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}induction\ i{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{case}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}Suc\ i{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}case\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ ltX{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}lower{\isacharunderscore}{\kern0pt}triangular{\isacharunderscore}{\kern0pt}mat\ {\isacharparenleft}{\kern0pt}matpow\ X\ i{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ Suc{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}\ lower{\isacharunderscore}{\kern0pt}triangular{\isacharunderscore}{\kern0pt}pow{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ ltY{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}lower{\isacharunderscore}{\kern0pt}triangular{\isacharunderscore}{\kern0pt}mat\ {\isacharparenleft}{\kern0pt}matpow\ Y\ i{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ Suc{\isacharparenleft}{\kern0pt}{\isadigit{3}}{\isacharparenright}{\kern0pt}\ lower{\isacharunderscore}{\kern0pt}triangular{\isacharunderscore}{\kern0pt}pow{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}\ {\isacharparenleft}{\kern0pt}{\isasymSum}k{\isasymin}UNIV{\isachardot}{\kern0pt}\ matpow\ X\ i\ {\isachardollar}{\kern0pt}\ s{\isacharprime}{\kern0pt}\ {\isachardollar}{\kern0pt}\ k\ {\isacharasterisk}{\kern0pt}\ X\ {\isachardollar}{\kern0pt}\ k\ {\isachardollar}{\kern0pt}\ j{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSum}k{\isasymin}UNIV{\isachardot}{\kern0pt}\ matpow\ Y\ i\ {\isachardollar}{\kern0pt}\ s{\isacharprime}{\kern0pt}\ {\isachardollar}{\kern0pt}\ k\ {\isacharasterisk}{\kern0pt}\ Y\ {\isachardollar}{\kern0pt}\ k\ {\isachardollar}{\kern0pt}\ j{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{for}\ j\isanewline
-\ \ \ \ \isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isasymSum}k{\isasymin}UNIV{\isachardot}{\kern0pt}\ matpow\ X\ i\ {\isachardollar}{\kern0pt}\ s{\isacharprime}{\kern0pt}\ {\isachardollar}{\kern0pt}\ k\ {\isacharasterisk}{\kern0pt}\ X\ {\isachardollar}{\kern0pt}\ k\ {\isachardollar}{\kern0pt}\ j{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSum}k{\isasymin}UNIV{\isachardot}{\kern0pt}\ if\ s{\isacharprime}{\kern0pt}\ {\isacharless}{\kern0pt}\ k\ then\ {\isadigit{0}}\ else\ matpow\ Y\ i\ {\isachardollar}{\kern0pt}\ s{\isacharprime}{\kern0pt}\ {\isachardollar}{\kern0pt}\ k\ {\isacharasterisk}{\kern0pt}\ X\ {\isachardollar}{\kern0pt}\ k\ {\isachardollar}{\kern0pt}\ j{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ Suc\ ltY\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ row{\isacharunderscore}{\kern0pt}def\ lower{\isacharunderscore}{\kern0pt}triangular{\isacharunderscore}{\kern0pt}mat{\isacharunderscore}{\kern0pt}def\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ sum{\isachardot}{\kern0pt}cong{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSum}k\ {\isasymin}\ UNIV\ {\isachardot}{\kern0pt}\ matpow\ Y\ i\ {\isachardollar}{\kern0pt}\ s{\isacharprime}{\kern0pt}\ {\isachardollar}{\kern0pt}\ k\ {\isacharasterisk}{\kern0pt}\ Y\ {\isachardollar}{\kern0pt}\ k\ {\isachardollar}{\kern0pt}\ j{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ Suc\ ltY\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ row{\isacharunderscore}{\kern0pt}def\ lower{\isacharunderscore}{\kern0pt}triangular{\isacharunderscore}{\kern0pt}mat{\isacharunderscore}{\kern0pt}def\ cong{\isacharcolon}{\kern0pt}\ if{\isacharunderscore}{\kern0pt}cong\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ sum{\isachardot}{\kern0pt}cong{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{finally}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isacommand{{\isachardot}{\kern0pt}}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ row{\isacharunderscore}{\kern0pt}def\ matrix{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}mult{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\isacommand{qed}\isamarkupfalse%
-\ simp%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ lower{\isacharunderscore}{\kern0pt}triangular{\isacharunderscore}{\kern0pt}mat{\isacharunderscore}{\kern0pt}mult{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}lower{\isacharunderscore}{\kern0pt}triangular{\isacharunderscore}{\kern0pt}mat\ M{\isachardoublequoteclose}\ {\isachardoublequoteopen}{\isasymAnd}i{\isachardot}{\kern0pt}\ i\ {\isasymle}\ j\ {\isasymLongrightarrow}\ v\ {\isachardollar}{\kern0pt}\ i\ {\isacharequal}{\kern0pt}\ v{\isacharprime}{\kern0pt}\ {\isachardollar}{\kern0pt}\ i{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}M\ {\isacharasterisk}{\kern0pt}v\ v{\isacharparenright}{\kern0pt}\ {\isachardollar}{\kern0pt}\ j\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}M\ {\isacharasterisk}{\kern0pt}v\ v{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isachardollar}{\kern0pt}\ j{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}M\ {\isacharasterisk}{\kern0pt}v\ v{\isacharparenright}{\kern0pt}\ {\isachardollar}{\kern0pt}\ j\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSum}i{\isasymin}UNIV{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}if\ j\ {\isacharless}{\kern0pt}\ i\ then\ {\isadigit{0}}\ else\ \ M\ {\isachardollar}{\kern0pt}\ j\ {\isachardollar}{\kern0pt}\ i\ {\isacharasterisk}{\kern0pt}\ v\ {\isachardollar}{\kern0pt}\ i{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assms\ \isacommand{unfolding}\isamarkupfalse%
-\ lower{\isacharunderscore}{\kern0pt}triangular{\isacharunderscore}{\kern0pt}mat{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ matrix{\isacharunderscore}{\kern0pt}vector{\isacharunderscore}{\kern0pt}mult{\isacharunderscore}{\kern0pt}def\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ sum{\isachardot}{\kern0pt}cong{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSum}i{\isasymin}UNIV{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}if\ j\ {\isacharless}{\kern0pt}\ i\ then\ {\isadigit{0}}\ else\ \ M\ {\isachardollar}{\kern0pt}\ j\ {\isachardollar}{\kern0pt}\ i\ {\isacharasterisk}{\kern0pt}\ v{\isacharprime}{\kern0pt}\ {\isachardollar}{\kern0pt}\ i{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assms\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ sum{\isachardot}{\kern0pt}cong{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}M\ {\isacharasterisk}{\kern0pt}v\ v{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isachardollar}{\kern0pt}\ j{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assms\ \isacommand{unfolding}\isamarkupfalse%
-\ lower{\isacharunderscore}{\kern0pt}triangular{\isacharunderscore}{\kern0pt}mat{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ matrix{\isacharunderscore}{\kern0pt}vector{\isacharunderscore}{\kern0pt}mult{\isacharunderscore}{\kern0pt}def\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ sum{\isachardot}{\kern0pt}cong{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{finally}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isacommand{{\isachardot}{\kern0pt}}\isamarkupfalse%
-\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Inverses%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{lemma}\isamarkupfalse%
-\ matrix{\isacharunderscore}{\kern0pt}inv{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}invertible\ M{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ matrix{\isacharunderscore}{\kern0pt}inv{\isacharunderscore}{\kern0pt}left{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}matrix{\isacharunderscore}{\kern0pt}inv\ M\ {\isacharasterisk}{\kern0pt}{\isacharasterisk}{\kern0pt}\ M\ {\isacharequal}{\kern0pt}\ mat\ {\isadigit{1}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isakeyword{and}\ matrix{\isacharunderscore}{\kern0pt}inv{\isacharunderscore}{\kern0pt}right{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}M\ {\isacharasterisk}{\kern0pt}{\isacharasterisk}{\kern0pt}\ matrix{\isacharunderscore}{\kern0pt}inv\ M\ {\isacharequal}{\kern0pt}\ mat\ {\isadigit{1}}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ {\isacartoucheopen}invertible\ M{\isacartoucheclose}\ \isakeyword{and}\ someI{\isacharunderscore}{\kern0pt}ex\ {\isacharbrackleft}{\kern0pt}of\ {\isachardoublequoteopen}{\isasymlambda}\ N{\isachardot}{\kern0pt}\ M\ {\isacharasterisk}{\kern0pt}{\isacharasterisk}{\kern0pt}\ N\ {\isacharequal}{\kern0pt}\ mat\ {\isadigit{1}}\ {\isasymand}\ N\ {\isacharasterisk}{\kern0pt}{\isacharasterisk}{\kern0pt}\ M\ {\isacharequal}{\kern0pt}\ mat\ {\isadigit{1}}{\isachardoublequoteclose}{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \isacommand{unfolding}\isamarkupfalse%
-\ invertible{\isacharunderscore}{\kern0pt}def\ \isakeyword{and}\ matrix{\isacharunderscore}{\kern0pt}inv{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ simp{\isacharunderscore}{\kern0pt}all%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ matrix{\isacharunderscore}{\kern0pt}inv{\isacharunderscore}{\kern0pt}unique{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{fixes}\ A{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}{\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}{\isacharbraceleft}{\kern0pt}semiring{\isacharunderscore}{\kern0pt}{\isadigit{1}}{\isacharbraceright}{\kern0pt}{\isacharcircum}{\kern0pt}{\isacharprime}{\kern0pt}n{\isacharcircum}{\kern0pt}{\isacharprime}{\kern0pt}n{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{assumes}\ AB{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}A\ {\isacharasterisk}{\kern0pt}{\isacharasterisk}{\kern0pt}\ B\ {\isacharequal}{\kern0pt}\ mat\ {\isadigit{1}}{\isachardoublequoteclose}\ \isakeyword{and}\ BA{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}B\ {\isacharasterisk}{\kern0pt}{\isacharasterisk}{\kern0pt}\ A\ {\isacharequal}{\kern0pt}\ mat\ {\isadigit{1}}{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}matrix{\isacharunderscore}{\kern0pt}inv\ A\ {\isacharequal}{\kern0pt}\ B{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ AB\ BA\ invertible{\isacharunderscore}{\kern0pt}def\ matrix{\isacharunderscore}{\kern0pt}inv{\isacharunderscore}{\kern0pt}right\ matrix{\isacharunderscore}{\kern0pt}mul{\isacharunderscore}{\kern0pt}assoc\ matrix{\isacharunderscore}{\kern0pt}mul{\isacharunderscore}{\kern0pt}lid{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\ \isanewline
-%
-\endisadelimproof
-%
-\isadelimtheory
-\isanewline
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{end}\isamarkupfalse%
-%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\end{isabellebody}%
-\endinput
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diff --git a/thys/MDP-Algorithms/output/outline/Modified_Policy_Iteration.tex b/thys/MDP-Algorithms/output/outline/Modified_Policy_Iteration.tex
deleted file mode 100644
--- a/thys/MDP-Algorithms/output/outline/Modified_Policy_Iteration.tex
+++ /dev/null
@@ -1,3615 +0,0 @@
-%
-\begin{isabellebody}%
-\setisabellecontext{Modified{\isacharunderscore}{\kern0pt}Policy{\isacharunderscore}{\kern0pt}Iteration}%
-%
-\isadelimtheory
-\isanewline
-\isanewline
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{theory}\isamarkupfalse%
-\ Modified{\isacharunderscore}{\kern0pt}Policy{\isacharunderscore}{\kern0pt}Iteration\isanewline
-\ \ \isakeyword{imports}\ \isanewline
-\ \ \ \ Policy{\isacharunderscore}{\kern0pt}Iteration\isanewline
-\ \ \ \ Value{\isacharunderscore}{\kern0pt}Iteration\isanewline
-\isakeyword{begin}%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsection{Modified Policy Iteration%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{locale}\isamarkupfalse%
-\ MDP{\isacharunderscore}{\kern0pt}MPI\ {\isacharequal}{\kern0pt}\ MDP{\isacharunderscore}{\kern0pt}finite{\isacharunderscore}{\kern0pt}type\ A\ K\ r\ l\ {\isacharplus}{\kern0pt}\ MDP{\isacharunderscore}{\kern0pt}act\ A\ K\ r\ l\ arb{\isacharunderscore}{\kern0pt}act\isanewline
-\ \ \isakeyword{for}\ A\ \isakeyword{and}\ K\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}s\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ finite\ {\isasymtimes}\ {\isacharprime}{\kern0pt}a\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ finite\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}s\ pmf{\isachardoublequoteclose}\ \isakeyword{and}\ r\ l\ arb{\isacharunderscore}{\kern0pt}act\isanewline
-\isakeyword{begin}%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{The Advantage Function \isa{B}%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{definition}\isamarkupfalse%
-\ {\isachardoublequoteopen}B\ v\ s\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSqunion}d\ {\isasymin}\ D\isactrlsub R{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}r{\isacharunderscore}{\kern0pt}dec\ d\ s\ {\isacharplus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isasymP}\isactrlsub {\isadigit{1}}\ d\ {\isacharminus}{\kern0pt}\ id{\isacharunderscore}{\kern0pt}blinfun{\isacharparenright}{\kern0pt}\ v\ s{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}%
-\begin{isamarkuptext}%
-The function \isa{B} denotes the advantage of choosing the optimal action vs.
-  the current value estimate%
-\end{isamarkuptext}\isamarkuptrue%
-\isacommand{lemma}\isamarkupfalse%
-\ B{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}{\isasymL}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}B\ v\ s\ {\isacharequal}{\kern0pt}\ {\isasymL}\ v\ s\ {\isacharminus}{\kern0pt}\ v\ s{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isacharasterisk}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}B\ v\ s\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSqunion}d\ {\isasymin}\ D\isactrlsub R{\isachardot}{\kern0pt}\ L\ d\ v\ s\ {\isacharminus}{\kern0pt}\ v\ s{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ B{\isacharunderscore}{\kern0pt}def\ L{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp\ add{\isacharcolon}{\kern0pt}\ blinfun{\isachardot}{\kern0pt}bilinear{\isacharunderscore}{\kern0pt}simps\ add{\isacharunderscore}{\kern0pt}diff{\isacharunderscore}{\kern0pt}eq{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ {\isacharasterisk}{\kern0pt}\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}rule\ antisym{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isasymSqunion}d{\isasymin}D\isactrlsub R{\isachardot}{\kern0pt}\ L\ d\ v\ s\ {\isacharminus}{\kern0pt}\ v\ s{\isacharparenright}{\kern0pt}\ {\isasymle}\ {\isasymL}\ v\ s\ {\isacharminus}{\kern0pt}\ v\ s{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ {\isasymL}{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ ex{\isacharunderscore}{\kern0pt}dec\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}fastforce\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ cSUP{\isacharunderscore}{\kern0pt}upper\ cSUP{\isacharunderscore}{\kern0pt}least{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}bdd{\isacharunderscore}{\kern0pt}above\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}{\isasymlambda}d{\isachardot}{\kern0pt}\ L\ d\ v\ s\ {\isacharminus}{\kern0pt}\ v\ s{\isacharparenright}{\kern0pt}\ {\isacharbackquote}{\kern0pt}\ D\isactrlsub R{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ bounded{\isacharunderscore}{\kern0pt}const\ bounded{\isacharunderscore}{\kern0pt}minus{\isacharunderscore}{\kern0pt}comp\ bounded{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}bdd{\isacharunderscore}{\kern0pt}above{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymL}\ v\ s\ {\isacharminus}{\kern0pt}\ v\ s\ {\isasymle}\ {\isacharparenleft}{\kern0pt}{\isasymSqunion}d\ {\isasymin}\ D\isactrlsub R{\isachardot}{\kern0pt}\ L\ d\ v\ s\ {\isacharminus}{\kern0pt}\ v\ s{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ {\isasymL}{\isacharunderscore}{\kern0pt}def\ diff{\isacharunderscore}{\kern0pt}le{\isacharunderscore}{\kern0pt}eq\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}intro\ cSUP{\isacharunderscore}{\kern0pt}least{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}auto\ intro{\isacharcolon}{\kern0pt}\ cSUP{\isacharunderscore}{\kern0pt}upper{\isadigit{2}}\ simp{\isacharcolon}{\kern0pt}\ diff{\isacharunderscore}{\kern0pt}le{\isacharunderscore}{\kern0pt}eq{\isacharbrackleft}{\kern0pt}symmetric{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\begin{isamarkuptext}%
-\isa{B} is a bounded function.%
-\end{isamarkuptext}\isamarkuptrue%
-\isacommand{lift{\isacharunderscore}{\kern0pt}definition}\isamarkupfalse%
-\ B\isactrlsub b\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}s\ {\isasymRightarrow}\isactrlsub b\ real{\isacharparenright}{\kern0pt}\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}s\ {\isasymRightarrow}\isactrlsub b\ real{\isachardoublequoteclose}\ \isakeyword{is}\ {\isachardoublequoteopen}B{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ {\isasymL}\isactrlsub b{\isachardot}{\kern0pt}rep{\isacharunderscore}{\kern0pt}eq{\isacharbrackleft}{\kern0pt}symmetric{\isacharbrackright}{\kern0pt}\ B{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}{\isasymL}\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ bfun{\isacharunderscore}{\kern0pt}normI\ order{\isachardot}{\kern0pt}trans{\isacharbrackleft}{\kern0pt}OF\ abs{\isacharunderscore}{\kern0pt}triangle{\isacharunderscore}{\kern0pt}ineq{\isadigit{4}}{\isacharbrackright}{\kern0pt}\ add{\isacharunderscore}{\kern0pt}mono\ abs{\isacharunderscore}{\kern0pt}le{\isacharunderscore}{\kern0pt}norm{\isacharunderscore}{\kern0pt}bfun{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ B\isactrlsub b{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}{\isasymL}\isactrlsub b{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}B\isactrlsub b\ v\ {\isacharequal}{\kern0pt}\ {\isasymL}\isactrlsub b\ v\ {\isacharminus}{\kern0pt}\ v{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ {\isasymL}\isactrlsub b{\isachardot}{\kern0pt}rep{\isacharunderscore}{\kern0pt}eq\ B\isactrlsub b{\isachardot}{\kern0pt}rep{\isacharunderscore}{\kern0pt}eq\ B{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}{\isasymL}{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ {\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}SUP{\isacharunderscore}{\kern0pt}L\isactrlsub a{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymL}\isactrlsub b\ v\ s\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSqunion}a\ {\isasymin}\ A\ s{\isachardot}{\kern0pt}\ L\isactrlsub a\ a\ v\ s{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ L{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}L\isactrlsub a{\isacharunderscore}{\kern0pt}det\ {\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}SUP{\isacharunderscore}{\kern0pt}det\ SUP{\isacharunderscore}{\kern0pt}step{\isacharunderscore}{\kern0pt}det{\isacharunderscore}{\kern0pt}eq\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Optimization of the Value Function over Multiple Steps%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{definition}\isamarkupfalse%
-\ {\isachardoublequoteopen}U\ m\ v\ s\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSqunion}d\ {\isasymin}\ D\isactrlsub R{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}fin\ {\isacharparenleft}{\kern0pt}mk{\isacharunderscore}{\kern0pt}stationary\ d{\isacharparenright}{\kern0pt}\ m\ {\isacharplus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isasymP}\isactrlsub {\isadigit{1}}\ d{\isacharparenright}{\kern0pt}{\isacharcircum}{\kern0pt}{\isacharcircum}{\kern0pt}m{\isacharparenright}{\kern0pt}\ v{\isacharparenright}{\kern0pt}\ s{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}%
-\begin{isamarkuptext}%
-\isa{U} expresses the value estimate obtained by optimizing the first \isa{m} steps and 
-  afterwards using the current estimate.%
-\end{isamarkuptext}\isamarkuptrue%
-\isacommand{lemma}\isamarkupfalse%
-\ U{\isacharunderscore}{\kern0pt}zero\ {\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}U\ {\isadigit{0}}\ v\ {\isacharequal}{\kern0pt}\ v{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ U{\isacharunderscore}{\kern0pt}def\ {\isasymL}{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}fin{\isachardot}{\kern0pt}rep{\isacharunderscore}{\kern0pt}eq{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ U{\isacharunderscore}{\kern0pt}one{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}{\isasymL}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}U\ {\isadigit{1}}\ v\ s\ {\isacharequal}{\kern0pt}\ {\isasymL}\ v\ s{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ U{\isacharunderscore}{\kern0pt}def\ {\isasymL}{\isacharunderscore}{\kern0pt}def\ \isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}fin{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}{\isasymP}\isactrlsub X\ L{\isacharunderscore}{\kern0pt}def\ blinfun{\isachardot}{\kern0pt}bilinear{\isacharunderscore}{\kern0pt}simps{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lift{\isacharunderscore}{\kern0pt}definition}\isamarkupfalse%
-\ U\isactrlsub b\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}nat\ {\isasymRightarrow}\ {\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}s\ {\isasymRightarrow}\isactrlsub b\ real{\isacharparenright}{\kern0pt}\ {\isasymRightarrow}\ {\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}s\ {\isasymRightarrow}\isactrlsub b\ real{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{is}\ U\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\ n\ v\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}norm\ {\isacharparenleft}{\kern0pt}{\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}fin\ {\isacharparenleft}{\kern0pt}mk{\isacharunderscore}{\kern0pt}stationary\ d{\isacharparenright}{\kern0pt}\ m{\isacharparenright}{\kern0pt}\ {\isasymle}\ {\isacharparenleft}{\kern0pt}{\isasymSum}i{\isacharless}{\kern0pt}m{\isachardot}{\kern0pt}\ l\ {\isacharcircum}{\kern0pt}\ i\ {\isacharasterisk}{\kern0pt}\ r\isactrlsub M{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{for}\ d\ m\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ abs{\isacharunderscore}{\kern0pt}{\isasymnu}{\isacharunderscore}{\kern0pt}fin{\isacharunderscore}{\kern0pt}le\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}fin{\isachardot}{\kern0pt}rep{\isacharunderscore}{\kern0pt}eq\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ norm{\isacharunderscore}{\kern0pt}bound{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}norm\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isasymP}\isactrlsub {\isadigit{1}}\ d{\isacharparenright}{\kern0pt}{\isacharcircum}{\kern0pt}{\isacharcircum}{\kern0pt}m{\isacharparenright}{\kern0pt}\ v{\isacharparenright}{\kern0pt}\ {\isasymle}\ l\ {\isacharcircum}{\kern0pt}\ m\ {\isacharasterisk}{\kern0pt}\ norm\ v{\isachardoublequoteclose}\ \isakeyword{for}\ d\ m\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ {\isasymP}\isactrlsub X{\isacharunderscore}{\kern0pt}const{\isacharbrackleft}{\kern0pt}symmetric{\isacharbrackright}{\kern0pt}\ blinfun{\isachardot}{\kern0pt}bilinear{\isacharunderscore}{\kern0pt}simps\ blincomp{\isacharunderscore}{\kern0pt}scaleR{\isacharunderscore}{\kern0pt}right\ simp\ del{\isacharcolon}{\kern0pt}\ {\isasymP}\isactrlsub X{\isacharunderscore}{\kern0pt}sconst\ \isanewline
-\ \ \ \ \ \ \ \ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ boundedI\ order{\isachardot}{\kern0pt}trans{\isacharbrackleft}{\kern0pt}OF\ abs{\isacharunderscore}{\kern0pt}le{\isacharunderscore}{\kern0pt}norm{\isacharunderscore}{\kern0pt}bfun{\isacharbrackright}{\kern0pt}\ mult{\isacharunderscore}{\kern0pt}left{\isacharunderscore}{\kern0pt}mono{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isacharasterisk}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}norm\ {\isacharparenleft}{\kern0pt}{\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}fin\ {\isacharparenleft}{\kern0pt}mk{\isacharunderscore}{\kern0pt}stationary\ d{\isacharparenright}{\kern0pt}\ m\ {\isacharplus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isasymP}\isactrlsub {\isadigit{1}}\ d{\isacharparenright}{\kern0pt}{\isacharcircum}{\kern0pt}{\isacharcircum}{\kern0pt}m{\isacharparenright}{\kern0pt}\ v{\isacharparenright}{\kern0pt}\ {\isasymle}\ {\isacharparenleft}{\kern0pt}{\isasymSum}i{\isacharless}{\kern0pt}m{\isachardot}{\kern0pt}\ l\ {\isacharcircum}{\kern0pt}\ i\ {\isacharasterisk}{\kern0pt}\ r\isactrlsub M{\isacharparenright}{\kern0pt}\ {\isacharplus}{\kern0pt}\ \ l\ {\isacharcircum}{\kern0pt}\ m\ {\isacharasterisk}{\kern0pt}\ norm\ v{\isachardoublequoteclose}\ \isakeyword{for}\ d\ m\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ norm{\isacharunderscore}{\kern0pt}triangle{\isacharunderscore}{\kern0pt}mono\ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}U\ n\ v\ {\isasymin}\ bfun{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ ex{\isacharunderscore}{\kern0pt}dec\ order{\isachardot}{\kern0pt}trans{\isacharbrackleft}{\kern0pt}OF\ abs{\isacharunderscore}{\kern0pt}le{\isacharunderscore}{\kern0pt}norm{\isacharunderscore}{\kern0pt}bfun\ {\isacharasterisk}{\kern0pt}{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}fastforce\ simp{\isacharcolon}{\kern0pt}\ U{\isacharunderscore}{\kern0pt}def\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ bfun{\isacharunderscore}{\kern0pt}normI\ cSup{\isacharunderscore}{\kern0pt}abs{\isacharunderscore}{\kern0pt}le{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ U\isactrlsub b{\isacharunderscore}{\kern0pt}contraction{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}dist\ {\isacharparenleft}{\kern0pt}U\isactrlsub b\ m\ v{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}U\isactrlsub b\ m\ u{\isacharparenright}{\kern0pt}\ {\isasymle}\ l\ {\isacharcircum}{\kern0pt}\ m\ {\isacharasterisk}{\kern0pt}\ dist\ v\ u{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ aux{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}dist\ {\isacharparenleft}{\kern0pt}U\isactrlsub b\ m\ v\ s{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}U\isactrlsub b\ m\ u\ s{\isacharparenright}{\kern0pt}\ {\isasymle}\ l\ {\isacharcircum}{\kern0pt}\ m\ {\isacharasterisk}{\kern0pt}\ dist\ v\ u{\isachardoublequoteclose}\ \isakeyword{if}\ le{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}U\isactrlsub b\ m\ u\ s\ {\isasymle}\ U\isactrlsub b\ m\ v\ s{\isachardoublequoteclose}\ \isakeyword{for}\ s\ v\ u\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}U\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}{\isasymlambda}m\ v\ d{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}fin\ {\isacharparenleft}{\kern0pt}mk{\isacharunderscore}{\kern0pt}stationary\ d{\isacharparenright}{\kern0pt}\ m\ {\isacharplus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isasymP}\isactrlsub {\isadigit{1}}\ d{\isacharparenright}{\kern0pt}\ {\isacharcircum}{\kern0pt}{\isacharcircum}{\kern0pt}\ m{\isacharparenright}{\kern0pt}\ v{\isacharparenright}{\kern0pt}\ s{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}U\isactrlsub b\ m\ v\ s\ {\isacharminus}{\kern0pt}\ U\isactrlsub b\ m\ u\ s\ {\isasymle}\ {\isacharparenleft}{\kern0pt}{\isasymSqunion}d\ {\isasymin}\ D\isactrlsub R{\isachardot}{\kern0pt}\ {\isacharquery}{\kern0pt}U\ m\ v\ d\ {\isacharminus}{\kern0pt}\ {\isacharquery}{\kern0pt}U\ m\ u\ d{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ bounded{\isacharunderscore}{\kern0pt}stationary{\isacharunderscore}{\kern0pt}{\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}fin\ bounded{\isacharunderscore}{\kern0pt}disc{\isacharunderscore}{\kern0pt}{\isasymP}\isactrlsub {\isadigit{1}}\ le\isanewline
-\ \ \ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ U\isactrlsub b{\isachardot}{\kern0pt}rep{\isacharunderscore}{\kern0pt}eq\ U{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}intro\ le{\isacharunderscore}{\kern0pt}SUP{\isacharunderscore}{\kern0pt}diff{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}auto\ intro{\isacharcolon}{\kern0pt}\ bounded{\isacharunderscore}{\kern0pt}plus{\isacharunderscore}{\kern0pt}comp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSqunion}d\ {\isasymin}\ D\isactrlsub R{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isasymP}\isactrlsub {\isadigit{1}}\ d{\isacharparenright}{\kern0pt}\ {\isacharcircum}{\kern0pt}{\isacharcircum}{\kern0pt}\ m{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}v\ {\isacharminus}{\kern0pt}\ u{\isacharparenright}{\kern0pt}\ s{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ L{\isacharunderscore}{\kern0pt}def\ scale{\isacharunderscore}{\kern0pt}right{\isacharunderscore}{\kern0pt}diff{\isacharunderscore}{\kern0pt}distrib\ blinfun{\isachardot}{\kern0pt}bilinear{\isacharunderscore}{\kern0pt}simps{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSqunion}d\ {\isasymin}\ D\isactrlsub R{\isachardot}{\kern0pt}\ l{\isacharcircum}{\kern0pt}m\ {\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}{\isasymP}\isactrlsub {\isadigit{1}}\ d\ {\isacharcircum}{\kern0pt}{\isacharcircum}{\kern0pt}\ m{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}v\ {\isacharminus}{\kern0pt}\ u{\isacharparenright}{\kern0pt}\ s{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ blincomp{\isacharunderscore}{\kern0pt}scaleR{\isacharunderscore}{\kern0pt}right\ blinfun{\isachardot}{\kern0pt}scaleR{\isacharunderscore}{\kern0pt}left{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isacharequal}{\kern0pt}\ l{\isacharcircum}{\kern0pt}m\ {\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSqunion}d\ {\isasymin}\ D\isactrlsub R{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}{\isasymP}\isactrlsub {\isadigit{1}}\ d\ {\isacharcircum}{\kern0pt}{\isacharcircum}{\kern0pt}\ m{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}v\ {\isacharminus}{\kern0pt}\ u{\isacharparenright}{\kern0pt}\ s{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ D\isactrlsub R{\isacharunderscore}{\kern0pt}ne\ bounded{\isacharunderscore}{\kern0pt}P\ bounded{\isacharunderscore}{\kern0pt}disc{\isacharunderscore}{\kern0pt}{\isasymP}\isactrlsub {\isadigit{1}}{\isacharprime}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ intro{\isacharcolon}{\kern0pt}\ bounded{\isacharunderscore}{\kern0pt}SUP{\isacharunderscore}{\kern0pt}mul{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isasymle}\ l{\isacharcircum}{\kern0pt}m\ {\isacharasterisk}{\kern0pt}\ norm\ {\isacharparenleft}{\kern0pt}{\isasymSqunion}d\ {\isasymin}\ D\isactrlsub R{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}{\isasymP}\isactrlsub {\isadigit{1}}\ d\ {\isacharcircum}{\kern0pt}{\isacharcircum}{\kern0pt}\ m{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}v\ {\isacharminus}{\kern0pt}\ u{\isacharparenright}{\kern0pt}\ s{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ mult{\isacharunderscore}{\kern0pt}left{\isacharunderscore}{\kern0pt}mono{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isasymle}\ l{\isacharcircum}{\kern0pt}m\ {\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSqunion}d\ {\isasymin}\ D\isactrlsub R{\isachardot}{\kern0pt}\ norm\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}{\isasymP}\isactrlsub {\isadigit{1}}\ d\ {\isacharcircum}{\kern0pt}{\isacharcircum}{\kern0pt}\ m{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}v\ {\isacharminus}{\kern0pt}\ u{\isacharparenright}{\kern0pt}\ s{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ D\isactrlsub R{\isacharunderscore}{\kern0pt}ne\ ex{\isacharunderscore}{\kern0pt}dec\ bounded{\isacharunderscore}{\kern0pt}norm{\isacharunderscore}{\kern0pt}comp\ bounded{\isacharunderscore}{\kern0pt}disc{\isacharunderscore}{\kern0pt}{\isasymP}\isactrlsub {\isadigit{1}}{\isacharprime}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}fastforce\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ mult{\isacharunderscore}{\kern0pt}left{\isacharunderscore}{\kern0pt}mono{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isasymle}\ l{\isacharcircum}{\kern0pt}m\ {\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSqunion}d\ {\isasymin}\ D\isactrlsub R{\isachardot}{\kern0pt}\ norm\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}{\isasymP}\isactrlsub {\isadigit{1}}\ d\ {\isacharcircum}{\kern0pt}{\isacharcircum}{\kern0pt}\ m{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}v\ {\isacharminus}{\kern0pt}\ u{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ ex{\isacharunderscore}{\kern0pt}dec\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}fastforce\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ order{\isachardot}{\kern0pt}trans{\isacharbrackleft}{\kern0pt}OF\ norm{\isacharunderscore}{\kern0pt}blinfun{\isacharbrackright}{\kern0pt}\ abs{\isacharunderscore}{\kern0pt}le{\isacharunderscore}{\kern0pt}norm{\isacharunderscore}{\kern0pt}bfun\ mult{\isacharunderscore}{\kern0pt}left{\isacharunderscore}{\kern0pt}mono\ cSUP{\isacharunderscore}{\kern0pt}mono{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isasymle}\ l{\isacharcircum}{\kern0pt}m\ {\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSqunion}d\ {\isasymin}\ D\isactrlsub R{\isachardot}{\kern0pt}\ norm\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}v\ {\isacharminus}{\kern0pt}\ u{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ norm{\isacharunderscore}{\kern0pt}{\isasymP}\isactrlsub X{\isacharunderscore}{\kern0pt}apply\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ {\isasymP}\isactrlsub X{\isacharunderscore}{\kern0pt}const{\isacharbrackleft}{\kern0pt}symmetric{\isacharbrackright}{\kern0pt}\ cSUP{\isacharunderscore}{\kern0pt}least\ mult{\isacharunderscore}{\kern0pt}left{\isacharunderscore}{\kern0pt}mono{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isacharequal}{\kern0pt}\ l\ {\isacharcircum}{\kern0pt}m\ {\isacharasterisk}{\kern0pt}\ dist\ v\ u{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ dist{\isacharunderscore}{\kern0pt}norm{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{finally}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}U\isactrlsub b\ m\ v\ s\ {\isacharminus}{\kern0pt}\ U\isactrlsub b\ m\ u\ s\ {\isasymle}\ l{\isacharcircum}{\kern0pt}m\ {\isacharasterisk}{\kern0pt}\ dist\ v\ u{\isachardoublequoteclose}\ \isacommand{{\isachardot}{\kern0pt}}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ dist{\isacharunderscore}{\kern0pt}real{\isacharunderscore}{\kern0pt}def\ le{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}U\isactrlsub b\ m\ v\ s\ {\isasymle}\ U\isactrlsub b\ m\ u\ s\ {\isasymLongrightarrow}\ dist\ {\isacharparenleft}{\kern0pt}U\isactrlsub b\ m\ v\ s{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}U\isactrlsub b\ m\ u\ s{\isacharparenright}{\kern0pt}\ {\isasymle}\ l{\isacharcircum}{\kern0pt}m\ {\isacharasterisk}{\kern0pt}\ dist\ v\ u{\isachardoublequoteclose}\ \isakeyword{for}\ u\ v\ s\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ aux\ dist{\isacharunderscore}{\kern0pt}commute{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}dist\ {\isacharparenleft}{\kern0pt}U\isactrlsub b\ m\ v\ s{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}U\isactrlsub b\ m\ u\ s{\isacharparenright}{\kern0pt}\ {\isasymle}\ l{\isacharcircum}{\kern0pt}m\ {\isacharasterisk}{\kern0pt}\ dist\ v\ u{\isachardoublequoteclose}\ \isakeyword{for}\ u\ v\ s\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ linear\ \isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}dist\ {\isacharparenleft}{\kern0pt}U\isactrlsub b\ m\ v{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}U\isactrlsub b\ m\ u{\isacharparenright}{\kern0pt}\ {\isasymle}\ l{\isacharcircum}{\kern0pt}m\ {\isacharasterisk}{\kern0pt}\ dist\ v\ u{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ dist{\isacharunderscore}{\kern0pt}bound{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ U\isactrlsub b{\isacharunderscore}{\kern0pt}conv{\isacharcolon}{\kern0pt}\isanewline
-\ \ {\isachardoublequoteopen}{\isasymexists}{\isacharbang}{\kern0pt}v{\isachardot}{\kern0pt}\ U\isactrlsub b\ {\isacharparenleft}{\kern0pt}Suc\ m{\isacharparenright}{\kern0pt}\ v\ {\isacharequal}{\kern0pt}\ v{\isachardoublequoteclose}\ \isanewline
-\ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isasymlambda}n{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}U\isactrlsub b\ {\isacharparenleft}{\kern0pt}Suc\ m{\isacharparenright}{\kern0pt}\ {\isacharcircum}{\kern0pt}{\isacharcircum}{\kern0pt}\ n{\isacharparenright}{\kern0pt}\ v{\isacharparenright}{\kern0pt}\ {\isasymlonglonglongrightarrow}\ {\isacharparenleft}{\kern0pt}THE\ v{\isachardot}{\kern0pt}\ U\isactrlsub b\ {\isacharparenleft}{\kern0pt}Suc\ m{\isacharparenright}{\kern0pt}\ v\ {\isacharequal}{\kern0pt}\ v{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isacharasterisk}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}is{\isacharunderscore}{\kern0pt}contraction\ {\isacharparenleft}{\kern0pt}U\isactrlsub b\ {\isacharparenleft}{\kern0pt}Suc\ m{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ is{\isacharunderscore}{\kern0pt}contraction{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ U\isactrlsub b{\isacharunderscore}{\kern0pt}contraction{\isacharbrackleft}{\kern0pt}of\ {\isachardoublequoteopen}Suc\ m{\isachardoublequoteclose}{\isacharbrackright}{\kern0pt}\ le{\isacharunderscore}{\kern0pt}neq{\isacharunderscore}{\kern0pt}trans{\isacharbrackleft}{\kern0pt}OF\ zero{\isacharunderscore}{\kern0pt}le{\isacharunderscore}{\kern0pt}disc{\isacharbrackright}{\kern0pt}\ \isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}cases\ {\isachardoublequoteopen}l\ {\isacharequal}{\kern0pt}\ {\isadigit{0}}{\isachardoublequoteclose}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}auto\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ power{\isacharunderscore}{\kern0pt}Suc{\isacharunderscore}{\kern0pt}less{\isacharunderscore}{\kern0pt}one\ intro{\isacharcolon}{\kern0pt}\ exI{\isacharbrackleft}{\kern0pt}of\ {\isacharunderscore}{\kern0pt}\ {\isachardoublequoteopen}l{\isacharcircum}{\kern0pt}{\isacharparenleft}{\kern0pt}Suc\ m{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymexists}{\isacharbang}{\kern0pt}v{\isachardot}{\kern0pt}\ U\isactrlsub b\ {\isacharparenleft}{\kern0pt}Suc\ m{\isacharparenright}{\kern0pt}\ v\ {\isacharequal}{\kern0pt}\ v{\isachardoublequoteclose}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isasymlambda}n{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}U\isactrlsub b\ {\isacharparenleft}{\kern0pt}Suc\ m{\isacharparenright}{\kern0pt}\ {\isacharcircum}{\kern0pt}{\isacharcircum}{\kern0pt}\ n{\isacharparenright}{\kern0pt}\ v{\isacharparenright}{\kern0pt}\ {\isasymlonglonglongrightarrow}\ {\isacharparenleft}{\kern0pt}THE\ v{\isachardot}{\kern0pt}\ U\isactrlsub b\ {\isacharparenleft}{\kern0pt}Suc\ m{\isacharparenright}{\kern0pt}\ v\ {\isacharequal}{\kern0pt}\ v{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ banach{\isacharprime}{\kern0pt}{\isacharbrackleft}{\kern0pt}OF\ {\isacharasterisk}{\kern0pt}{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ U\isactrlsub b{\isacharunderscore}{\kern0pt}convergent{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}convergent\ {\isacharparenleft}{\kern0pt}{\isasymlambda}n{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}U\isactrlsub b\ {\isacharparenleft}{\kern0pt}Suc\ m{\isacharparenright}{\kern0pt}\ {\isacharcircum}{\kern0pt}{\isacharcircum}{\kern0pt}\ n{\isacharparenright}{\kern0pt}\ v{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}intro\ convergentI{\isacharbrackleft}{\kern0pt}OF\ U\isactrlsub b{\isacharunderscore}{\kern0pt}conv{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ U\isactrlsub b{\isacharunderscore}{\kern0pt}mono{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}v\ {\isasymle}\ u{\isachardoublequoteclose}\ \isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}U\isactrlsub b\ m\ v\ {\isasymle}\ U\isactrlsub b\ m\ u{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ \ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}U\isactrlsub b\ m\ v\ s\ {\isasymle}\ U\isactrlsub b\ m\ u\ s{\isachardoublequoteclose}\ \isakeyword{for}\ s\isanewline
-\ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ U\isactrlsub b{\isachardot}{\kern0pt}rep{\isacharunderscore}{\kern0pt}eq\ U{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}intro\ cSUP{\isacharunderscore}{\kern0pt}mono{\isacharcomma}{\kern0pt}\ goal{\isacharunderscore}{\kern0pt}cases{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{case}\isamarkupfalse%
-\ {\isadigit{2}}\isanewline
-\ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}case\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ bounded{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}bdd{\isacharunderscore}{\kern0pt}above\ bounded{\isacharunderscore}{\kern0pt}disc{\isacharunderscore}{\kern0pt}{\isasymP}\isactrlsub {\isadigit{1}}\ bounded{\isacharunderscore}{\kern0pt}plus{\isacharunderscore}{\kern0pt}comp\ bounded{\isacharunderscore}{\kern0pt}stationary{\isacharunderscore}{\kern0pt}{\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}fin{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{case}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}{\isadigit{3}}\ n{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}case\ \isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ less{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}bfunD{\isacharbrackleft}{\kern0pt}OF\ {\isasymP}\isactrlsub X{\isacharunderscore}{\kern0pt}mono{\isacharbrackleft}{\kern0pt}OF\ assms{\isacharbrackright}{\kern0pt}{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ {\isasymP}\isactrlsub X{\isacharunderscore}{\kern0pt}const{\isacharbrackleft}{\kern0pt}symmetric{\isacharbrackright}{\kern0pt}\ blincomp{\isacharunderscore}{\kern0pt}scaleR{\isacharunderscore}{\kern0pt}right\ blinfun{\isachardot}{\kern0pt}scaleR{\isacharunderscore}{\kern0pt}left\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ mult{\isacharunderscore}{\kern0pt}left{\isacharunderscore}{\kern0pt}mono\ exI{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assms\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ U\isactrlsub b{\isacharunderscore}{\kern0pt}le{\isacharunderscore}{\kern0pt}{\isasymL}\isactrlsub b{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}U\isactrlsub b\ m\ v\ {\isasymle}\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ {\isacharcircum}{\kern0pt}{\isacharcircum}{\kern0pt}\ m{\isacharparenright}{\kern0pt}\ v{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}U\isactrlsub b\ m\ v\ s\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSqunion}d\ {\isasymin}\ D\isactrlsub R{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}L\ d{\isacharcircum}{\kern0pt}{\isacharcircum}{\kern0pt}\ m{\isacharparenright}{\kern0pt}\ v\ s{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{for}\ m\ v\ s\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ L{\isacharunderscore}{\kern0pt}iter\ U\isactrlsub b{\isachardot}{\kern0pt}rep{\isacharunderscore}{\kern0pt}eq\ {\isasymL}\isactrlsub b{\isachardot}{\kern0pt}rep{\isacharunderscore}{\kern0pt}eq\ U{\isacharunderscore}{\kern0pt}def\ {\isasymL}{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ L{\isacharunderscore}{\kern0pt}iter{\isacharunderscore}{\kern0pt}le{\isacharunderscore}{\kern0pt}{\isasymL}\isactrlsub b\ ex{\isacharunderscore}{\kern0pt}dec\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}fastforce\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ cSUP{\isacharunderscore}{\kern0pt}least{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ L{\isacharunderscore}{\kern0pt}iter{\isacharunderscore}{\kern0pt}le{\isacharunderscore}{\kern0pt}U\isactrlsub b{\isacharcolon}{\kern0pt}\ \isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}d\ {\isasymin}\ D\isactrlsub R{\isachardoublequoteclose}\ \isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}L\ d{\isacharcircum}{\kern0pt}{\isacharcircum}{\kern0pt}m{\isacharparenright}{\kern0pt}\ v\ {\isasymle}\ U\isactrlsub b\ m\ v{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ assms\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}fastforce\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ cSUP{\isacharunderscore}{\kern0pt}upper\ bounded{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}bdd{\isacharunderscore}{\kern0pt}above\isanewline
-\ \ \ \ \ \ simp{\isacharcolon}{\kern0pt}\ L{\isacharunderscore}{\kern0pt}iter\ U\isactrlsub b{\isachardot}{\kern0pt}rep{\isacharunderscore}{\kern0pt}eq\ U{\isacharunderscore}{\kern0pt}def\ bounded{\isacharunderscore}{\kern0pt}disc{\isacharunderscore}{\kern0pt}{\isasymP}\isactrlsub {\isadigit{1}}\ bounded{\isacharunderscore}{\kern0pt}plus{\isacharunderscore}{\kern0pt}comp\ bounded{\isacharunderscore}{\kern0pt}stationary{\isacharunderscore}{\kern0pt}{\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}fin{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ lim{\isacharunderscore}{\kern0pt}U\isactrlsub b{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}lim\ {\isacharparenleft}{\kern0pt}{\isasymlambda}n{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}U\isactrlsub b\ {\isacharparenleft}{\kern0pt}Suc\ m{\isacharparenright}{\kern0pt}\ {\isacharcircum}{\kern0pt}{\isacharcircum}{\kern0pt}\ n{\isacharparenright}{\kern0pt}\ v{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ le{\isacharunderscore}{\kern0pt}U{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt\ {\isasymle}\ U\isactrlsub b\ m\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt{\isachardoublequoteclose}\ \isakeyword{for}\ m\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{obtain}\isamarkupfalse%
-\ d\ \isakeyword{where}\ d{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymnu}{\isacharunderscore}{\kern0pt}improving\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt\ {\isacharparenleft}{\kern0pt}mk{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\ d{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ {\isachardoublequoteopen}d\ {\isasymin}\ D\isactrlsub D{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ ex{\isacharunderscore}{\kern0pt}improving{\isacharunderscore}{\kern0pt}det\ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}L\ {\isacharparenleft}{\kern0pt}mk{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\ d{\isacharparenright}{\kern0pt}\ {\isacharcircum}{\kern0pt}{\isacharcircum}{\kern0pt}\ m{\isacharparenright}{\kern0pt}\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}induction\ m{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}metis\ L{\isacharunderscore}{\kern0pt}{\isasymnu}{\isacharunderscore}{\kern0pt}fix{\isacharunderscore}{\kern0pt}iff\ {\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}opt\ {\isasymnu}{\isacharunderscore}{\kern0pt}improving{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}{\isasymL}\isactrlsub b\ d{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}\ funpow{\isacharunderscore}{\kern0pt}swap{\isadigit{1}}{\isacharparenright}{\kern0pt}{\isacharplus}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isacartoucheopen}d\ {\isasymin}\ D\isactrlsub D{\isacartoucheclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ order{\isachardot}{\kern0pt}trans{\isacharbrackleft}{\kern0pt}OF\ {\isacharunderscore}{\kern0pt}\ L{\isacharunderscore}{\kern0pt}iter{\isacharunderscore}{\kern0pt}le{\isacharunderscore}{\kern0pt}U\isactrlsub b{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}U\isactrlsub b\ m\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt\ {\isasymle}\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt{\isachardoublequoteclose}\ \isakeyword{for}\ m\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isasymL}{\isacharunderscore}{\kern0pt}inc{\isacharunderscore}{\kern0pt}le{\isacharunderscore}{\kern0pt}opt\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ order{\isachardot}{\kern0pt}trans{\isacharbrackleft}{\kern0pt}OF\ U\isactrlsub b{\isacharunderscore}{\kern0pt}le{\isacharunderscore}{\kern0pt}{\isasymL}\isactrlsub b{\isacharbrackright}{\kern0pt}\ simp{\isacharcolon}{\kern0pt}\ funpow{\isacharunderscore}{\kern0pt}swap{\isadigit{1}}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}U\isactrlsub b\ {\isacharparenleft}{\kern0pt}Suc\ m{\isacharparenright}{\kern0pt}\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt\ {\isacharequal}{\kern0pt}\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ le{\isacharunderscore}{\kern0pt}U\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ antisym{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}lim\ {\isacharparenleft}{\kern0pt}{\isasymlambda}n{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}U\isactrlsub b\ {\isacharparenleft}{\kern0pt}Suc\ m{\isacharparenright}{\kern0pt}\ {\isacharcircum}{\kern0pt}{\isacharcircum}{\kern0pt}n{\isacharparenright}{\kern0pt}\ v{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ U\isactrlsub b\ {\isacharparenleft}{\kern0pt}Suc\ m{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}lim\ {\isacharparenleft}{\kern0pt}{\isasymlambda}n{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}U\isactrlsub b\ {\isacharparenleft}{\kern0pt}Suc\ m{\isacharparenright}{\kern0pt}\ {\isacharcircum}{\kern0pt}{\isacharcircum}{\kern0pt}n{\isacharparenright}{\kern0pt}\ v{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ limI{\isacharbrackleft}{\kern0pt}OF\ U\isactrlsub b{\isacharunderscore}{\kern0pt}conv{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isacharbrackright}{\kern0pt}\ theI{\isacharprime}{\kern0pt}{\isacharbrackleft}{\kern0pt}OF\ U\isactrlsub b{\isacharunderscore}{\kern0pt}conv{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ U\isactrlsub b{\isacharunderscore}{\kern0pt}conv{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ U\isactrlsub b{\isacharunderscore}{\kern0pt}tendsto{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isasymlambda}n{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}U\isactrlsub b\ {\isacharparenleft}{\kern0pt}Suc\ m{\isacharparenright}{\kern0pt}\ {\isacharcircum}{\kern0pt}{\isacharcircum}{\kern0pt}\ n{\isacharparenright}{\kern0pt}\ v{\isacharparenright}{\kern0pt}\ {\isasymlonglonglongrightarrow}\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ lim{\isacharunderscore}{\kern0pt}U\isactrlsub b\ U\isactrlsub b{\isacharunderscore}{\kern0pt}convergent\ convergent{\isacharunderscore}{\kern0pt}LIMSEQ{\isacharunderscore}{\kern0pt}iff\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ metis%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ U\isactrlsub b{\isacharunderscore}{\kern0pt}fix{\isacharunderscore}{\kern0pt}unique{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}U\isactrlsub b\ {\isacharparenleft}{\kern0pt}Suc\ m{\isacharparenright}{\kern0pt}\ v\ {\isacharequal}{\kern0pt}\ v\ {\isasymlongleftrightarrow}\ v\ {\isacharequal}{\kern0pt}\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt{\isachardoublequoteclose}\ \isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ theI{\isacharprime}{\kern0pt}{\isacharbrackleft}{\kern0pt}OF\ U\isactrlsub b{\isacharunderscore}{\kern0pt}conv{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}{\isacharbrackright}{\kern0pt}\ U\isactrlsub b{\isacharunderscore}{\kern0pt}conv{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ LIMSEQ{\isacharunderscore}{\kern0pt}unique{\isacharbrackleft}{\kern0pt}OF\ U\isactrlsub b{\isacharunderscore}{\kern0pt}tendsto\ U\isactrlsub b{\isacharunderscore}{\kern0pt}conv{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isacharbrackleft}{\kern0pt}of\ m{\isacharbrackright}{\kern0pt}{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ dist{\isacharunderscore}{\kern0pt}U\isactrlsub b{\isacharunderscore}{\kern0pt}opt{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}dist\ {\isacharparenleft}{\kern0pt}U\isactrlsub b\ m\ v{\isacharparenright}{\kern0pt}\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt\ {\isasymle}\ l{\isacharcircum}{\kern0pt}m\ {\isacharasterisk}{\kern0pt}\ dist\ v\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}dist\ {\isacharparenleft}{\kern0pt}U\isactrlsub b\ m\ v{\isacharparenright}{\kern0pt}\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt\ {\isacharequal}{\kern0pt}\ dist\ {\isacharparenleft}{\kern0pt}U\isactrlsub b\ m\ v{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}U\isactrlsub b\ m\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ U\isactrlsub b{\isachardot}{\kern0pt}abs{\isacharunderscore}{\kern0pt}eq\ U\isactrlsub b{\isacharunderscore}{\kern0pt}fix{\isacharunderscore}{\kern0pt}unique\ U{\isacharunderscore}{\kern0pt}zero\ apply{\isacharunderscore}{\kern0pt}bfun{\isacharunderscore}{\kern0pt}inverse\ not{\isadigit{0}}{\isacharunderscore}{\kern0pt}implies{\isacharunderscore}{\kern0pt}Suc{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isasymle}\ l{\isacharcircum}{\kern0pt}m\ {\isacharasterisk}{\kern0pt}\ dist\ v\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}meson\ U\isactrlsub b{\isacharunderscore}{\kern0pt}contraction{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{finally}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\ \isacommand{{\isachardot}{\kern0pt}}\isamarkupfalse%
-\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Expressing a Single Step of Modified Policy Iteration%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\begin{isamarkuptext}%
-The function \isa{W} equals the value computed by the Modified Policy Iteration Algorithm
-  in a single iteration.
-  The right hand addend in the definition describes the advantage of using the optimal action for 
-  the first m steps.%
-\end{isamarkuptext}\isamarkuptrue%
-\isacommand{definition}\isamarkupfalse%
-\ {\isachardoublequoteopen}W\ d\ m\ v\ {\isacharequal}{\kern0pt}\ v\ {\isacharplus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSum}i\ {\isacharless}{\kern0pt}\ m{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isasymP}\isactrlsub {\isadigit{1}}\ d{\isacharparenright}{\kern0pt}{\isacharcircum}{\kern0pt}{\isacharcircum}{\kern0pt}i{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}B\isactrlsub b\ v{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ W{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}L{\isacharunderscore}{\kern0pt}iter{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}{\isasymnu}{\isacharunderscore}{\kern0pt}improving\ v\ d{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}W\ d\ m\ v\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}L\ d{\isacharcircum}{\kern0pt}{\isacharcircum}{\kern0pt}m{\isacharparenright}{\kern0pt}\ v{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isasymSum}i{\isacharless}{\kern0pt}m{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isasymP}\isactrlsub {\isadigit{1}}\ d{\isacharparenright}{\kern0pt}{\isacharcircum}{\kern0pt}{\isacharcircum}{\kern0pt}i{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ v{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSum}i{\isacharless}{\kern0pt}m{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isasymP}\isactrlsub {\isadigit{1}}\ d{\isacharparenright}{\kern0pt}{\isacharcircum}{\kern0pt}{\isacharcircum}{\kern0pt}i{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}L\ d\ v{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isasymnu}{\isacharunderscore}{\kern0pt}improving{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}{\isasymL}\isactrlsub b\ assms\ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}W\ d\ m\ v\ {\isacharequal}{\kern0pt}\ \ v\ {\isacharplus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}{\isasymSum}i{\isacharless}{\kern0pt}m{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isasymP}\isactrlsub {\isadigit{1}}\ d{\isacharparenright}{\kern0pt}{\isacharcircum}{\kern0pt}{\isacharcircum}{\kern0pt}i{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}L\ d\ v{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharminus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSum}i{\isacharless}{\kern0pt}m{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isasymP}\isactrlsub {\isadigit{1}}\ d{\isacharparenright}{\kern0pt}{\isacharcircum}{\kern0pt}{\isacharcircum}{\kern0pt}i{\isacharparenright}{\kern0pt}\ v{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ W{\isacharunderscore}{\kern0pt}def\ B\isactrlsub b{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}{\isasymL}\isactrlsub b\ blinfun{\isachardot}{\kern0pt}bilinear{\isacharunderscore}{\kern0pt}simps\ algebra{\isacharunderscore}{\kern0pt}simps\ {\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isacharequal}{\kern0pt}\ v\ {\isacharplus}{\kern0pt}\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}fin\ {\isacharparenleft}{\kern0pt}mk{\isacharunderscore}{\kern0pt}stationary\ d{\isacharparenright}{\kern0pt}\ m\ {\isacharplus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSum}i{\isacharless}{\kern0pt}m{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isasymP}\isactrlsub {\isadigit{1}}\ d{\isacharparenright}{\kern0pt}{\isacharcircum}{\kern0pt}{\isacharcircum}{\kern0pt}i{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isasymP}\isactrlsub {\isadigit{1}}\ d{\isacharparenright}{\kern0pt}\ v{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharminus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSum}i{\isacharless}{\kern0pt}m{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isasymP}\isactrlsub {\isadigit{1}}\ d{\isacharparenright}{\kern0pt}{\isacharcircum}{\kern0pt}{\isacharcircum}{\kern0pt}i{\isacharparenright}{\kern0pt}\ v{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ L{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}fin{\isacharunderscore}{\kern0pt}eq\ blinfun{\isachardot}{\kern0pt}bilinear{\isacharunderscore}{\kern0pt}simps\ blinfun{\isachardot}{\kern0pt}sum{\isacharunderscore}{\kern0pt}left\ scaleR{\isacharunderscore}{\kern0pt}right{\isachardot}{\kern0pt}sum{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isacharequal}{\kern0pt}\ v\ {\isacharplus}{\kern0pt}\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}fin\ {\isacharparenleft}{\kern0pt}mk{\isacharunderscore}{\kern0pt}stationary\ d{\isacharparenright}{\kern0pt}\ m\ {\isacharplus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSum}i{\isacharless}{\kern0pt}m{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isasymP}\isactrlsub {\isadigit{1}}\ d{\isacharparenright}{\kern0pt}{\isacharcircum}{\kern0pt}{\isacharcircum}{\kern0pt}Suc\ i{\isacharparenright}{\kern0pt}\ v{\isacharparenright}{\kern0pt}\ {\isacharminus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSum}i{\isacharless}{\kern0pt}m{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isasymP}\isactrlsub {\isadigit{1}}\ d{\isacharparenright}{\kern0pt}{\isacharcircum}{\kern0pt}{\isacharcircum}{\kern0pt}i{\isacharparenright}{\kern0pt}\ v{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp\ del{\isacharcolon}{\kern0pt}\ blinfunpow{\isachardot}{\kern0pt}simps\ simp{\isacharcolon}{\kern0pt}\ blinfunpow{\isacharunderscore}{\kern0pt}assoc{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isacharequal}{\kern0pt}\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}fin\ {\isacharparenleft}{\kern0pt}mk{\isacharunderscore}{\kern0pt}stationary\ d{\isacharparenright}{\kern0pt}\ m\ {\isacharplus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSum}i{\isacharless}{\kern0pt}Suc\ m{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isasymP}\isactrlsub {\isadigit{1}}\ d{\isacharparenright}{\kern0pt}{\isacharcircum}{\kern0pt}{\isacharcircum}{\kern0pt}\ i{\isacharparenright}{\kern0pt}\ v{\isacharparenright}{\kern0pt}\ \ {\isacharminus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSum}i{\isacharless}{\kern0pt}m{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isasymP}\isactrlsub {\isadigit{1}}\ d{\isacharparenright}{\kern0pt}{\isacharcircum}{\kern0pt}{\isacharcircum}{\kern0pt}\ i{\isacharparenright}{\kern0pt}\ v{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}subst\ sum{\isachardot}{\kern0pt}lessThan{\isacharunderscore}{\kern0pt}Suc{\isacharunderscore}{\kern0pt}shift{\isacharparenright}{\kern0pt}\ auto\isanewline
-\ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isacharequal}{\kern0pt}\ \ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}fin\ {\isacharparenleft}{\kern0pt}mk{\isacharunderscore}{\kern0pt}stationary\ d{\isacharparenright}{\kern0pt}\ m\ {\isacharplus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isasymP}\isactrlsub {\isadigit{1}}\ d{\isacharparenright}{\kern0pt}{\isacharcircum}{\kern0pt}{\isacharcircum}{\kern0pt}m{\isacharparenright}{\kern0pt}\ v{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ blinfun{\isachardot}{\kern0pt}sum{\isacharunderscore}{\kern0pt}left{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}L\ d\ {\isacharcircum}{\kern0pt}{\isacharcircum}{\kern0pt}\ m{\isacharparenright}{\kern0pt}\ v{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ L{\isacharunderscore}{\kern0pt}iter\ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{finally}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\ \isacommand{{\isachardot}{\kern0pt}}\isamarkupfalse%
-\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ W{\isacharunderscore}{\kern0pt}le{\isacharunderscore}{\kern0pt}U\isactrlsub b{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}v\ {\isasymle}\ u{\isachardoublequoteclose}\ {\isachardoublequoteopen}{\isasymnu}{\isacharunderscore}{\kern0pt}improving\ v\ d{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}W\ d\ m\ v\ {\isasymle}\ U\isactrlsub b\ m\ u{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}U\isactrlsub b\ m\ u\ {\isacharminus}{\kern0pt}\ W\ d\ m\ v\ {\isasymge}\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}fin\ {\isacharparenleft}{\kern0pt}mk{\isacharunderscore}{\kern0pt}stationary\ d{\isacharparenright}{\kern0pt}\ m\ {\isacharplus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isasymP}\isactrlsub {\isadigit{1}}\ d{\isacharparenright}{\kern0pt}\ {\isacharcircum}{\kern0pt}{\isacharcircum}{\kern0pt}\ m{\isacharparenright}{\kern0pt}\ u\ {\isacharminus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}fin\ {\isacharparenleft}{\kern0pt}mk{\isacharunderscore}{\kern0pt}stationary\ d{\isacharparenright}{\kern0pt}\ m\ {\isacharplus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isasymP}\isactrlsub {\isadigit{1}}\ d{\isacharparenright}{\kern0pt}{\isacharcircum}{\kern0pt}{\isacharcircum}{\kern0pt}m{\isacharparenright}{\kern0pt}\ v{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isasymnu}{\isacharunderscore}{\kern0pt}improving{\isacharunderscore}{\kern0pt}D{\isacharunderscore}{\kern0pt}MR\ assms{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}\ bounded{\isacharunderscore}{\kern0pt}stationary{\isacharunderscore}{\kern0pt}{\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}fin\ bounded{\isacharunderscore}{\kern0pt}disc{\isacharunderscore}{\kern0pt}{\isasymP}\isactrlsub {\isadigit{1}}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}fastforce\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ diff{\isacharunderscore}{\kern0pt}mono\ bounded{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}bdd{\isacharunderscore}{\kern0pt}above\ cSUP{\isacharunderscore}{\kern0pt}upper\ bounded{\isacharunderscore}{\kern0pt}plus{\isacharunderscore}{\kern0pt}comp\ simp{\isacharcolon}{\kern0pt}\ U\isactrlsub b{\isachardot}{\kern0pt}rep{\isacharunderscore}{\kern0pt}eq\ U{\isacharunderscore}{\kern0pt}def\ L{\isacharunderscore}{\kern0pt}iter\ W{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}L{\isacharunderscore}{\kern0pt}iter{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isacharasterisk}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}U\isactrlsub b\ m\ u\ {\isacharminus}{\kern0pt}\ W\ d\ m\ v\ {\isasymge}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isasymP}\isactrlsub {\isadigit{1}}\ d{\isacharparenright}{\kern0pt}\ {\isacharcircum}{\kern0pt}{\isacharcircum}{\kern0pt}\ m{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}u\ {\isacharminus}{\kern0pt}\ v{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ blinfun{\isachardot}{\kern0pt}diff{\isacharunderscore}{\kern0pt}right{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}W\ d\ m\ v\ {\isasymle}\ U\isactrlsub b\ m\ u{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ order{\isachardot}{\kern0pt}trans{\isacharbrackleft}{\kern0pt}OF\ {\isasymP}\isactrlsub {\isadigit{1}}{\isacharunderscore}{\kern0pt}n{\isacharunderscore}{\kern0pt}disc{\isacharunderscore}{\kern0pt}pos{\isacharbrackleft}{\kern0pt}unfolded\ blincomp{\isacharunderscore}{\kern0pt}scaleR{\isacharunderscore}{\kern0pt}right{\isacharbrackleft}{\kern0pt}symmetric{\isacharbrackright}{\kern0pt}{\isacharbrackright}{\kern0pt}\ {\isacharasterisk}{\kern0pt}{\isacharbrackright}{\kern0pt}\ assms\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ W{\isacharunderscore}{\kern0pt}ge{\isacharunderscore}{\kern0pt}{\isasymL}\isactrlsub b{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}v\ {\isasymle}\ u{\isachardoublequoteclose}\ {\isachardoublequoteopen}{\isadigit{0}}\ {\isasymle}\ B\isactrlsub b\ u{\isachardoublequoteclose}\ {\isachardoublequoteopen}{\isasymnu}{\isacharunderscore}{\kern0pt}improving\ u\ d{\isacharprime}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}{\isasymL}\isactrlsub b\ v\ {\isasymle}\ W\ d{\isacharprime}{\kern0pt}\ {\isacharparenleft}{\kern0pt}Suc\ m{\isacharparenright}{\kern0pt}\ u{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymL}\isactrlsub b\ v\ {\isasymle}\ u\ {\isacharplus}{\kern0pt}\ B\isactrlsub b\ u{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assms{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}mono\ B\isactrlsub b{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}{\isasymL}\isactrlsub b\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isasymle}\ W\ d{\isacharprime}{\kern0pt}\ {\isacharparenleft}{\kern0pt}Suc\ m{\isacharparenright}{\kern0pt}\ u{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ L{\isacharunderscore}{\kern0pt}mono\ {\isasymnu}{\isacharunderscore}{\kern0pt}improving{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}{\isasymL}\isactrlsub b\ assms{\isacharparenleft}{\kern0pt}{\isadigit{3}}{\isacharparenright}{\kern0pt}\ assms\ \isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}induction\ m{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ W{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}L{\isacharunderscore}{\kern0pt}iter\ B\isactrlsub b{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}{\isasymL}\isactrlsub b{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{finally}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\ \isacommand{{\isachardot}{\kern0pt}}\isamarkupfalse%
-\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ B\isactrlsub b{\isacharunderscore}{\kern0pt}le{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}{\isasymnu}{\isacharunderscore}{\kern0pt}improving\ v\ d{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}B\isactrlsub b\ v\ {\isacharplus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isasymP}\isactrlsub {\isadigit{1}}\ d\ {\isacharminus}{\kern0pt}\ id{\isacharunderscore}{\kern0pt}blinfun{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}u\ {\isacharminus}{\kern0pt}\ v{\isacharparenright}{\kern0pt}\ {\isasymle}\ B\isactrlsub b\ u{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}r{\isacharunderscore}{\kern0pt}dec\isactrlsub b\ d\ {\isacharplus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isasymP}\isactrlsub {\isadigit{1}}\ d\ {\isacharminus}{\kern0pt}\ id{\isacharunderscore}{\kern0pt}blinfun{\isacharparenright}{\kern0pt}\ u\ {\isasymle}\ B\isactrlsub b\ u{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ L{\isacharunderscore}{\kern0pt}def\ L{\isacharunderscore}{\kern0pt}le{\isacharunderscore}{\kern0pt}{\isasymL}\isactrlsub b\ assms\ \ \ \ \ \isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ B\isactrlsub b{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}{\isasymL}\isactrlsub b\ {\isasymL}\isactrlsub b{\isachardot}{\kern0pt}rep{\isacharunderscore}{\kern0pt}eq\ {\isasymL}{\isacharunderscore}{\kern0pt}def\ blinfun{\isachardot}{\kern0pt}bilinear{\isacharunderscore}{\kern0pt}simps{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}B\isactrlsub b\ v\ {\isacharequal}{\kern0pt}\ r{\isacharunderscore}{\kern0pt}dec\isactrlsub b\ d\ {\isacharplus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isasymP}\isactrlsub {\isadigit{1}}\ d\ {\isacharminus}{\kern0pt}\ id{\isacharunderscore}{\kern0pt}blinfun{\isacharparenright}{\kern0pt}\ v{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assms\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ B\isactrlsub b{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}{\isasymL}\isactrlsub b\ {\isasymnu}{\isacharunderscore}{\kern0pt}improving{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}{\isasymL}\isactrlsub b{\isacharbrackleft}{\kern0pt}of\ {\isacharunderscore}{\kern0pt}\ d{\isacharbrackright}{\kern0pt}\ L{\isacharunderscore}{\kern0pt}def\ blinfun{\isachardot}{\kern0pt}bilinear{\isacharunderscore}{\kern0pt}simps{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ blinfun{\isachardot}{\kern0pt}diff{\isacharunderscore}{\kern0pt}right{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ {\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}W{\isacharunderscore}{\kern0pt}ge{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}u\ {\isasymle}\ {\isasymL}\isactrlsub b\ u{\isachardoublequoteclose}\ {\isachardoublequoteopen}{\isasymnu}{\isacharunderscore}{\kern0pt}improving\ u\ d{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}W\ d\ m\ u\ {\isasymle}\ {\isasymL}\isactrlsub b\ {\isacharparenleft}{\kern0pt}W\ d\ m\ u{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isadigit{0}}\ {\isasymle}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isasymP}\isactrlsub {\isadigit{1}}\ d{\isacharparenright}{\kern0pt}\ {\isacharcircum}{\kern0pt}{\isacharcircum}{\kern0pt}\ m{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}B\isactrlsub b\ u{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ B\isactrlsub b{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}{\isasymL}\isactrlsub b\ {\isasymP}\isactrlsub {\isadigit{1}}{\isacharunderscore}{\kern0pt}n{\isacharunderscore}{\kern0pt}disc{\isacharunderscore}{\kern0pt}pos\ assms{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}\ blincomp{\isacharunderscore}{\kern0pt}scaleR{\isacharunderscore}{\kern0pt}right\ diff{\isacharunderscore}{\kern0pt}ge{\isacharunderscore}{\kern0pt}{\isadigit{0}}{\isacharunderscore}{\kern0pt}iff{\isacharunderscore}{\kern0pt}ge{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isasymP}\isactrlsub {\isadigit{1}}\ d{\isacharparenright}{\kern0pt}{\isacharcircum}{\kern0pt}{\isacharcircum}{\kern0pt}{\isadigit{0}}\ {\isacharplus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSum}i\ {\isacharless}{\kern0pt}\ m{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isasymP}\isactrlsub {\isadigit{1}}\ d{\isacharparenright}{\kern0pt}{\isacharcircum}{\kern0pt}{\isacharcircum}{\kern0pt}{\isacharparenleft}{\kern0pt}Suc\ i{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}B\isactrlsub b\ u{\isacharparenright}{\kern0pt}\ {\isacharminus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSum}i\ {\isacharless}{\kern0pt}\ m{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isasymP}\isactrlsub {\isadigit{1}}\ d{\isacharparenright}{\kern0pt}{\isacharcircum}{\kern0pt}{\isacharcircum}{\kern0pt}\ i{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}B\isactrlsub b\ u{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}subst\ sum{\isachardot}{\kern0pt}lessThan{\isacharunderscore}{\kern0pt}Suc{\isacharunderscore}{\kern0pt}shift{\isacharbrackleft}{\kern0pt}symmetric{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ blinfun{\isachardot}{\kern0pt}diff{\isacharunderscore}{\kern0pt}left{\isacharbrackleft}{\kern0pt}symmetric{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isacharequal}{\kern0pt}\ B\isactrlsub b\ u\ {\isacharplus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isasymP}\isactrlsub {\isadigit{1}}\ d\ {\isacharminus}{\kern0pt}\ id{\isacharunderscore}{\kern0pt}blinfun{\isacharparenright}{\kern0pt}\ o\isactrlsub L\ {\isacharparenleft}{\kern0pt}{\isasymSum}i\ {\isacharless}{\kern0pt}\ m{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isasymP}\isactrlsub {\isadigit{1}}\ d{\isacharparenright}{\kern0pt}{\isacharcircum}{\kern0pt}{\isacharcircum}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}B\isactrlsub b\ u{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ blinfun{\isachardot}{\kern0pt}bilinear{\isacharunderscore}{\kern0pt}simps\ sum{\isacharunderscore}{\kern0pt}subtractf{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isacharequal}{\kern0pt}\ B\isactrlsub b\ u\ {\isacharplus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isasymP}\isactrlsub {\isadigit{1}}\ d\ {\isacharminus}{\kern0pt}\ id{\isacharunderscore}{\kern0pt}blinfun{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}W\ d\ m\ u\ {\isacharminus}{\kern0pt}\ u{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ W{\isacharunderscore}{\kern0pt}def\ sum{\isachardot}{\kern0pt}lessThan{\isacharunderscore}{\kern0pt}Suc{\isacharbrackleft}{\kern0pt}unfolded\ lessThan{\isacharunderscore}{\kern0pt}Suc{\isacharunderscore}{\kern0pt}atMost{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isasymle}\ B\isactrlsub b\ {\isacharparenleft}{\kern0pt}W\ d\ m\ u{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ B\isactrlsub b{\isacharunderscore}{\kern0pt}le\ assms{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}\ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{finally}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isadigit{0}}\ {\isasymle}\ B\isactrlsub b\ {\isacharparenleft}{\kern0pt}W\ d\ m\ u{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isacommand{{\isachardot}{\kern0pt}}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\ \isacommand{using}\isamarkupfalse%
-\ B\isactrlsub b{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}{\isasymL}\isactrlsub b\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Computing the Bellman Operator over Multiple Steps%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{definition}\isamarkupfalse%
-\ L{\isacharunderscore}{\kern0pt}pow\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}s\ {\isasymRightarrow}\isactrlsub b\ real{\isacharparenright}{\kern0pt}\ {\isasymRightarrow}\ {\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}s\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a{\isacharparenright}{\kern0pt}\ {\isasymRightarrow}\ nat\ {\isasymRightarrow}\ {\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}s\ {\isasymRightarrow}\isactrlsub b\ real{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}L{\isacharunderscore}{\kern0pt}pow\ v\ d\ m\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}L\ {\isacharparenleft}{\kern0pt}mk{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\ d{\isacharparenright}{\kern0pt}\ {\isacharcircum}{\kern0pt}{\isacharcircum}{\kern0pt}\ Suc\ m{\isacharparenright}{\kern0pt}\ v{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ sum{\isacharunderscore}{\kern0pt}telescope{\isacharprime}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isasymSum}i{\isasymle}k{\isachardot}{\kern0pt}\ f\ {\isacharparenleft}{\kern0pt}Suc\ i{\isacharparenright}{\kern0pt}\ {\isacharminus}{\kern0pt}\ f\ i\ {\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ f\ {\isacharparenleft}{\kern0pt}Suc\ k{\isacharparenright}{\kern0pt}\ {\isacharminus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}f\ {\isadigit{0}}\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isacharprime}{\kern0pt}c\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ ab{\isacharunderscore}{\kern0pt}group{\isacharunderscore}{\kern0pt}add{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ sum{\isacharunderscore}{\kern0pt}telescope{\isacharbrackleft}{\kern0pt}of\ {\isachardoublequoteopen}{\isacharminus}{\kern0pt}f{\isachardoublequoteclose}\ k{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ L{\isacharunderscore}{\kern0pt}pow{\isacharunderscore}{\kern0pt}eq{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}{\isasymnu}{\isacharunderscore}{\kern0pt}improving\ v\ {\isacharparenleft}{\kern0pt}mk{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\ d{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}L{\isacharunderscore}{\kern0pt}pow\ v\ d\ m\ {\isacharequal}{\kern0pt}\ v\ {\isacharplus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSum}i\ {\isasymle}\ m{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isasymP}\isactrlsub {\isadigit{1}}\ {\isacharparenleft}{\kern0pt}mk{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\ d{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharcircum}{\kern0pt}{\isacharcircum}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}B\isactrlsub b\ v{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}d\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}mk{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\ d{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isasymSum}i\ {\isasymle}\ m{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isasymP}\isactrlsub {\isadigit{1}}\ {\isacharquery}{\kern0pt}d{\isacharparenright}{\kern0pt}{\isacharcircum}{\kern0pt}{\isacharcircum}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}B\isactrlsub b\ v{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSum}i\ {\isasymle}\ m{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isasymP}\isactrlsub {\isadigit{1}}\ {\isacharquery}{\kern0pt}d{\isacharparenright}{\kern0pt}{\isacharcircum}{\kern0pt}{\isacharcircum}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}L\ {\isacharquery}{\kern0pt}d\ v{\isacharparenright}{\kern0pt}\ {\isacharminus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSum}i\ {\isasymle}\ m{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isasymP}\isactrlsub {\isadigit{1}}\ {\isacharquery}{\kern0pt}d{\isacharparenright}{\kern0pt}{\isacharcircum}{\kern0pt}{\isacharcircum}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ v{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assms\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ B\isactrlsub b{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}{\isasymL}\isactrlsub b\ blinfun{\isachardot}{\kern0pt}bilinear{\isacharunderscore}{\kern0pt}simps\ {\isasymnu}{\isacharunderscore}{\kern0pt}improving{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}{\isasymL}\isactrlsub b{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSum}i\ {\isasymle}\ m{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isasymP}\isactrlsub {\isadigit{1}}\ {\isacharquery}{\kern0pt}d{\isacharparenright}{\kern0pt}{\isacharcircum}{\kern0pt}{\isacharcircum}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}r{\isacharunderscore}{\kern0pt}dec\isactrlsub b\ {\isacharquery}{\kern0pt}d{\isacharparenright}{\kern0pt}\ {\isacharplus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSum}i\ {\isasymle}\ m{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isasymP}\isactrlsub {\isadigit{1}}\ {\isacharquery}{\kern0pt}d{\isacharparenright}{\kern0pt}{\isacharcircum}{\kern0pt}{\isacharcircum}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isasymP}\isactrlsub {\isadigit{1}}\ {\isacharquery}{\kern0pt}d{\isacharparenright}{\kern0pt}\ v{\isacharparenright}{\kern0pt}\ {\isacharminus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSum}i\ {\isasymle}\ m{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isasymP}\isactrlsub {\isadigit{1}}\ {\isacharquery}{\kern0pt}d{\isacharparenright}{\kern0pt}{\isacharcircum}{\kern0pt}{\isacharcircum}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ v{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ L{\isacharunderscore}{\kern0pt}def\ blinfun{\isachardot}{\kern0pt}bilinear{\isacharunderscore}{\kern0pt}simps{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSum}i\ {\isasymle}\ m{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isasymP}\isactrlsub {\isadigit{1}}\ {\isacharquery}{\kern0pt}d{\isacharparenright}{\kern0pt}{\isacharcircum}{\kern0pt}{\isacharcircum}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}r{\isacharunderscore}{\kern0pt}dec\isactrlsub b\ {\isacharquery}{\kern0pt}d{\isacharparenright}{\kern0pt}\ {\isacharplus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSum}i\ {\isasymle}\ m{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isasymP}\isactrlsub {\isadigit{1}}\ {\isacharquery}{\kern0pt}d{\isacharparenright}{\kern0pt}{\isacharcircum}{\kern0pt}{\isacharcircum}{\kern0pt}Suc\ i{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ v\ {\isacharminus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSum}i\ {\isasymle}\ m{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isasymP}\isactrlsub {\isadigit{1}}\ {\isacharquery}{\kern0pt}d{\isacharparenright}{\kern0pt}{\isacharcircum}{\kern0pt}{\isacharcircum}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ v{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ blinfun{\isachardot}{\kern0pt}sum{\isacharunderscore}{\kern0pt}left\ blinfunpow{\isacharunderscore}{\kern0pt}assoc\ simp\ del{\isacharcolon}{\kern0pt}\ blinfunpow{\isachardot}{\kern0pt}simps{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSum}i\ {\isasymle}\ m{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isasymP}\isactrlsub {\isadigit{1}}\ {\isacharquery}{\kern0pt}d{\isacharparenright}{\kern0pt}{\isacharcircum}{\kern0pt}{\isacharcircum}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}r{\isacharunderscore}{\kern0pt}dec\isactrlsub b\ {\isacharquery}{\kern0pt}d{\isacharparenright}{\kern0pt}\ {\isacharplus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSum}i\ {\isasymle}\ m{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isasymP}\isactrlsub {\isadigit{1}}\ {\isacharquery}{\kern0pt}d{\isacharparenright}{\kern0pt}{\isacharcircum}{\kern0pt}{\isacharcircum}{\kern0pt}Suc\ i{\isacharparenright}{\kern0pt}\ {\isacharminus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isasymP}\isactrlsub {\isadigit{1}}\ {\isacharquery}{\kern0pt}d{\isacharparenright}{\kern0pt}{\isacharcircum}{\kern0pt}{\isacharcircum}{\kern0pt}i{\isacharparenright}{\kern0pt}\ v{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ blinfun{\isachardot}{\kern0pt}diff{\isacharunderscore}{\kern0pt}left\ sum{\isacharunderscore}{\kern0pt}subtractf{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSum}i\ {\isasymle}\ m{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isasymP}\isactrlsub {\isadigit{1}}\ {\isacharquery}{\kern0pt}d{\isacharparenright}{\kern0pt}{\isacharcircum}{\kern0pt}{\isacharcircum}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}r{\isacharunderscore}{\kern0pt}dec\isactrlsub b\ {\isacharquery}{\kern0pt}d{\isacharparenright}{\kern0pt}\ {\isacharplus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isasymP}\isactrlsub {\isadigit{1}}\ {\isacharquery}{\kern0pt}d{\isacharparenright}{\kern0pt}{\isacharcircum}{\kern0pt}{\isacharcircum}{\kern0pt}Suc\ m{\isacharparenright}{\kern0pt}\ v\ {\isacharminus}{\kern0pt}\ v{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}subst\ sum{\isacharunderscore}{\kern0pt}telescope{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ blinfun{\isachardot}{\kern0pt}bilinear{\isacharunderscore}{\kern0pt}simps{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{finally}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isasymSum}i\ {\isasymle}\ m{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isasymP}\isactrlsub {\isadigit{1}}\ {\isacharquery}{\kern0pt}d{\isacharparenright}{\kern0pt}{\isacharcircum}{\kern0pt}{\isacharcircum}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}B\isactrlsub b\ v{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSum}i\ {\isasymle}\ m{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isasymP}\isactrlsub {\isadigit{1}}\ {\isacharquery}{\kern0pt}d{\isacharparenright}{\kern0pt}{\isacharcircum}{\kern0pt}{\isacharcircum}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}r{\isacharunderscore}{\kern0pt}dec\isactrlsub b\ {\isacharquery}{\kern0pt}d{\isacharparenright}{\kern0pt}\ {\isacharplus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isasymP}\isactrlsub {\isadigit{1}}\ {\isacharquery}{\kern0pt}d{\isacharparenright}{\kern0pt}{\isacharcircum}{\kern0pt}{\isacharcircum}{\kern0pt}Suc\ m{\isacharparenright}{\kern0pt}\ v\ {\isacharminus}{\kern0pt}\ v{\isachardoublequoteclose}\ \isacommand{{\isachardot}{\kern0pt}}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}L{\isacharunderscore}{\kern0pt}pow\ v\ d\ m\ {\isacharequal}{\kern0pt}\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}fin\ {\isacharparenleft}{\kern0pt}mk{\isacharunderscore}{\kern0pt}stationary{\isacharunderscore}{\kern0pt}det\ d{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}Suc\ m{\isacharparenright}{\kern0pt}\ {\isacharplus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isasymP}\isactrlsub {\isadigit{1}}\ {\isacharquery}{\kern0pt}d{\isacharparenright}{\kern0pt}{\isacharcircum}{\kern0pt}{\isacharcircum}{\kern0pt}Suc\ m{\isacharparenright}{\kern0pt}\ v{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ only{\isacharcolon}{\kern0pt}\ L{\isacharunderscore}{\kern0pt}pow{\isacharunderscore}{\kern0pt}def\ L{\isacharunderscore}{\kern0pt}iter\ lessThan{\isacharunderscore}{\kern0pt}Suc{\isacharunderscore}{\kern0pt}atMost{\isacharbrackleft}{\kern0pt}symmetric{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}fin{\isacharunderscore}{\kern0pt}eq\ lessThan{\isacharunderscore}{\kern0pt}Suc{\isacharunderscore}{\kern0pt}atMost{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ L{\isacharunderscore}{\kern0pt}pow{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}W{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}d\ {\isasymin}\ D\isactrlsub D{\isachardoublequoteclose}\ \isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}L{\isacharunderscore}{\kern0pt}pow\ v\ {\isacharparenleft}{\kern0pt}policy{\isacharunderscore}{\kern0pt}improvement\ d\ v{\isacharparenright}{\kern0pt}\ m\ {\isacharequal}{\kern0pt}\ W\ {\isacharparenleft}{\kern0pt}mk{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\ {\isacharparenleft}{\kern0pt}policy{\isacharunderscore}{\kern0pt}improvement\ d\ v{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}Suc\ m{\isacharparenright}{\kern0pt}\ v{\isachardoublequoteclose}\ \isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ assms\ policy{\isacharunderscore}{\kern0pt}improvement{\isacharunderscore}{\kern0pt}improving\ \isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ W{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}L{\isacharunderscore}{\kern0pt}iter\ L{\isacharunderscore}{\kern0pt}pow{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ L{\isacharunderscore}{\kern0pt}pow{\isacharunderscore}{\kern0pt}{\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}mono{\isacharunderscore}{\kern0pt}inv{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}d\ {\isasymin}\ D\isactrlsub D{\isachardoublequoteclose}\ {\isachardoublequoteopen}v\ {\isasymle}\ {\isasymL}\isactrlsub b\ v{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}L{\isacharunderscore}{\kern0pt}pow\ v\ {\isacharparenleft}{\kern0pt}policy{\isacharunderscore}{\kern0pt}improvement\ d\ v{\isacharparenright}{\kern0pt}\ m\ {\isasymle}\ {\isasymL}\isactrlsub b\ {\isacharparenleft}{\kern0pt}L{\isacharunderscore}{\kern0pt}pow\ v\ {\isacharparenleft}{\kern0pt}policy{\isacharunderscore}{\kern0pt}improvement\ d\ v{\isacharparenright}{\kern0pt}\ m{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ assms\ L{\isacharunderscore}{\kern0pt}pow{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}W\ {\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}W{\isacharunderscore}{\kern0pt}ge\ policy{\isacharunderscore}{\kern0pt}improvement{\isacharunderscore}{\kern0pt}improving\ \isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{The Modified Policy Iteration Algorithm%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{context}\isamarkupfalse%
-\isanewline
-\ \ \isakeyword{fixes}\ d{\isadigit{0}}\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}s\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{fixes}\ v{\isadigit{0}}\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}s\ {\isasymRightarrow}\isactrlsub b\ real{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{fixes}\ m\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}nat\ {\isasymRightarrow}\ {\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}s\ {\isasymRightarrow}\isactrlsub b\ real{\isacharparenright}{\kern0pt}\ {\isasymRightarrow}\ nat{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{assumes}\ d{\isadigit{0}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}d{\isadigit{0}}\ {\isasymin}\ D\isactrlsub D{\isachardoublequoteclose}\isanewline
-\isakeyword{begin}%
-\begin{isamarkuptext}%
-We first define a function that executes the algorithm for n steps.%
-\end{isamarkuptext}\isamarkuptrue%
-\isacommand{fun}\isamarkupfalse%
-\ mpi\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}nat\ {\isasymRightarrow}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}s\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a{\isacharparenright}{\kern0pt}\ {\isasymtimes}\ {\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}s\ {\isasymRightarrow}\isactrlsub b\ real{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}mpi\ {\isadigit{0}}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}policy{\isacharunderscore}{\kern0pt}improvement\ d{\isadigit{0}}\ v{\isadigit{0}}{\isacharcomma}{\kern0pt}\ v{\isadigit{0}}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ {\isacharbar}{\kern0pt}\isanewline
-\ \ {\isachardoublequoteopen}mpi\ {\isacharparenleft}{\kern0pt}Suc\ n{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\isanewline
-\ \ {\isacharparenleft}{\kern0pt}let\ {\isacharparenleft}{\kern0pt}d{\isacharcomma}{\kern0pt}\ v{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ mpi\ n{\isacharsemicolon}{\kern0pt}\ v{\isacharprime}{\kern0pt}\ {\isacharequal}{\kern0pt}\ L{\isacharunderscore}{\kern0pt}pow\ v\ d\ {\isacharparenleft}{\kern0pt}m\ n\ v{\isacharparenright}{\kern0pt}\ in\isanewline
-\ \ {\isacharparenleft}{\kern0pt}policy{\isacharunderscore}{\kern0pt}improvement\ d\ v{\isacharprime}{\kern0pt}{\isacharcomma}{\kern0pt}\ v{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ {\isachardoublequoteopen}mpi{\isacharunderscore}{\kern0pt}val\ n\ {\isacharequal}{\kern0pt}\ snd\ {\isacharparenleft}{\kern0pt}mpi\ n{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ {\isachardoublequoteopen}mpi{\isacharunderscore}{\kern0pt}pol\ n\ {\isacharequal}{\kern0pt}\ fst\ {\isacharparenleft}{\kern0pt}mpi\ n{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ mpi{\isacharunderscore}{\kern0pt}pol{\isacharunderscore}{\kern0pt}zero{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}mpi{\isacharunderscore}{\kern0pt}pol\ {\isadigit{0}}\ {\isacharequal}{\kern0pt}\ policy{\isacharunderscore}{\kern0pt}improvement\ d{\isadigit{0}}\ v{\isadigit{0}}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ mpi{\isacharunderscore}{\kern0pt}pol{\isacharunderscore}{\kern0pt}def\ \isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ mpi{\isacharunderscore}{\kern0pt}pol{\isacharunderscore}{\kern0pt}Suc{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}mpi{\isacharunderscore}{\kern0pt}pol\ {\isacharparenleft}{\kern0pt}Suc\ n{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ policy{\isacharunderscore}{\kern0pt}improvement\ {\isacharparenleft}{\kern0pt}mpi{\isacharunderscore}{\kern0pt}pol\ n{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}mpi{\isacharunderscore}{\kern0pt}val\ {\isacharparenleft}{\kern0pt}Suc\ n{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ case{\isacharunderscore}{\kern0pt}prod{\isacharunderscore}{\kern0pt}beta{\isacharprime}{\kern0pt}\ Let{\isacharunderscore}{\kern0pt}def\ mpi{\isacharunderscore}{\kern0pt}pol{\isacharunderscore}{\kern0pt}def\ mpi{\isacharunderscore}{\kern0pt}val{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ mpi{\isacharunderscore}{\kern0pt}pol{\isacharunderscore}{\kern0pt}is{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}mpi{\isacharunderscore}{\kern0pt}pol\ n\ {\isasymin}\ D\isactrlsub D{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ mpi{\isacharunderscore}{\kern0pt}pol{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \isacommand{using}\isamarkupfalse%
-\ policy{\isacharunderscore}{\kern0pt}improvement{\isacharunderscore}{\kern0pt}is{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\ d{\isadigit{0}}\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}induction\ n{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ Let{\isacharunderscore}{\kern0pt}def\ split{\isacharcolon}{\kern0pt}\ prod{\isachardot}{\kern0pt}splits{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ {\isasymnu}{\isacharunderscore}{\kern0pt}improving{\isacharunderscore}{\kern0pt}mpi{\isacharunderscore}{\kern0pt}pol{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymnu}{\isacharunderscore}{\kern0pt}improving\ {\isacharparenleft}{\kern0pt}mpi{\isacharunderscore}{\kern0pt}val\ n{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}mk{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\ {\isacharparenleft}{\kern0pt}mpi{\isacharunderscore}{\kern0pt}pol\ n{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ d{\isadigit{0}}\ policy{\isacharunderscore}{\kern0pt}improvement{\isacharunderscore}{\kern0pt}improving\ mpi{\isacharunderscore}{\kern0pt}pol{\isacharunderscore}{\kern0pt}is{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\ mpi{\isacharunderscore}{\kern0pt}pol{\isacharunderscore}{\kern0pt}Suc\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}cases\ n{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ mpi{\isacharunderscore}{\kern0pt}pol{\isacharunderscore}{\kern0pt}def\ mpi{\isacharunderscore}{\kern0pt}val{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ mpi{\isacharunderscore}{\kern0pt}val{\isacharunderscore}{\kern0pt}zero{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}mpi{\isacharunderscore}{\kern0pt}val\ {\isadigit{0}}\ {\isacharequal}{\kern0pt}\ v{\isadigit{0}}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ mpi{\isacharunderscore}{\kern0pt}val{\isacharunderscore}{\kern0pt}def\ \isacommand{by}\isamarkupfalse%
-\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ mpi{\isacharunderscore}{\kern0pt}val{\isacharunderscore}{\kern0pt}Suc{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}mpi{\isacharunderscore}{\kern0pt}val\ {\isacharparenleft}{\kern0pt}Suc\ n{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ L{\isacharunderscore}{\kern0pt}pow\ {\isacharparenleft}{\kern0pt}mpi{\isacharunderscore}{\kern0pt}val\ n{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}mpi{\isacharunderscore}{\kern0pt}pol\ n{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}m\ n\ {\isacharparenleft}{\kern0pt}mpi{\isacharunderscore}{\kern0pt}val\ n{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ mpi{\isacharunderscore}{\kern0pt}val{\isacharunderscore}{\kern0pt}def\ mpi{\isacharunderscore}{\kern0pt}pol{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ case{\isacharunderscore}{\kern0pt}prod{\isacharunderscore}{\kern0pt}beta{\isacharprime}{\kern0pt}\ Let{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ mpi{\isacharunderscore}{\kern0pt}val{\isacharunderscore}{\kern0pt}eq{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}mpi{\isacharunderscore}{\kern0pt}val\ {\isacharparenleft}{\kern0pt}Suc\ n{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ \isanewline
-\ \ mpi{\isacharunderscore}{\kern0pt}val\ n\ {\isacharplus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSum}i\ {\isasymle}\ m\ n\ {\isacharparenleft}{\kern0pt}mpi{\isacharunderscore}{\kern0pt}val\ n{\isacharparenright}{\kern0pt}{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isasymP}\isactrlsub {\isadigit{1}}\ {\isacharparenleft}{\kern0pt}mk{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\ {\isacharparenleft}{\kern0pt}mpi{\isacharunderscore}{\kern0pt}pol\ n{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharcircum}{\kern0pt}{\isacharcircum}{\kern0pt}\ i{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}B\isactrlsub b\ {\isacharparenleft}{\kern0pt}mpi{\isacharunderscore}{\kern0pt}val\ n{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \ \isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ L{\isacharunderscore}{\kern0pt}pow{\isacharunderscore}{\kern0pt}eq{\isacharbrackleft}{\kern0pt}OF\ {\isasymnu}{\isacharunderscore}{\kern0pt}improving{\isacharunderscore}{\kern0pt}mpi{\isacharunderscore}{\kern0pt}pol{\isacharbrackright}{\kern0pt}\ mpi{\isacharunderscore}{\kern0pt}val{\isacharunderscore}{\kern0pt}Suc\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\begin{isamarkuptext}%
-Value Iteration is a special case of MPI where \isa{{\isasymforall}n\ v{\isachardot}{\kern0pt}\ m\ n\ v\ {\isacharequal}{\kern0pt}\ {\isadigit{0}}}.%
-\end{isamarkuptext}\isamarkuptrue%
-\isacommand{lemma}\isamarkupfalse%
-\ mpi{\isacharunderscore}{\kern0pt}includes{\isacharunderscore}{\kern0pt}value{\isacharunderscore}{\kern0pt}it{\isacharcolon}{\kern0pt}\ \isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}{\isasymforall}n\ v{\isachardot}{\kern0pt}\ m\ n\ v\ {\isacharequal}{\kern0pt}\ {\isadigit{0}}{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}mpi{\isacharunderscore}{\kern0pt}val\ {\isacharparenleft}{\kern0pt}Suc\ n{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isasymL}\isactrlsub b\ {\isacharparenleft}{\kern0pt}mpi{\isacharunderscore}{\kern0pt}val\ n{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ assms\ B\isactrlsub b{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}{\isasymL}\isactrlsub b\ mpi{\isacharunderscore}{\kern0pt}val{\isacharunderscore}{\kern0pt}eq\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Convergence Proof%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\begin{isamarkuptext}%
-We define the sequence \isa{w} as an upper bound for the values of MPI.%
-\end{isamarkuptext}\isamarkuptrue%
-\isacommand{fun}\isamarkupfalse%
-\ w\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}w\ {\isadigit{0}}\ {\isacharequal}{\kern0pt}\ v{\isadigit{0}}{\isachardoublequoteclose}\ {\isacharbar}{\kern0pt}\isanewline
-\ \ {\isachardoublequoteopen}w\ {\isacharparenleft}{\kern0pt}Suc\ n{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ U\isactrlsub b\ {\isacharparenleft}{\kern0pt}Suc\ {\isacharparenleft}{\kern0pt}m\ n\ {\isacharparenleft}{\kern0pt}mpi{\isacharunderscore}{\kern0pt}val\ n{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}w\ n{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ dist{\isacharunderscore}{\kern0pt}{\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}dist\ {\isacharparenleft}{\kern0pt}w\ {\isacharparenleft}{\kern0pt}Suc\ n{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt\ {\isasymle}\ l\ {\isacharasterisk}{\kern0pt}\ dist\ {\isacharparenleft}{\kern0pt}w\ n{\isacharparenright}{\kern0pt}\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}fastforce\ simp{\isacharcolon}{\kern0pt}\ algebra{\isacharunderscore}{\kern0pt}simps\ intro{\isacharcolon}{\kern0pt}\ order{\isachardot}{\kern0pt}trans{\isacharbrackleft}{\kern0pt}OF\ dist{\isacharunderscore}{\kern0pt}U\isactrlsub b{\isacharunderscore}{\kern0pt}opt{\isacharbrackright}{\kern0pt}\ mult{\isacharunderscore}{\kern0pt}left{\isacharunderscore}{\kern0pt}mono\ power{\isacharunderscore}{\kern0pt}le{\isacharunderscore}{\kern0pt}one\isanewline
-\ \ \ \ \ \ mult{\isacharunderscore}{\kern0pt}left{\isacharunderscore}{\kern0pt}le{\isacharunderscore}{\kern0pt}one{\isacharunderscore}{\kern0pt}le\ order{\isachardot}{\kern0pt}strict{\isacharunderscore}{\kern0pt}implies{\isacharunderscore}{\kern0pt}order{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ dist{\isacharunderscore}{\kern0pt}{\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt{\isacharunderscore}{\kern0pt}n{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}dist\ {\isacharparenleft}{\kern0pt}w\ n{\isacharparenright}{\kern0pt}\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt\ {\isasymle}\ l{\isacharcircum}{\kern0pt}n\ {\isacharasterisk}{\kern0pt}\ dist\ v{\isadigit{0}}\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}induction\ n{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}fastforce\ simp{\isacharcolon}{\kern0pt}\ algebra{\isacharunderscore}{\kern0pt}simps\ intro{\isacharcolon}{\kern0pt}\ order{\isachardot}{\kern0pt}trans{\isacharbrackleft}{\kern0pt}OF\ dist{\isacharunderscore}{\kern0pt}{\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt{\isacharbrackright}{\kern0pt}\ mult{\isacharunderscore}{\kern0pt}left{\isacharunderscore}{\kern0pt}mono{\isacharparenright}{\kern0pt}{\isacharplus}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ w{\isacharunderscore}{\kern0pt}conv{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}w\ {\isasymlonglonglongrightarrow}\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isasymlambda}n{\isachardot}{\kern0pt}\ l{\isacharcircum}{\kern0pt}n\ {\isacharasterisk}{\kern0pt}\ dist\ v{\isadigit{0}}\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt{\isacharparenright}{\kern0pt}\ {\isasymlonglonglongrightarrow}\ {\isadigit{0}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ LIMSEQ{\isacharunderscore}{\kern0pt}realpow{\isacharunderscore}{\kern0pt}zero\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}cases\ {\isachardoublequoteopen}v{\isadigit{0}}\ {\isacharequal}{\kern0pt}\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt{\isachardoublequoteclose}{\isacharparenright}{\kern0pt}\ auto\isanewline
-\ \ \isacommand{then}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}fastforce\ intro{\isacharcolon}{\kern0pt}\ metric{\isacharunderscore}{\kern0pt}LIMSEQ{\isacharunderscore}{\kern0pt}I\ order{\isachardot}{\kern0pt}strict{\isacharunderscore}{\kern0pt}trans{\isadigit{1}}{\isacharbrackleft}{\kern0pt}OF\ dist{\isacharunderscore}{\kern0pt}{\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt{\isacharunderscore}{\kern0pt}n{\isacharbrackright}{\kern0pt}\ simp{\isacharcolon}{\kern0pt}\ LIMSEQ{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\begin{isamarkuptext}%
-MPI converges monotonically to the optimal value from below. 
-  The iterates are sandwiched between \isa{{\isasymL}\isactrlsub b} from below and \isa{U\isactrlsub b} from above.%
-\end{isamarkuptext}\isamarkuptrue%
-\isacommand{theorem}\isamarkupfalse%
-\ mpi{\isacharunderscore}{\kern0pt}conv{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}v{\isadigit{0}}\ {\isasymle}\ {\isasymL}\isactrlsub b\ v{\isadigit{0}}{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}mpi{\isacharunderscore}{\kern0pt}val\ {\isasymlonglonglongrightarrow}\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt{\isachardoublequoteclose}\ \isakeyword{and}\ {\isachardoublequoteopen}{\isasymAnd}n{\isachardot}{\kern0pt}\ mpi{\isacharunderscore}{\kern0pt}val\ n\ {\isasymle}\ mpi{\isacharunderscore}{\kern0pt}val\ {\isacharparenleft}{\kern0pt}Suc\ n{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{define}\isamarkupfalse%
-\ y\ \isakeyword{where}\ {\isachardoublequoteopen}y\ n\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b{\isacharcircum}{\kern0pt}{\isacharcircum}{\kern0pt}n{\isacharparenright}{\kern0pt}\ v{\isadigit{0}}{\isachardoublequoteclose}\ \isakeyword{for}\ n\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ aux{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}mpi{\isacharunderscore}{\kern0pt}val\ n\ {\isasymle}\ {\isasymL}\isactrlsub b\ {\isacharparenleft}{\kern0pt}mpi{\isacharunderscore}{\kern0pt}val\ n{\isacharparenright}{\kern0pt}\ {\isasymand}\ mpi{\isacharunderscore}{\kern0pt}val\ n\ {\isasymle}\ mpi{\isacharunderscore}{\kern0pt}val\ {\isacharparenleft}{\kern0pt}Suc\ n{\isacharparenright}{\kern0pt}\ {\isasymand}\ y\ n\ {\isasymle}\ mpi{\isacharunderscore}{\kern0pt}val\ n\ {\isasymand}\ mpi{\isacharunderscore}{\kern0pt}val\ n\ {\isasymle}\ w\ n{\isachardoublequoteclose}\ \isakeyword{for}\ n\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}induction\ n{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{case}\isamarkupfalse%
-\ {\isadigit{0}}\isanewline
-\ \ \ \ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}case\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assms\ B\isactrlsub b{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}{\isasymL}\isactrlsub b\isanewline
-\ \ \ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ y{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ mpi{\isacharunderscore}{\kern0pt}val{\isacharunderscore}{\kern0pt}eq\ blinfun{\isachardot}{\kern0pt}sum{\isacharunderscore}{\kern0pt}left\ {\isasymP}\isactrlsub {\isadigit{1}}{\isacharunderscore}{\kern0pt}n{\isacharunderscore}{\kern0pt}disc{\isacharunderscore}{\kern0pt}pos\ blincomp{\isacharunderscore}{\kern0pt}scaleR{\isacharunderscore}{\kern0pt}right\ sum{\isacharunderscore}{\kern0pt}nonneg{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{case}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}Suc\ n{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ val{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}W{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}mpi{\isacharunderscore}{\kern0pt}val\ {\isacharparenleft}{\kern0pt}Suc\ n{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ W\ {\isacharparenleft}{\kern0pt}mk{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\ {\isacharparenleft}{\kern0pt}mpi{\isacharunderscore}{\kern0pt}pol\ n{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}Suc\ {\isacharparenleft}{\kern0pt}m\ n\ {\isacharparenleft}{\kern0pt}mpi{\isacharunderscore}{\kern0pt}val\ n{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}mpi{\isacharunderscore}{\kern0pt}val\ n{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isasymnu}{\isacharunderscore}{\kern0pt}improving{\isacharunderscore}{\kern0pt}mpi{\isacharunderscore}{\kern0pt}pol\ mpi{\isacharunderscore}{\kern0pt}val{\isacharunderscore}{\kern0pt}Suc\ W{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}L{\isacharunderscore}{\kern0pt}iter\ L{\isacharunderscore}{\kern0pt}pow{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ {\isacharasterisk}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}mpi{\isacharunderscore}{\kern0pt}val\ {\isacharparenleft}{\kern0pt}Suc\ n{\isacharparenright}{\kern0pt}\ {\isasymle}\ {\isasymL}\isactrlsub b\ {\isacharparenleft}{\kern0pt}mpi{\isacharunderscore}{\kern0pt}val\ {\isacharparenleft}{\kern0pt}Suc\ n{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ Suc{\isachardot}{\kern0pt}IH\ {\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}W{\isacharunderscore}{\kern0pt}ge\ {\isasymnu}{\isacharunderscore}{\kern0pt}improving{\isacharunderscore}{\kern0pt}mpi{\isacharunderscore}{\kern0pt}pol\ \isacommand{by}\isamarkupfalse%
-\ presburger\isanewline
-\ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}mpi{\isacharunderscore}{\kern0pt}val\ {\isacharparenleft}{\kern0pt}Suc\ n{\isacharparenright}{\kern0pt}\ {\isasymle}\ mpi{\isacharunderscore}{\kern0pt}val\ {\isacharparenleft}{\kern0pt}Suc\ {\isacharparenleft}{\kern0pt}Suc\ n{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isacharasterisk}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ B\isactrlsub b{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}{\isasymL}\isactrlsub b\ mpi{\isacharunderscore}{\kern0pt}val{\isacharunderscore}{\kern0pt}eq\ {\isasymP}\isactrlsub {\isadigit{1}}{\isacharunderscore}{\kern0pt}n{\isacharunderscore}{\kern0pt}disc{\isacharunderscore}{\kern0pt}pos\ blincomp{\isacharunderscore}{\kern0pt}scaleR{\isacharunderscore}{\kern0pt}right\ blinfun{\isachardot}{\kern0pt}sum{\isacharunderscore}{\kern0pt}left\ sum{\isacharunderscore}{\kern0pt}nonneg{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}mpi{\isacharunderscore}{\kern0pt}val\ {\isacharparenleft}{\kern0pt}Suc\ n{\isacharparenright}{\kern0pt}\ {\isasymle}\ w\ {\isacharparenleft}{\kern0pt}Suc\ n{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ Suc{\isachardot}{\kern0pt}IH\ {\isasymnu}{\isacharunderscore}{\kern0pt}improving{\isacharunderscore}{\kern0pt}mpi{\isacharunderscore}{\kern0pt}pol\ \isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ val{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}W\ intro{\isacharcolon}{\kern0pt}\ order{\isachardot}{\kern0pt}trans{\isacharbrackleft}{\kern0pt}OF\ {\isacharunderscore}{\kern0pt}\ W{\isacharunderscore}{\kern0pt}le{\isacharunderscore}{\kern0pt}U\isactrlsub b{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}y\ {\isacharparenleft}{\kern0pt}Suc\ n{\isacharparenright}{\kern0pt}\ {\isasymle}\ mpi{\isacharunderscore}{\kern0pt}val\ {\isacharparenleft}{\kern0pt}Suc\ n{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ Suc{\isachardot}{\kern0pt}IH\ {\isasymnu}{\isacharunderscore}{\kern0pt}improving{\isacharunderscore}{\kern0pt}mpi{\isacharunderscore}{\kern0pt}pol\ W{\isacharunderscore}{\kern0pt}ge{\isacharunderscore}{\kern0pt}{\isasymL}\isactrlsub b\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ y{\isacharunderscore}{\kern0pt}def\ B\isactrlsub b{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}{\isasymL}\isactrlsub b\ val{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}W{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}case\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}mpi{\isacharunderscore}{\kern0pt}val\ n\ {\isasymle}\ mpi{\isacharunderscore}{\kern0pt}val\ {\isacharparenleft}{\kern0pt}Suc\ n{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{for}\ n\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}y\ {\isasymlonglonglongrightarrow}\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}lim\ y{\isacharunderscore}{\kern0pt}def\ \isacommand{by}\isamarkupfalse%
-\ presburger\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}mpi{\isacharunderscore}{\kern0pt}val\ {\isasymlonglonglongrightarrow}\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ aux\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ intro{\isacharcolon}{\kern0pt}\ tendsto{\isacharunderscore}{\kern0pt}bfun{\isacharunderscore}{\kern0pt}sandwich{\isacharbrackleft}{\kern0pt}OF\ {\isacharunderscore}{\kern0pt}\ w{\isacharunderscore}{\kern0pt}conv{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{$\epsilon$-Optimality%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\begin{isamarkuptext}%
-This gives an upper bound on the error of MPI.%
-\end{isamarkuptext}\isamarkuptrue%
-\isacommand{lemma}\isamarkupfalse%
-\ mpi{\isacharunderscore}{\kern0pt}pol{\isacharunderscore}{\kern0pt}eps{\isacharunderscore}{\kern0pt}opt{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}{\isadigit{2}}\ {\isacharasterisk}{\kern0pt}\ l\ {\isacharasterisk}{\kern0pt}\ dist\ {\isacharparenleft}{\kern0pt}mpi{\isacharunderscore}{\kern0pt}val\ n{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ {\isacharparenleft}{\kern0pt}mpi{\isacharunderscore}{\kern0pt}val\ n{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharless}{\kern0pt}\ eps\ {\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}\ {\isacharminus}{\kern0pt}\ l{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ {\isachardoublequoteopen}eps\ {\isachargreater}{\kern0pt}\ {\isadigit{0}}{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}dist\ {\isacharparenleft}{\kern0pt}{\isasymnu}\isactrlsub b\ {\isacharparenleft}{\kern0pt}mk{\isacharunderscore}{\kern0pt}stationary{\isacharunderscore}{\kern0pt}det\ {\isacharparenleft}{\kern0pt}mpi{\isacharunderscore}{\kern0pt}pol\ n{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ {\isacharparenleft}{\kern0pt}mpi{\isacharunderscore}{\kern0pt}val\ n{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymle}\ eps\ {\isacharslash}{\kern0pt}\ {\isadigit{2}}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}p\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}mk{\isacharunderscore}{\kern0pt}stationary{\isacharunderscore}{\kern0pt}det\ {\isacharparenleft}{\kern0pt}mpi{\isacharunderscore}{\kern0pt}pol\ n{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}d\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}mk{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\ {\isacharparenleft}{\kern0pt}mpi{\isacharunderscore}{\kern0pt}pol\ n{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}v\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}mpi{\isacharunderscore}{\kern0pt}val\ n{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}dist\ {\isacharparenleft}{\kern0pt}{\isasymnu}\isactrlsub b\ {\isacharquery}{\kern0pt}p{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ {\isacharquery}{\kern0pt}v{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ dist\ {\isacharparenleft}{\kern0pt}L\ {\isacharquery}{\kern0pt}d\ {\isacharparenleft}{\kern0pt}{\isasymnu}\isactrlsub b\ {\isacharquery}{\kern0pt}p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ {\isacharquery}{\kern0pt}v{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ L{\isacharunderscore}{\kern0pt}{\isasymnu}{\isacharunderscore}{\kern0pt}fix\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ force\isanewline
-\ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isacharequal}{\kern0pt}\ dist\ {\isacharparenleft}{\kern0pt}L\ {\isacharquery}{\kern0pt}d\ {\isacharparenleft}{\kern0pt}{\isasymnu}\isactrlsub b\ {\isacharquery}{\kern0pt}p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}L\ {\isacharquery}{\kern0pt}d\ {\isacharquery}{\kern0pt}v{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isasymnu}{\isacharunderscore}{\kern0pt}improving{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}{\isasymL}\isactrlsub b\ {\isasymnu}{\isacharunderscore}{\kern0pt}improving{\isacharunderscore}{\kern0pt}mpi{\isacharunderscore}{\kern0pt}pol{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isasymle}\ dist\ {\isacharparenleft}{\kern0pt}L\ {\isacharquery}{\kern0pt}d\ {\isacharparenleft}{\kern0pt}{\isasymnu}\isactrlsub b\ {\isacharquery}{\kern0pt}p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}L\ {\isacharquery}{\kern0pt}d\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ {\isacharquery}{\kern0pt}v{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharplus}{\kern0pt}\ dist\ {\isacharparenleft}{\kern0pt}L\ {\isacharquery}{\kern0pt}d\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ {\isacharquery}{\kern0pt}v{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}L\ {\isacharquery}{\kern0pt}d\ {\isacharquery}{\kern0pt}v{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ dist{\isacharunderscore}{\kern0pt}triangle\ \isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isasymle}\ l\ {\isacharasterisk}{\kern0pt}\ dist\ {\isacharparenleft}{\kern0pt}{\isasymnu}\isactrlsub b\ {\isacharquery}{\kern0pt}p{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ {\isacharquery}{\kern0pt}v{\isacharparenright}{\kern0pt}\ {\isacharplus}{\kern0pt}\ dist\ {\isacharparenleft}{\kern0pt}L\ {\isacharquery}{\kern0pt}d\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ {\isacharquery}{\kern0pt}v{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}L\ {\isacharquery}{\kern0pt}d\ {\isacharquery}{\kern0pt}v{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ contraction{\isacharunderscore}{\kern0pt}L\ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isasymle}\ l\ {\isacharasterisk}{\kern0pt}\ dist\ {\isacharparenleft}{\kern0pt}{\isasymnu}\isactrlsub b\ {\isacharquery}{\kern0pt}p{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ {\isacharquery}{\kern0pt}v{\isacharparenright}{\kern0pt}\ {\isacharplus}{\kern0pt}\ l\ {\isacharasterisk}{\kern0pt}\ dist\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ {\isacharquery}{\kern0pt}v{\isacharparenright}{\kern0pt}\ {\isacharquery}{\kern0pt}v{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ contraction{\isacharunderscore}{\kern0pt}L\ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{finally}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}dist\ {\isacharparenleft}{\kern0pt}{\isasymnu}\isactrlsub b\ {\isacharquery}{\kern0pt}p{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ {\isacharquery}{\kern0pt}v{\isacharparenright}{\kern0pt}\ {\isasymle}\ l\ {\isacharasterisk}{\kern0pt}\ dist\ {\isacharparenleft}{\kern0pt}{\isasymnu}\isactrlsub b\ {\isacharquery}{\kern0pt}p{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ {\isacharquery}{\kern0pt}v{\isacharparenright}{\kern0pt}\ {\isacharplus}{\kern0pt}\ l\ {\isacharasterisk}{\kern0pt}\ dist\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ {\isacharquery}{\kern0pt}v{\isacharparenright}{\kern0pt}\ {\isacharquery}{\kern0pt}v{\isachardoublequoteclose}\isacommand{{\isachardot}{\kern0pt}}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isacharasterisk}{\kern0pt}{\isacharcolon}{\kern0pt}{\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharminus}{\kern0pt}l{\isacharparenright}{\kern0pt}\ {\isacharasterisk}{\kern0pt}\ dist\ {\isacharparenleft}{\kern0pt}{\isasymnu}\isactrlsub b\ {\isacharquery}{\kern0pt}p{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ {\isacharquery}{\kern0pt}v{\isacharparenright}{\kern0pt}\ {\isasymle}\ l\ {\isacharasterisk}{\kern0pt}\ dist\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ {\isacharquery}{\kern0pt}v{\isacharparenright}{\kern0pt}\ {\isacharquery}{\kern0pt}v{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ left{\isacharunderscore}{\kern0pt}diff{\isacharunderscore}{\kern0pt}distrib{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}cases\ {\isachardoublequoteopen}l\ {\isacharequal}{\kern0pt}\ {\isadigit{0}}{\isachardoublequoteclose}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{case}\isamarkupfalse%
-\ True\isanewline
-\ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assms\ {\isacharasterisk}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{case}\isamarkupfalse%
-\ False\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isacharasterisk}{\kern0pt}{\isacharasterisk}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}dist\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ {\isacharquery}{\kern0pt}v{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}mpi{\isacharunderscore}{\kern0pt}val\ n{\isacharparenright}{\kern0pt}\ {\isacharless}{\kern0pt}\ eps\ {\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}\ {\isacharminus}{\kern0pt}\ l{\isacharparenright}{\kern0pt}\ {\isacharslash}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isadigit{2}}\ {\isacharasterisk}{\kern0pt}\ l{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ False\ le{\isacharunderscore}{\kern0pt}neq{\isacharunderscore}{\kern0pt}trans{\isacharbrackleft}{\kern0pt}OF\ zero{\isacharunderscore}{\kern0pt}le{\isacharunderscore}{\kern0pt}disc\ False{\isacharbrackleft}{\kern0pt}symmetric{\isacharbrackright}{\kern0pt}{\isacharbrackright}{\kern0pt}\ assms\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ dist{\isacharunderscore}{\kern0pt}commute\ pos{\isacharunderscore}{\kern0pt}less{\isacharunderscore}{\kern0pt}divide{\isacharunderscore}{\kern0pt}eq\ Groups{\isachardot}{\kern0pt}mult{\isacharunderscore}{\kern0pt}ac{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}dist\ {\isacharparenleft}{\kern0pt}{\isasymnu}\isactrlsub b\ {\isacharquery}{\kern0pt}p{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ {\isacharquery}{\kern0pt}v{\isacharparenright}{\kern0pt}\ {\isasymle}\ {\isacharparenleft}{\kern0pt}l{\isacharslash}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharminus}{\kern0pt}l{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharasterisk}{\kern0pt}\ dist\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ {\isacharquery}{\kern0pt}v{\isacharparenright}{\kern0pt}\ {\isacharquery}{\kern0pt}v{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isacharasterisk}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ mult{\isachardot}{\kern0pt}commute\ pos{\isacharunderscore}{\kern0pt}le{\isacharunderscore}{\kern0pt}divide{\isacharunderscore}{\kern0pt}eq{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isasymle}\ {\isacharparenleft}{\kern0pt}l{\isacharslash}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharminus}{\kern0pt}l{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}eps\ {\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}\ {\isacharminus}{\kern0pt}\ l{\isacharparenright}{\kern0pt}\ {\isacharslash}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isadigit{2}}\ {\isacharasterisk}{\kern0pt}\ l{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isacharasterisk}{\kern0pt}{\isacharasterisk}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}fastforce\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ mult{\isacharunderscore}{\kern0pt}left{\isacharunderscore}{\kern0pt}mono\ simp{\isacharcolon}{\kern0pt}\ divide{\isacharunderscore}{\kern0pt}nonneg{\isacharunderscore}{\kern0pt}pos{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isacharequal}{\kern0pt}\ eps\ {\isacharslash}{\kern0pt}\ {\isadigit{2}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ False\ disc{\isacharunderscore}{\kern0pt}lt{\isacharunderscore}{\kern0pt}one\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ order{\isachardot}{\kern0pt}strict{\isacharunderscore}{\kern0pt}iff{\isacharunderscore}{\kern0pt}order{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{finally}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}dist\ {\isacharparenleft}{\kern0pt}{\isasymnu}\isactrlsub b\ {\isacharquery}{\kern0pt}p{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ {\isacharquery}{\kern0pt}v{\isacharparenright}{\kern0pt}\ {\isasymle}\ eps\ {\isacharslash}{\kern0pt}\ {\isadigit{2}}{\isachardoublequoteclose}\isacommand{{\isachardot}{\kern0pt}}\isamarkupfalse%
-\ \ \ \ \isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ mpi{\isacharunderscore}{\kern0pt}pol{\isacharunderscore}{\kern0pt}opt{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}{\isadigit{2}}\ {\isacharasterisk}{\kern0pt}\ l\ {\isacharasterisk}{\kern0pt}\ dist\ {\isacharparenleft}{\kern0pt}mpi{\isacharunderscore}{\kern0pt}val\ n{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ {\isacharparenleft}{\kern0pt}mpi{\isacharunderscore}{\kern0pt}val\ n{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharless}{\kern0pt}\ eps\ {\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}\ {\isacharminus}{\kern0pt}\ l{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ {\isachardoublequoteopen}eps\ {\isachargreater}{\kern0pt}\ {\isadigit{0}}{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}dist\ {\isacharparenleft}{\kern0pt}{\isasymnu}\isactrlsub b\ {\isacharparenleft}{\kern0pt}mk{\isacharunderscore}{\kern0pt}stationary{\isacharunderscore}{\kern0pt}det\ {\isacharparenleft}{\kern0pt}mpi{\isacharunderscore}{\kern0pt}pol\ n{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt{\isacharparenright}{\kern0pt}\ {\isacharless}{\kern0pt}\ eps{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}dist\ {\isacharparenleft}{\kern0pt}{\isasymnu}\isactrlsub b\ {\isacharparenleft}{\kern0pt}mk{\isacharunderscore}{\kern0pt}stationary{\isacharunderscore}{\kern0pt}det\ {\isacharparenleft}{\kern0pt}mpi{\isacharunderscore}{\kern0pt}pol\ n{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt{\isacharparenright}{\kern0pt}\ {\isasymle}\ eps{\isacharslash}{\kern0pt}{\isadigit{2}}\ {\isacharplus}{\kern0pt}\ dist\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ {\isacharparenleft}{\kern0pt}mpi{\isacharunderscore}{\kern0pt}val\ n{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ mpi{\isacharunderscore}{\kern0pt}pol{\isacharunderscore}{\kern0pt}eps{\isacharunderscore}{\kern0pt}opt{\isacharbrackleft}{\kern0pt}OF\ assms{\isacharbrackright}{\kern0pt}\ dist{\isacharunderscore}{\kern0pt}commute\ dist{\isacharunderscore}{\kern0pt}triangle{\isacharunderscore}{\kern0pt}le\ add{\isacharunderscore}{\kern0pt}right{\isacharunderscore}{\kern0pt}mono{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ dist{\isacharunderscore}{\kern0pt}{\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}opt{\isacharunderscore}{\kern0pt}eps\ assms\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ fastforce\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ mpi{\isacharunderscore}{\kern0pt}val{\isacharunderscore}{\kern0pt}term{\isacharunderscore}{\kern0pt}ex{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}v{\isadigit{0}}\ {\isasymle}\ {\isasymL}\isactrlsub b\ v{\isadigit{0}}{\isachardoublequoteclose}\ {\isachardoublequoteopen}eps\ {\isachargreater}{\kern0pt}\ {\isadigit{0}}{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}{\isasymexists}n{\isachardot}{\kern0pt}\ {\isadigit{2}}\ {\isacharasterisk}{\kern0pt}\ l\ {\isacharasterisk}{\kern0pt}\ dist\ {\isacharparenleft}{\kern0pt}mpi{\isacharunderscore}{\kern0pt}val\ n{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ {\isacharparenleft}{\kern0pt}mpi{\isacharunderscore}{\kern0pt}val\ n{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharless}{\kern0pt}\ eps\ {\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}\ {\isacharminus}{\kern0pt}\ l{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{note}\isamarkupfalse%
-\ dist{\isacharunderscore}{\kern0pt}{\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}lt{\isacharunderscore}{\kern0pt}dist{\isacharunderscore}{\kern0pt}opt\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isasymlambda}n{\isachardot}{\kern0pt}\ dist\ {\isacharparenleft}{\kern0pt}mpi{\isacharunderscore}{\kern0pt}val\ n{\isacharparenright}{\kern0pt}\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt{\isacharparenright}{\kern0pt}\ {\isasymlonglonglongrightarrow}\ {\isadigit{0}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ mpi{\isacharunderscore}{\kern0pt}conv{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}{\isacharbrackleft}{\kern0pt}OF\ assms{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}{\isacharbrackright}{\kern0pt}\ tendsto{\isacharunderscore}{\kern0pt}dist{\isacharunderscore}{\kern0pt}iff\ \isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isasymlambda}n{\isachardot}{\kern0pt}\ dist\ {\isacharparenleft}{\kern0pt}mpi{\isacharunderscore}{\kern0pt}val\ n{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ {\isacharparenleft}{\kern0pt}mpi{\isacharunderscore}{\kern0pt}val\ n{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymlonglonglongrightarrow}\ {\isadigit{0}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ dist{\isacharunderscore}{\kern0pt}{\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}lt{\isacharunderscore}{\kern0pt}dist{\isacharunderscore}{\kern0pt}opt\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ metric{\isacharunderscore}{\kern0pt}LIMSEQ{\isacharunderscore}{\kern0pt}I\ intro{\isacharcolon}{\kern0pt}\ tendsto{\isacharunderscore}{\kern0pt}sandwich{\isacharbrackleft}{\kern0pt}of\ {\isachardoublequoteopen}{\isasymlambda}{\isacharunderscore}{\kern0pt}{\isachardot}{\kern0pt}\ {\isadigit{0}}{\isachardoublequoteclose}\ {\isacharunderscore}{\kern0pt}\ {\isacharunderscore}{\kern0pt}\ {\isachardoublequoteopen}{\isasymlambda}n{\isachardot}{\kern0pt}\ {\isadigit{2}}\ {\isacharasterisk}{\kern0pt}\ dist\ {\isacharparenleft}{\kern0pt}mpi{\isacharunderscore}{\kern0pt}val\ n{\isacharparenright}{\kern0pt}\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt{\isachardoublequoteclose}{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymforall}e\ {\isachargreater}{\kern0pt}{\isadigit{0}}{\isachardot}{\kern0pt}\ {\isasymexists}n{\isachardot}{\kern0pt}\ dist\ {\isacharparenleft}{\kern0pt}mpi{\isacharunderscore}{\kern0pt}val\ n{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ {\isacharparenleft}{\kern0pt}mpi{\isacharunderscore}{\kern0pt}val\ n{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharless}{\kern0pt}\ e{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}fastforce\ dest{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ metric{\isacharunderscore}{\kern0pt}LIMSEQ{\isacharunderscore}{\kern0pt}D{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}l\ {\isasymnoteq}\ {\isadigit{0}}\ {\isasymLongrightarrow}\ {\isasymexists}n{\isachardot}{\kern0pt}\ dist\ {\isacharparenleft}{\kern0pt}mpi{\isacharunderscore}{\kern0pt}val\ n{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ {\isacharparenleft}{\kern0pt}mpi{\isacharunderscore}{\kern0pt}val\ n{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharless}{\kern0pt}\ eps\ {\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}\ {\isacharminus}{\kern0pt}\ l{\isacharparenright}{\kern0pt}\ {\isacharslash}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isadigit{2}}\ {\isacharasterisk}{\kern0pt}\ l{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ assms\ order{\isachardot}{\kern0pt}not{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}order{\isacharunderscore}{\kern0pt}implies{\isacharunderscore}{\kern0pt}strict{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymexists}n{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isadigit{2}}\ {\isacharasterisk}{\kern0pt}\ l{\isacharparenright}{\kern0pt}\ {\isacharasterisk}{\kern0pt}\ dist\ {\isacharparenleft}{\kern0pt}mpi{\isacharunderscore}{\kern0pt}val\ n{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ {\isacharparenleft}{\kern0pt}mpi{\isacharunderscore}{\kern0pt}val\ n{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharless}{\kern0pt}\ eps\ {\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}\ {\isacharminus}{\kern0pt}\ l{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assms\ le{\isacharunderscore}{\kern0pt}neq{\isacharunderscore}{\kern0pt}trans{\isacharbrackleft}{\kern0pt}OF\ zero{\isacharunderscore}{\kern0pt}le{\isacharunderscore}{\kern0pt}disc{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}cases\ {\isachardoublequoteopen}l\ {\isacharequal}{\kern0pt}\ {\isadigit{0}}{\isachardoublequoteclose}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ mult{\isachardot}{\kern0pt}commute\ pos{\isacharunderscore}{\kern0pt}less{\isacharunderscore}{\kern0pt}divide{\isacharunderscore}{\kern0pt}eq{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isacommand{end}\isamarkupfalse%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Unbounded MPI%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{context}\isamarkupfalse%
-\isanewline
-\ \ \isakeyword{fixes}\ eps\ {\isasymdelta}\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ real\ \isakeyword{and}\ M\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ nat\isanewline
-\isakeyword{begin}\isanewline
-\isanewline
-\isacommand{function}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}domintros{\isacharparenright}{\kern0pt}\ mpi{\isacharunderscore}{\kern0pt}algo\ \isakeyword{where}\ {\isachardoublequoteopen}mpi{\isacharunderscore}{\kern0pt}algo\ d\ v\ m\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}\isanewline
-\ \ if\ {\isadigit{2}}\ {\isacharasterisk}{\kern0pt}\ l\ {\isacharasterisk}{\kern0pt}\ dist\ v\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ v{\isacharparenright}{\kern0pt}\ {\isacharless}{\kern0pt}\ \ eps\ {\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}\ {\isacharminus}{\kern0pt}\ l{\isacharparenright}{\kern0pt}\isanewline
-\ \ then\ {\isacharparenleft}{\kern0pt}policy{\isacharunderscore}{\kern0pt}improvement\ d\ v{\isacharcomma}{\kern0pt}\ v{\isacharparenright}{\kern0pt}\isanewline
-\ \ else\ mpi{\isacharunderscore}{\kern0pt}algo\ {\isacharparenleft}{\kern0pt}policy{\isacharunderscore}{\kern0pt}improvement\ d\ v{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}L{\isacharunderscore}{\kern0pt}pow\ v\ {\isacharparenleft}{\kern0pt}policy{\isacharunderscore}{\kern0pt}improvement\ d\ v{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}m\ {\isadigit{0}}\ v{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymlambda}n{\isachardot}{\kern0pt}\ m\ {\isacharparenleft}{\kern0pt}Suc\ n{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\begin{isamarkuptext}%
-We define a tailrecursive version of \isa{mpi} which more closely resembles \isa{mpi{\isacharunderscore}{\kern0pt}algo}.%
-\end{isamarkuptext}\isamarkuptrue%
-\isacommand{fun}\isamarkupfalse%
-\ mpi{\isacharprime}{\kern0pt}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}mpi{\isacharprime}{\kern0pt}\ d\ v\ {\isadigit{0}}\ m\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}policy{\isacharunderscore}{\kern0pt}improvement\ d\ v{\isacharcomma}{\kern0pt}\ v{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ {\isacharbar}{\kern0pt}\isanewline
-\ \ {\isachardoublequoteopen}mpi{\isacharprime}{\kern0pt}\ d\ v\ {\isacharparenleft}{\kern0pt}Suc\ n{\isacharparenright}{\kern0pt}\ m\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}\isanewline
-\ \ let\ d{\isacharprime}{\kern0pt}\ {\isacharequal}{\kern0pt}\ policy{\isacharunderscore}{\kern0pt}improvement\ d\ v{\isacharsemicolon}{\kern0pt}\ v{\isacharprime}{\kern0pt}\ {\isacharequal}{\kern0pt}\ L{\isacharunderscore}{\kern0pt}pow\ v\ d{\isacharprime}{\kern0pt}\ {\isacharparenleft}{\kern0pt}m\ {\isadigit{0}}\ v{\isacharparenright}{\kern0pt}\ in\ mpi{\isacharprime}{\kern0pt}\ d{\isacharprime}{\kern0pt}\ v{\isacharprime}{\kern0pt}\ n\ {\isacharparenleft}{\kern0pt}{\isasymlambda}n{\isachardot}{\kern0pt}\ m\ {\isacharparenleft}{\kern0pt}Suc\ n{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ mpi{\isacharunderscore}{\kern0pt}Suc{\isacharprime}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}d\ {\isasymin}\ D\isactrlsub D{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}mpi\ d\ v\ m\ {\isacharparenleft}{\kern0pt}Suc\ n{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ mpi\ {\isacharparenleft}{\kern0pt}policy{\isacharunderscore}{\kern0pt}improvement\ d\ v{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}L{\isacharunderscore}{\kern0pt}pow\ v\ {\isacharparenleft}{\kern0pt}policy{\isacharunderscore}{\kern0pt}improvement\ d\ v{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}m\ {\isadigit{0}}\ v{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymlambda}a{\isachardot}{\kern0pt}\ m\ {\isacharparenleft}{\kern0pt}Suc\ a{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ n{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ assms\ policy{\isacharunderscore}{\kern0pt}improvement{\isacharunderscore}{\kern0pt}is{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}induction\ n\ rule{\isacharcolon}{\kern0pt}\ nat{\isachardot}{\kern0pt}induct{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ Let{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ \isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}d\ {\isasymin}\ D\isactrlsub D{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}mpi\ d\ v\ m\ n\ {\isacharequal}{\kern0pt}\ mpi{\isacharprime}{\kern0pt}\ d\ v\ n\ m{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ assms\isanewline
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}induction\ n\ arbitrary{\isacharcolon}{\kern0pt}\ d\ v\ m\ rule{\isacharcolon}{\kern0pt}\ nat{\isachardot}{\kern0pt}induct{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{case}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}Suc\ nat{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}case\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ policy{\isacharunderscore}{\kern0pt}improvement{\isacharunderscore}{\kern0pt}is{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ Let{\isacharunderscore}{\kern0pt}def\ mpi{\isacharunderscore}{\kern0pt}Suc{\isacharprime}{\kern0pt}{\isacharbrackleft}{\kern0pt}OF\ Suc{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isacharbrackright}{\kern0pt}\ Suc{\isachardot}{\kern0pt}IH{\isacharbrackleft}{\kern0pt}symmetric{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ termination{\isacharunderscore}{\kern0pt}mpi{\isacharunderscore}{\kern0pt}algo{\isacharcolon}{\kern0pt}\ \isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}eps\ {\isachargreater}{\kern0pt}\ {\isadigit{0}}{\isachardoublequoteclose}\ {\isachardoublequoteopen}d\ {\isasymin}\ D\isactrlsub D{\isachardoublequoteclose}\ {\isachardoublequoteopen}v\ {\isasymle}\ {\isasymL}\isactrlsub b\ v{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}mpi{\isacharunderscore}{\kern0pt}algo{\isacharunderscore}{\kern0pt}dom\ {\isacharparenleft}{\kern0pt}d{\isacharcomma}{\kern0pt}\ v{\isacharcomma}{\kern0pt}\ m{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{define}\isamarkupfalse%
-\ n\ \isakeyword{where}\ {\isachardoublequoteopen}n\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}LEAST\ n{\isachardot}{\kern0pt}\ {\isadigit{2}}\ {\isacharasterisk}{\kern0pt}\ l\ {\isacharasterisk}{\kern0pt}\ dist\ {\isacharparenleft}{\kern0pt}mpi{\isacharunderscore}{\kern0pt}val\ d\ v\ m\ n{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ {\isacharparenleft}{\kern0pt}mpi{\isacharunderscore}{\kern0pt}val\ d\ v\ m\ n{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharless}{\kern0pt}\ eps\ {\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}\ {\isacharminus}{\kern0pt}\ l{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ {\isacharparenleft}{\kern0pt}\isakeyword{is}\ {\isachardoublequoteopen}n\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}LEAST\ n{\isachardot}{\kern0pt}\ {\isacharquery}{\kern0pt}P\ d\ v\ m\ n{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ least{\isadigit{0}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymexists}n{\isachardot}{\kern0pt}\ P\ n\ {\isasymLongrightarrow}\ {\isacharparenleft}{\kern0pt}LEAST\ n{\isachardot}{\kern0pt}\ P\ n{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isadigit{0}}\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ nat{\isacharparenright}{\kern0pt}\ {\isasymLongrightarrow}\ P\ {\isadigit{0}}{\isachardoublequoteclose}\ \ \isakeyword{for}\ P\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ LeastI{\isacharunderscore}{\kern0pt}ex{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ n{\isacharunderscore}{\kern0pt}def\ assms\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}induction\ n\ arbitrary{\isacharcolon}{\kern0pt}\ v\ d\ m{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{case}\isamarkupfalse%
-\ {\isadigit{0}}\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isadigit{2}}\ {\isacharasterisk}{\kern0pt}\ l\ {\isacharasterisk}{\kern0pt}\ dist\ {\isacharparenleft}{\kern0pt}mpi{\isacharunderscore}{\kern0pt}val\ d\ v\ m\ {\isadigit{0}}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ {\isacharparenleft}{\kern0pt}mpi{\isacharunderscore}{\kern0pt}val\ d\ v\ m\ {\isadigit{0}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharless}{\kern0pt}\ eps\ {\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}\ {\isacharminus}{\kern0pt}\ l{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ least{\isadigit{0}}\ mpi{\isacharunderscore}{\kern0pt}val{\isacharunderscore}{\kern0pt}term{\isacharunderscore}{\kern0pt}ex\ {\isadigit{0}}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharcomma}{\kern0pt}\ lifting{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}case\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isadigit{0}}\ mpi{\isacharunderscore}{\kern0pt}algo{\isachardot}{\kern0pt}domintros\ mpi{\isacharunderscore}{\kern0pt}val{\isacharunderscore}{\kern0pt}zero\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharcomma}{\kern0pt}\ opaque{\isacharunderscore}{\kern0pt}lifting{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{case}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}Suc\ n\ v\ d\ m{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}d\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}policy{\isacharunderscore}{\kern0pt}improvement\ d\ v{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}Suc\ n\ {\isacharequal}{\kern0pt}\ Suc\ {\isacharparenleft}{\kern0pt}LEAST\ n{\isachardot}{\kern0pt}\ {\isadigit{2}}\ {\isacharasterisk}{\kern0pt}\ l\ {\isacharasterisk}{\kern0pt}\ dist\ {\isacharparenleft}{\kern0pt}mpi{\isacharunderscore}{\kern0pt}val\ d\ v\ m\ {\isacharparenleft}{\kern0pt}Suc\ n{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ {\isacharparenleft}{\kern0pt}mpi{\isacharunderscore}{\kern0pt}val\ d\ v\ m\ {\isacharparenleft}{\kern0pt}Suc\ n{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharless}{\kern0pt}\ eps\ {\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}\ {\isacharminus}{\kern0pt}\ l{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ mpi{\isacharunderscore}{\kern0pt}val{\isacharunderscore}{\kern0pt}term{\isacharunderscore}{\kern0pt}ex{\isacharbrackleft}{\kern0pt}OF\ Suc{\isachardot}{\kern0pt}prems{\isacharparenleft}{\kern0pt}{\isadigit{3}}{\isacharparenright}{\kern0pt}\ {\isacartoucheopen}v\ {\isasymle}\ {\isasymL}\isactrlsub b\ v{\isacartoucheclose}\ {\isacartoucheopen}{\isadigit{0}}\ {\isacharless}{\kern0pt}\ eps{\isacartoucheclose}{\isacharcomma}{\kern0pt}\ of\ m{\isacharbrackright}{\kern0pt}\ Suc{\isachardot}{\kern0pt}prems\ \isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}subst\ Nat{\isachardot}{\kern0pt}Least{\isacharunderscore}{\kern0pt}Suc{\isacharbrackleft}{\kern0pt}symmetric{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}auto\ intro{\isacharcolon}{\kern0pt}\ LeastI{\isacharunderscore}{\kern0pt}ex{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}n\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}LEAST\ n{\isachardot}{\kern0pt}\ {\isadigit{2}}\ {\isacharasterisk}{\kern0pt}\ l\ {\isacharasterisk}{\kern0pt}\ dist\ {\isacharparenleft}{\kern0pt}mpi{\isacharunderscore}{\kern0pt}val\ d\ v\ m\ {\isacharparenleft}{\kern0pt}Suc\ n{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ {\isacharparenleft}{\kern0pt}mpi{\isacharunderscore}{\kern0pt}val\ d\ v\ m\ {\isacharparenleft}{\kern0pt}Suc\ n{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharless}{\kern0pt}\ eps\ {\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}\ {\isacharminus}{\kern0pt}\ l{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ n{\isacharunderscore}{\kern0pt}eq{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}n\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ {\isacharparenleft}{\kern0pt}LEAST\ n{\isachardot}{\kern0pt}\ {\isadigit{2}}\ {\isacharasterisk}{\kern0pt}\ l\ {\isacharasterisk}{\kern0pt}\ dist\ {\isacharparenleft}{\kern0pt}mpi{\isacharunderscore}{\kern0pt}val\ {\isacharquery}{\kern0pt}d\ {\isacharparenleft}{\kern0pt}L{\isacharunderscore}{\kern0pt}pow\ v\ {\isacharquery}{\kern0pt}d\ {\isacharparenleft}{\kern0pt}m\ {\isadigit{0}}\ v{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymlambda}a{\isachardot}{\kern0pt}\ m\ {\isacharparenleft}{\kern0pt}Suc\ a{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ n{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ {\isacharparenleft}{\kern0pt}mpi{\isacharunderscore}{\kern0pt}val\ {\isacharquery}{\kern0pt}d\ {\isacharparenleft}{\kern0pt}L{\isacharunderscore}{\kern0pt}pow\ v\ {\isacharquery}{\kern0pt}d\ {\isacharparenleft}{\kern0pt}m\ {\isadigit{0}}\ v{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymlambda}a{\isachardot}{\kern0pt}\ m\ {\isacharparenleft}{\kern0pt}Suc\ a{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ n{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isacharless}{\kern0pt}\ eps\ {\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}\ {\isacharminus}{\kern0pt}\ l{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ Suc{\isachardot}{\kern0pt}prems\ mpi{\isacharunderscore}{\kern0pt}Suc{\isacharprime}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ is{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det{\isacharunderscore}{\kern0pt}pi\ mpi{\isacharunderscore}{\kern0pt}val{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymnot}\ {\isadigit{2}}\ {\isacharasterisk}{\kern0pt}\ l\ {\isacharasterisk}{\kern0pt}\ dist\ v\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ v{\isacharparenright}{\kern0pt}\ {\isacharless}{\kern0pt}\ eps\ {\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}\ {\isacharminus}{\kern0pt}\ l{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ Suc\ mpi{\isacharunderscore}{\kern0pt}val{\isacharunderscore}{\kern0pt}zero\ \isacommand{by}\isamarkupfalse%
-\ force\isanewline
-\ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}mpi{\isacharunderscore}{\kern0pt}algo{\isacharunderscore}{\kern0pt}dom\ {\isacharparenleft}{\kern0pt}{\isacharquery}{\kern0pt}d{\isacharcomma}{\kern0pt}\ L{\isacharunderscore}{\kern0pt}pow\ v\ {\isacharquery}{\kern0pt}d\ {\isacharparenleft}{\kern0pt}m\ {\isadigit{0}}\ v{\isacharparenright}{\kern0pt}{\isacharcomma}{\kern0pt}\ {\isasymlambda}a{\isachardot}{\kern0pt}\ m\ {\isacharparenleft}{\kern0pt}Suc\ a{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ Suc{\isachardot}{\kern0pt}IH{\isacharbrackleft}{\kern0pt}OF\ n{\isacharunderscore}{\kern0pt}eq\ {\isacartoucheopen}{\isadigit{0}}\ {\isacharless}{\kern0pt}\ eps{\isacartoucheclose}{\isacharbrackright}{\kern0pt}\ Suc{\isachardot}{\kern0pt}prems\ is{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det{\isacharunderscore}{\kern0pt}pi\ L{\isacharunderscore}{\kern0pt}pow{\isacharunderscore}{\kern0pt}{\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}mono{\isacharunderscore}{\kern0pt}inv\ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}case\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ mpi{\isacharunderscore}{\kern0pt}algo{\isachardot}{\kern0pt}domintros\ \isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{abbreviation}\isamarkupfalse%
-\ {\isachardoublequoteopen}mpi{\isacharunderscore}{\kern0pt}alg{\isacharunderscore}{\kern0pt}rec\ d\ v\ m\ {\isasymequiv}\ \isanewline
-\ \ \ \ {\isacharparenleft}{\kern0pt}if\ {\isadigit{2}}\ {\isacharasterisk}{\kern0pt}\ l\ {\isacharasterisk}{\kern0pt}\ dist\ v\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ v{\isacharparenright}{\kern0pt}\ {\isacharless}{\kern0pt}\ eps\ {\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}\ {\isacharminus}{\kern0pt}\ l{\isacharparenright}{\kern0pt}\ then\ {\isacharparenleft}{\kern0pt}policy{\isacharunderscore}{\kern0pt}improvement\ d\ v{\isacharcomma}{\kern0pt}\ v{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ else\ mpi{\isacharunderscore}{\kern0pt}algo\ {\isacharparenleft}{\kern0pt}policy{\isacharunderscore}{\kern0pt}improvement\ d\ v{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}L{\isacharunderscore}{\kern0pt}pow\ v\ {\isacharparenleft}{\kern0pt}policy{\isacharunderscore}{\kern0pt}improvement\ d\ v{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}m\ {\isadigit{0}}\ v{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymlambda}n{\isachardot}{\kern0pt}\ m\ {\isacharparenleft}{\kern0pt}Suc\ n{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ mpi{\isacharunderscore}{\kern0pt}algo{\isacharunderscore}{\kern0pt}def{\isacharprime}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}d\ {\isasymin}\ D\isactrlsub D{\isachardoublequoteclose}\ {\isachardoublequoteopen}v\ {\isasymle}\ {\isasymL}\isactrlsub b\ v{\isachardoublequoteclose}\ {\isachardoublequoteopen}eps\ {\isachargreater}{\kern0pt}\ {\isadigit{0}}{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}mpi{\isacharunderscore}{\kern0pt}algo\ d\ v\ m\ {\isacharequal}{\kern0pt}\ mpi{\isacharunderscore}{\kern0pt}alg{\isacharunderscore}{\kern0pt}rec\ d\ v\ m{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ mpi{\isacharunderscore}{\kern0pt}algo{\isachardot}{\kern0pt}psimps\ termination{\isacharunderscore}{\kern0pt}mpi{\isacharunderscore}{\kern0pt}algo\ assms\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ mpi{\isacharunderscore}{\kern0pt}algo{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}mpi{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}d\ {\isasymin}\ D\isactrlsub D{\isachardoublequoteclose}\ {\isachardoublequoteopen}v\ {\isasymle}\ {\isasymL}\isactrlsub b\ v{\isachardoublequoteclose}\ {\isachardoublequoteopen}eps\ {\isachargreater}{\kern0pt}\ {\isadigit{0}}{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}mpi{\isacharunderscore}{\kern0pt}algo\ d\ v\ m\ {\isacharequal}{\kern0pt}\ mpi\ d\ v\ m\ {\isacharparenleft}{\kern0pt}LEAST\ n{\isachardot}{\kern0pt}\ {\isadigit{2}}\ {\isacharasterisk}{\kern0pt}\ l\ {\isacharasterisk}{\kern0pt}\ dist\ {\isacharparenleft}{\kern0pt}mpi{\isacharunderscore}{\kern0pt}val\ d\ v\ m\ n{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ {\isacharparenleft}{\kern0pt}mpi{\isacharunderscore}{\kern0pt}val\ d\ v\ m\ n{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharless}{\kern0pt}\ eps\ {\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}\ {\isacharminus}{\kern0pt}\ l{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{define}\isamarkupfalse%
-\ n\ \isakeyword{where}\ {\isachardoublequoteopen}n\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}LEAST\ n{\isachardot}{\kern0pt}\ {\isadigit{2}}\ {\isacharasterisk}{\kern0pt}\ l\ {\isacharasterisk}{\kern0pt}\ dist\ {\isacharparenleft}{\kern0pt}mpi{\isacharunderscore}{\kern0pt}val\ d\ v\ m\ n{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ {\isacharparenleft}{\kern0pt}mpi{\isacharunderscore}{\kern0pt}val\ d\ v\ m\ n{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharless}{\kern0pt}\ eps\ {\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}\ {\isacharminus}{\kern0pt}\ l{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ {\isacharparenleft}{\kern0pt}\isakeyword{is}\ {\isachardoublequoteopen}n\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}LEAST\ n{\isachardot}{\kern0pt}\ {\isacharquery}{\kern0pt}P\ d\ v\ m\ n{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ n{\isacharunderscore}{\kern0pt}def\ assms\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}induction\ n\ arbitrary{\isacharcolon}{\kern0pt}\ d\ v\ m{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{case}\isamarkupfalse%
-\ {\isadigit{0}}\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharquery}{\kern0pt}P\ d\ v\ m\ {\isadigit{0}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharcomma}{\kern0pt}\ lifting{\isacharparenright}{\kern0pt}\ assms{\isacharparenleft}{\kern0pt}{\isadigit{3}}{\isacharparenright}{\kern0pt}\ LeastI{\isacharunderscore}{\kern0pt}ex\ {\isadigit{0}}\ mpi{\isacharunderscore}{\kern0pt}val{\isacharunderscore}{\kern0pt}term{\isacharunderscore}{\kern0pt}ex{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}case\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assms\ {\isadigit{0}}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ mpi{\isacharunderscore}{\kern0pt}val{\isacharunderscore}{\kern0pt}def\ mpi{\isacharunderscore}{\kern0pt}algo{\isacharunderscore}{\kern0pt}def{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{case}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}Suc\ n{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ not{\isadigit{0}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymnot}\ {\isacharparenleft}{\kern0pt}{\isadigit{2}}\ {\isacharasterisk}{\kern0pt}\ l\ {\isacharasterisk}{\kern0pt}\ dist\ v\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ v{\isacharparenright}{\kern0pt}\ {\isacharless}{\kern0pt}\ eps\ {\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}\ {\isacharminus}{\kern0pt}\ l{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ Suc{\isacharparenleft}{\kern0pt}{\isadigit{3}}{\isacharparenright}{\kern0pt}\ mpi{\isacharunderscore}{\kern0pt}val{\isacharunderscore}{\kern0pt}zero\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \isacommand{obtain}\isamarkupfalse%
-\ n{\isacharprime}{\kern0pt}\ \isakeyword{where}\ {\isachardoublequoteopen}{\isadigit{2}}\ {\isacharasterisk}{\kern0pt}\ l\ {\isacharasterisk}{\kern0pt}\ dist\ {\isacharparenleft}{\kern0pt}mpi{\isacharunderscore}{\kern0pt}val\ d\ v\ m\ n{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ {\isacharparenleft}{\kern0pt}mpi{\isacharunderscore}{\kern0pt}val\ d\ v\ m\ n{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharless}{\kern0pt}\ eps\ {\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}\ {\isacharminus}{\kern0pt}\ l{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ mpi{\isacharunderscore}{\kern0pt}val{\isacharunderscore}{\kern0pt}term{\isacharunderscore}{\kern0pt}ex{\isacharbrackleft}{\kern0pt}OF\ Suc{\isacharparenleft}{\kern0pt}{\isadigit{3}}{\isacharparenright}{\kern0pt}\ Suc{\isacharparenleft}{\kern0pt}{\isadigit{4}}{\isacharparenright}{\kern0pt}{\isacharcomma}{\kern0pt}\ of\ {\isacharunderscore}{\kern0pt}\ m{\isacharbrackright}{\kern0pt}\ assms\ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}n\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}LEAST\ n{\isachardot}{\kern0pt}\ {\isacharquery}{\kern0pt}P\ d\ v\ m\ {\isacharparenleft}{\kern0pt}Suc\ n{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ Suc{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}\ Suc\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}subst\ {\isacharparenleft}{\kern0pt}asm{\isacharparenright}{\kern0pt}\ Least{\isacharunderscore}{\kern0pt}Suc{\isacharparenright}{\kern0pt}\ auto\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}n\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}LEAST\ n{\isachardot}{\kern0pt}\ {\isacharquery}{\kern0pt}P\ {\isacharparenleft}{\kern0pt}policy{\isacharunderscore}{\kern0pt}improvement\ d\ v{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}L{\isacharunderscore}{\kern0pt}pow\ v\ {\isacharparenleft}{\kern0pt}policy{\isacharunderscore}{\kern0pt}improvement\ d\ v{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}m\ {\isadigit{0}}\ v{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymlambda}n{\isachardot}{\kern0pt}\ m\ {\isacharparenleft}{\kern0pt}Suc\ n{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ n{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ Suc{\isacharparenleft}{\kern0pt}{\isadigit{3}}{\isacharparenright}{\kern0pt}\ policy{\isacharunderscore}{\kern0pt}improvement{\isacharunderscore}{\kern0pt}is{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\ mpi{\isacharunderscore}{\kern0pt}Suc{\isacharprime}{\kern0pt}\ \isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ mpi{\isacharunderscore}{\kern0pt}val{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}mpi{\isacharunderscore}{\kern0pt}algo\ d\ v\ m\ {\isacharequal}{\kern0pt}\ mpi\ d\ v\ m\ {\isacharparenleft}{\kern0pt}Suc\ n{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ mpi{\isacharunderscore}{\kern0pt}algo{\isacharunderscore}{\kern0pt}def{\isacharprime}{\kern0pt}{\isacharbrackleft}{\kern0pt}OF\ Suc{\isachardot}{\kern0pt}prems{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharminus}{\kern0pt}{\isadigit{4}}{\isacharparenright}{\kern0pt}{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ Suc{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}\ Suc{\isachardot}{\kern0pt}prems{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharminus}{\kern0pt}{\isadigit{4}}{\isacharparenright}{\kern0pt}\ is{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det{\isacharunderscore}{\kern0pt}pi\ mpi{\isacharunderscore}{\kern0pt}Suc{\isacharprime}{\kern0pt}\ not{\isadigit{0}}\ L{\isacharunderscore}{\kern0pt}pow{\isacharunderscore}{\kern0pt}{\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}mono{\isacharunderscore}{\kern0pt}inv\ \isacommand{by}\isamarkupfalse%
-\ force\isanewline
-\ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}case\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ Suc{\isachardot}{\kern0pt}prems{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}\ \isacommand{by}\isamarkupfalse%
-\ presburger\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ mpi{\isacharunderscore}{\kern0pt}algo{\isacharunderscore}{\kern0pt}opt{\isacharcolon}{\kern0pt}\ \isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}v{\isadigit{0}}\ {\isasymle}\ {\isasymL}\isactrlsub b\ v{\isadigit{0}}{\isachardoublequoteclose}\ {\isachardoublequoteopen}eps\ {\isachargreater}{\kern0pt}\ {\isadigit{0}}{\isachardoublequoteclose}\ {\isachardoublequoteopen}d\ {\isasymin}\ D\isactrlsub D{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}dist\ {\isacharparenleft}{\kern0pt}{\isasymnu}\isactrlsub b\ {\isacharparenleft}{\kern0pt}mk{\isacharunderscore}{\kern0pt}stationary{\isacharunderscore}{\kern0pt}det\ {\isacharparenleft}{\kern0pt}fst\ {\isacharparenleft}{\kern0pt}mpi{\isacharunderscore}{\kern0pt}algo\ d\ v{\isadigit{0}}\ m{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt\ {\isacharless}{\kern0pt}\ eps{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}P\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}{\isasymlambda}n{\isachardot}{\kern0pt}\ {\isadigit{2}}\ {\isacharasterisk}{\kern0pt}\ l\ {\isacharasterisk}{\kern0pt}\ dist\ {\isacharparenleft}{\kern0pt}mpi{\isacharunderscore}{\kern0pt}val\ d\ v{\isadigit{0}}\ m\ n{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ {\isacharparenleft}{\kern0pt}mpi{\isacharunderscore}{\kern0pt}val\ d\ v{\isadigit{0}}\ m\ n{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharless}{\kern0pt}\ \ eps\ {\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}\ {\isacharminus}{\kern0pt}\ l{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}n\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}Least\ {\isacharquery}{\kern0pt}P{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}mpi{\isacharunderscore}{\kern0pt}algo\ d\ v{\isadigit{0}}\ m\ {\isacharequal}{\kern0pt}\ mpi\ d\ v{\isadigit{0}}\ m\ {\isacharquery}{\kern0pt}n{\isachardoublequoteclose}\ \isakeyword{and}\ {\isachardoublequoteopen}{\isacharquery}{\kern0pt}P\ {\isacharquery}{\kern0pt}n{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ mpi{\isacharunderscore}{\kern0pt}algo{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}mpi\ LeastI{\isacharunderscore}{\kern0pt}ex{\isacharbrackleft}{\kern0pt}OF\ mpi{\isacharunderscore}{\kern0pt}val{\isacharunderscore}{\kern0pt}term{\isacharunderscore}{\kern0pt}ex{\isacharbrackright}{\kern0pt}\ assms\ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assms\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ mpi{\isacharunderscore}{\kern0pt}pol{\isacharunderscore}{\kern0pt}opt\ mpi{\isacharunderscore}{\kern0pt}pol{\isacharunderscore}{\kern0pt}def{\isacharbrackleft}{\kern0pt}symmetric{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{end}\isamarkupfalse%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Initial Value Estimate \isa{v{\isadigit{0}}{\isacharunderscore}{\kern0pt}mpi}%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\begin{isamarkuptext}%
-We define an initial estimate of the value function for which Modified Policy Iteration 
-  always terminates.%
-\end{isamarkuptext}\isamarkuptrue%
-\isacommand{abbreviation}\isamarkupfalse%
-\ {\isachardoublequoteopen}r{\isacharunderscore}{\kern0pt}min\ {\isasymequiv}\ {\isacharparenleft}{\kern0pt}{\isasymSqinter}s{\isacharprime}{\kern0pt}\ a{\isachardot}{\kern0pt}\ r\ {\isacharparenleft}{\kern0pt}s{\isacharprime}{\kern0pt}{\isacharcomma}{\kern0pt}\ a{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ {\isachardoublequoteopen}v{\isadigit{0}}{\isacharunderscore}{\kern0pt}mpi\ s\ {\isacharequal}{\kern0pt}\ r{\isacharunderscore}{\kern0pt}min\ {\isacharslash}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}\ {\isacharminus}{\kern0pt}\ l{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{lift{\isacharunderscore}{\kern0pt}definition}\isamarkupfalse%
-\ v{\isadigit{0}}{\isacharunderscore}{\kern0pt}mpi\isactrlsub b\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}s\ {\isasymRightarrow}\isactrlsub b\ real{\isachardoublequoteclose}\ \isakeyword{is}\ {\isachardoublequoteopen}v{\isadigit{0}}{\isacharunderscore}{\kern0pt}mpi{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ fastforce%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ v{\isadigit{0}}{\isacharunderscore}{\kern0pt}mpi\isactrlsub b{\isacharunderscore}{\kern0pt}le{\isacharunderscore}{\kern0pt}{\isasymL}\isactrlsub b{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}v{\isadigit{0}}{\isacharunderscore}{\kern0pt}mpi\isactrlsub b\ {\isasymle}\ {\isasymL}\isactrlsub b\ v{\isadigit{0}}{\isacharunderscore}{\kern0pt}mpi\isactrlsub b{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}rule\ less{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}bfunI{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\ x\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}r{\isacharunderscore}{\kern0pt}min\ {\isasymle}\ r\ {\isacharparenleft}{\kern0pt}s{\isacharcomma}{\kern0pt}\ a{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{for}\ s\ a\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}fastforce\ intro{\isacharcolon}{\kern0pt}\ cInf{\isacharunderscore}{\kern0pt}lower{\isadigit{2}}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}r{\isacharunderscore}{\kern0pt}min\ {\isasymle}\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharminus}{\kern0pt}l{\isacharparenright}{\kern0pt}\ {\isacharasterisk}{\kern0pt}\ r\ {\isacharparenleft}{\kern0pt}s{\isacharcomma}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isacharplus}{\kern0pt}\ l\ {\isacharasterisk}{\kern0pt}\ r{\isacharunderscore}{\kern0pt}min{\isachardoublequoteclose}\ \isakeyword{for}\ s\ a\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ disc{\isacharunderscore}{\kern0pt}lt{\isacharunderscore}{\kern0pt}one\ zero{\isacharunderscore}{\kern0pt}le{\isacharunderscore}{\kern0pt}disc\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}meson\ order{\isacharunderscore}{\kern0pt}less{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}le\ order{\isacharunderscore}{\kern0pt}refl\ segment{\isacharunderscore}{\kern0pt}bound{\isacharunderscore}{\kern0pt}lemma{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}r{\isacharunderscore}{\kern0pt}min\ {\isacharslash}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}\ {\isacharminus}{\kern0pt}\ l{\isacharparenright}{\kern0pt}\ {\isasymle}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharminus}{\kern0pt}l{\isacharparenright}{\kern0pt}\ {\isacharasterisk}{\kern0pt}\ r\ {\isacharparenleft}{\kern0pt}s{\isacharcomma}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isacharplus}{\kern0pt}\ l\ {\isacharasterisk}{\kern0pt}\ r{\isacharunderscore}{\kern0pt}min{\isacharparenright}{\kern0pt}\ {\isacharslash}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}\ {\isacharminus}{\kern0pt}\ l{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{for}\ s\ a\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ order{\isacharunderscore}{\kern0pt}less{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}le{\isacharbrackleft}{\kern0pt}OF\ disc{\isacharunderscore}{\kern0pt}lt{\isacharunderscore}{\kern0pt}one{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ divide{\isacharunderscore}{\kern0pt}right{\isacharunderscore}{\kern0pt}mono{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}r{\isacharunderscore}{\kern0pt}min\ {\isacharslash}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}\ {\isacharminus}{\kern0pt}\ l{\isacharparenright}{\kern0pt}\ {\isasymle}\ r\ {\isacharparenleft}{\kern0pt}s{\isacharcomma}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isacharplus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}l\ {\isacharasterisk}{\kern0pt}\ r{\isacharunderscore}{\kern0pt}min{\isacharparenright}{\kern0pt}\ {\isacharslash}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}\ {\isacharminus}{\kern0pt}\ l{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{for}\ s\ a\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ disc{\isacharunderscore}{\kern0pt}lt{\isacharunderscore}{\kern0pt}one\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ add{\isacharunderscore}{\kern0pt}divide{\isacharunderscore}{\kern0pt}distrib{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}v{\isadigit{0}}{\isacharunderscore}{\kern0pt}mpi\isactrlsub b\ x\ {\isasymle}\ {\isasymL}\isactrlsub b\ v{\isadigit{0}}{\isacharunderscore}{\kern0pt}mpi\isactrlsub b\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ {\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}SUP{\isacharunderscore}{\kern0pt}L\isactrlsub a\ v{\isadigit{0}}{\isacharunderscore}{\kern0pt}mpi\isactrlsub b{\isachardot}{\kern0pt}rep{\isacharunderscore}{\kern0pt}eq\ v{\isadigit{0}}{\isacharunderscore}{\kern0pt}mpi{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ A{\isacharunderscore}{\kern0pt}ne\ intro{\isacharcolon}{\kern0pt}\ cSUP{\isacharunderscore}{\kern0pt}upper{\isadigit{2}}{\isacharbrackleft}{\kern0pt}\isakeyword{where}\ x\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}arb{\isacharunderscore}{\kern0pt}act\ {\isacharparenleft}{\kern0pt}A\ x{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{An Instance of Modified Policy Iteration with a Valid Conservative Initial Value Estimate%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{definition}\isamarkupfalse%
-\ {\isachardoublequoteopen}mpi{\isacharunderscore}{\kern0pt}user\ eps\ m\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}\isanewline
-\ \ if\ eps\ {\isasymle}\ {\isadigit{0}}\ then\ undefined\ else\ mpi{\isacharunderscore}{\kern0pt}algo\ eps\ {\isacharparenleft}{\kern0pt}{\isasymlambda}x{\isachardot}{\kern0pt}\ arb{\isacharunderscore}{\kern0pt}act\ {\isacharparenleft}{\kern0pt}A\ x{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ v{\isadigit{0}}{\isacharunderscore}{\kern0pt}mpi\isactrlsub b\ m{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ mpi{\isacharunderscore}{\kern0pt}user{\isacharunderscore}{\kern0pt}eq{\isacharcolon}{\kern0pt}\ \isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}eps\ {\isachargreater}{\kern0pt}\ {\isadigit{0}}{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}mpi{\isacharunderscore}{\kern0pt}user\ eps\ {\isacharequal}{\kern0pt}\ mpi{\isacharunderscore}{\kern0pt}alg{\isacharunderscore}{\kern0pt}rec\ eps\ {\isacharparenleft}{\kern0pt}{\isasymlambda}x{\isachardot}{\kern0pt}\ arb{\isacharunderscore}{\kern0pt}act\ {\isacharparenleft}{\kern0pt}A\ x{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ v{\isadigit{0}}{\isacharunderscore}{\kern0pt}mpi\isactrlsub b{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ v{\isadigit{0}}{\isacharunderscore}{\kern0pt}mpi\isactrlsub b{\isacharunderscore}{\kern0pt}le{\isacharunderscore}{\kern0pt}{\isasymL}\isactrlsub b\ assms\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ mpi{\isacharunderscore}{\kern0pt}user{\isacharunderscore}{\kern0pt}def\ mpi{\isacharunderscore}{\kern0pt}algo{\isacharunderscore}{\kern0pt}def{\isacharprime}{\kern0pt}\ A{\isacharunderscore}{\kern0pt}ne\ is{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ mpi{\isacharunderscore}{\kern0pt}user{\isacharunderscore}{\kern0pt}opt{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}eps\ {\isachargreater}{\kern0pt}\ {\isadigit{0}}{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}dist\ {\isacharparenleft}{\kern0pt}{\isasymnu}\isactrlsub b\ {\isacharparenleft}{\kern0pt}mk{\isacharunderscore}{\kern0pt}stationary{\isacharunderscore}{\kern0pt}det\ {\isacharparenleft}{\kern0pt}fst\ {\isacharparenleft}{\kern0pt}mpi{\isacharunderscore}{\kern0pt}user\ eps\ n{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt\ {\isacharless}{\kern0pt}\ eps{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ mpi{\isacharunderscore}{\kern0pt}user{\isacharunderscore}{\kern0pt}def\ \isacommand{using}\isamarkupfalse%
-\ assms\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ intro{\isacharcolon}{\kern0pt}\ mpi{\isacharunderscore}{\kern0pt}algo{\isacharunderscore}{\kern0pt}opt\ simp{\isacharcolon}{\kern0pt}\ is{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det{\isacharunderscore}{\kern0pt}def\ A{\isacharunderscore}{\kern0pt}ne\ v{\isadigit{0}}{\isacharunderscore}{\kern0pt}mpi\isactrlsub b{\isacharunderscore}{\kern0pt}le{\isacharunderscore}{\kern0pt}{\isasymL}\isactrlsub b{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{end}\isamarkupfalse%
-\isanewline
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-\isadelimtheory
-\isanewline
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-\endisadelimtheory
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-\isatagtheory
-\isacommand{end}\isamarkupfalse%
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-\endisatagtheory
-{\isafoldtheory}%
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-\isadelimtheory
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-\endisadelimtheory
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-\end{isabellebody}%
-\endinput
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diff --git a/thys/MDP-Algorithms/output/outline/Policy_Iteration.tex b/thys/MDP-Algorithms/output/outline/Policy_Iteration.tex
deleted file mode 100644
--- a/thys/MDP-Algorithms/output/outline/Policy_Iteration.tex
+++ /dev/null
@@ -1,1335 +0,0 @@
-%
-\begin{isabellebody}%
-\setisabellecontext{Policy{\isacharunderscore}{\kern0pt}Iteration}%
-%
-\isadelimtheory
-\isanewline
-\isanewline
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{theory}\isamarkupfalse%
-\ Policy{\isacharunderscore}{\kern0pt}Iteration\isanewline
-\ \ \isakeyword{imports}\ {\isachardoublequoteopen}MDP{\isacharminus}{\kern0pt}Rewards{\isachardot}{\kern0pt}MDP{\isacharunderscore}{\kern0pt}reward{\isachardoublequoteclose}\isanewline
-\isanewline
-\isakeyword{begin}%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsection{Policy Iteration%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\begin{isamarkuptext}%
-The Policy Iteration algorithms provides another way to find optimal policies under the expected 
-total reward criterion.
-It differs from Value Iteration in that it continuously improves an initial guess for an optimal 
-decision rule. Its execution can be subdivided into two alternating steps: policy evaluation and 
-policy improvement.
-
-Policy evaluation means the calculation of the value of the current decision rule.
-
-During the improvement phase, we choose the decision rule with the maximum value for L, 
-while we prefer to keep the old action selection in case of ties.%
-\end{isamarkuptext}\isamarkuptrue%
-\isacommand{context}\isamarkupfalse%
-\ MDP{\isacharunderscore}{\kern0pt}att{\isacharunderscore}{\kern0pt}{\isasymL}\ \isakeyword{begin}\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ {\isachardoublequoteopen}policy{\isacharunderscore}{\kern0pt}eval\ d\ {\isacharequal}{\kern0pt}\ {\isasymnu}\isactrlsub b\ {\isacharparenleft}{\kern0pt}mk{\isacharunderscore}{\kern0pt}stationary{\isacharunderscore}{\kern0pt}det\ d{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isacommand{end}\isamarkupfalse%
-\isanewline
-\isanewline
-\isacommand{context}\isamarkupfalse%
-\ MDP{\isacharunderscore}{\kern0pt}act\isanewline
-\isakeyword{begin}\isanewline
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ {\isachardoublequoteopen}policy{\isacharunderscore}{\kern0pt}improvement\ d\ v\ s\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}\isanewline
-\ \ if\ is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max\ {\isacharparenleft}{\kern0pt}{\isasymlambda}a{\isachardot}{\kern0pt}\ L\isactrlsub a\ a\ {\isacharparenleft}{\kern0pt}apply{\isacharunderscore}{\kern0pt}bfun\ v{\isacharparenright}{\kern0pt}\ s{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymlambda}a{\isachardot}{\kern0pt}\ a\ {\isasymin}\ A\ s{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}d\ s{\isacharparenright}{\kern0pt}\ \isanewline
-\ \ then\ d\ s\isanewline
-\ \ else\ arb{\isacharunderscore}{\kern0pt}act\ {\isacharparenleft}{\kern0pt}opt{\isacharunderscore}{\kern0pt}acts\ v\ s{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ {\isachardoublequoteopen}policy{\isacharunderscore}{\kern0pt}step\ d\ {\isacharequal}{\kern0pt}\ policy{\isacharunderscore}{\kern0pt}improvement\ d\ {\isacharparenleft}{\kern0pt}policy{\isacharunderscore}{\kern0pt}eval\ d{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isanewline
-\isacommand{function}\isamarkupfalse%
-\ policy{\isacharunderscore}{\kern0pt}iteration\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}s\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a{\isacharparenright}{\kern0pt}\ {\isasymRightarrow}\ {\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}s\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}policy{\isacharunderscore}{\kern0pt}iteration\ d\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}\isanewline
-\ \ let\ d{\isacharprime}{\kern0pt}\ {\isacharequal}{\kern0pt}\ policy{\isacharunderscore}{\kern0pt}step\ d\ in\isanewline
-\ \ if\ d\ {\isacharequal}{\kern0pt}\ d{\isacharprime}{\kern0pt}\ {\isasymor}\ {\isasymnot}is{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\ d\ then\ d\ else\ policy{\isacharunderscore}{\kern0pt}iteration\ d{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\begin{isamarkuptext}%
-The policy iteration algorithm as stated above does require that the supremum in \isa{{\isasymL}\isactrlsub b} is
-always attained.%
-\end{isamarkuptext}\isamarkuptrue%
-%
-\begin{isamarkuptext}%
-Each policy improvement returns a valid decision rule.%
-\end{isamarkuptext}\isamarkuptrue%
-\isacommand{lemma}\isamarkupfalse%
-\ is{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det{\isacharunderscore}{\kern0pt}pi{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}is{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\ {\isacharparenleft}{\kern0pt}policy{\isacharunderscore}{\kern0pt}improvement\ d\ v{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ policy{\isacharunderscore}{\kern0pt}improvement{\isacharunderscore}{\kern0pt}def\ is{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det{\isacharunderscore}{\kern0pt}def\ is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ some{\isacharunderscore}{\kern0pt}opt{\isacharunderscore}{\kern0pt}acts{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ policy{\isacharunderscore}{\kern0pt}improvement{\isacharunderscore}{\kern0pt}is{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}d\ {\isasymin}\ D\isactrlsub D\ {\isasymLongrightarrow}\ policy{\isacharunderscore}{\kern0pt}improvement\ d\ v\ {\isasymin}\ D\isactrlsub D{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ policy{\isacharunderscore}{\kern0pt}improvement{\isacharunderscore}{\kern0pt}def\ is{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \isacommand{using}\isamarkupfalse%
-\ some{\isacharunderscore}{\kern0pt}opt{\isacharunderscore}{\kern0pt}acts{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ policy{\isacharunderscore}{\kern0pt}improvement{\isacharunderscore}{\kern0pt}improving{\isacharcolon}{\kern0pt}\ \isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}d\ {\isasymin}\ D\isactrlsub D{\isachardoublequoteclose}\ \isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}{\isasymnu}{\isacharunderscore}{\kern0pt}improving\ v\ {\isacharparenleft}{\kern0pt}mk{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\ {\isacharparenleft}{\kern0pt}policy{\isacharunderscore}{\kern0pt}improvement\ d\ v{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymL}\isactrlsub b\ v\ x\ {\isacharequal}{\kern0pt}\ L\ {\isacharparenleft}{\kern0pt}mk{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\ {\isacharparenleft}{\kern0pt}policy{\isacharunderscore}{\kern0pt}improvement\ d\ v{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ v\ x{\isachardoublequoteclose}\ \isakeyword{for}\ x\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ is{\isacharunderscore}{\kern0pt}opt{\isacharunderscore}{\kern0pt}act{\isacharunderscore}{\kern0pt}some\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}fastforce\ simp{\isacharcolon}{\kern0pt}\ thm{\isacharunderscore}{\kern0pt}{\isadigit{6}}{\isacharunderscore}{\kern0pt}{\isadigit{2}}{\isacharunderscore}{\kern0pt}{\isadigit{1}}{\isadigit{0}}{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}aux{\isacharprime}{\kern0pt}\ L{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}L\isactrlsub a{\isacharunderscore}{\kern0pt}det\ is{\isacharunderscore}{\kern0pt}opt{\isacharunderscore}{\kern0pt}act{\isacharunderscore}{\kern0pt}def\ policy{\isacharunderscore}{\kern0pt}improvement{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}SUP{\isacharbrackleft}{\kern0pt}symmetric{\isacharcomma}{\kern0pt}\ of\ {\isacharunderscore}{\kern0pt}\ {\isacharunderscore}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}policy{\isacharunderscore}{\kern0pt}improvement\ d\ v\ x{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}{\isacharbrackright}{\kern0pt}\ {\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ policy{\isacharunderscore}{\kern0pt}improvement{\isacharunderscore}{\kern0pt}is{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\ assms\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ {\isasymnu}{\isacharunderscore}{\kern0pt}improving{\isacharunderscore}{\kern0pt}alt{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ eval{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}step{\isacharunderscore}{\kern0pt}L{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}is{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\ d{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}L\ {\isacharparenleft}{\kern0pt}mk{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\ {\isacharparenleft}{\kern0pt}policy{\isacharunderscore}{\kern0pt}step\ d{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}policy{\isacharunderscore}{\kern0pt}eval\ d{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isasymL}\isactrlsub b\ {\isacharparenleft}{\kern0pt}policy{\isacharunderscore}{\kern0pt}eval\ d{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ policy{\isacharunderscore}{\kern0pt}step{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \isacommand{using}\isamarkupfalse%
-\ assms\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ {\isasymnu}{\isacharunderscore}{\kern0pt}improving{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}{\isasymL}\isactrlsub b{\isacharbrackleft}{\kern0pt}OF\ policy{\isacharunderscore}{\kern0pt}improvement{\isacharunderscore}{\kern0pt}improving{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\begin{isamarkuptext}%
-The sequence of policies generated by policy iteration has monotonically increasing 
-discounted reward.%
-\end{isamarkuptext}\isamarkuptrue%
-\isacommand{lemma}\isamarkupfalse%
-\ policy{\isacharunderscore}{\kern0pt}eval{\isacharunderscore}{\kern0pt}mon{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}is{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\ d{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}policy{\isacharunderscore}{\kern0pt}eval\ d\ {\isasymle}\ policy{\isacharunderscore}{\kern0pt}eval\ {\isacharparenleft}{\kern0pt}policy{\isacharunderscore}{\kern0pt}step\ d{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}d{\isacharprime}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}mk{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\ {\isacharparenleft}{\kern0pt}policy{\isacharunderscore}{\kern0pt}step\ d{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}dp\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}mk{\isacharunderscore}{\kern0pt}stationary{\isacharunderscore}{\kern0pt}det\ d{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}P\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}{\isasymSum}t{\isachardot}{\kern0pt}\ l\ {\isacharcircum}{\kern0pt}\ t\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isasymP}\isactrlsub {\isadigit{1}}\ {\isacharquery}{\kern0pt}d{\isacharprime}{\kern0pt}\ {\isacharcircum}{\kern0pt}{\isacharcircum}{\kern0pt}\ t{\isachardoublequoteclose}\isanewline
-\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}L\ {\isacharparenleft}{\kern0pt}mk{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\ d{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}policy{\isacharunderscore}{\kern0pt}eval\ d{\isacharparenright}{\kern0pt}\ {\isasymle}\ L\ {\isacharquery}{\kern0pt}d{\isacharprime}{\kern0pt}\ {\isacharparenleft}{\kern0pt}policy{\isacharunderscore}{\kern0pt}eval\ d{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assms\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ L{\isacharunderscore}{\kern0pt}le{\isacharunderscore}{\kern0pt}{\isasymL}\isactrlsub b\ eval{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}step{\isacharunderscore}{\kern0pt}L{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}policy{\isacharunderscore}{\kern0pt}eval\ d\ {\isasymle}\ L\ {\isacharquery}{\kern0pt}d{\isacharprime}{\kern0pt}\ {\isacharparenleft}{\kern0pt}policy{\isacharunderscore}{\kern0pt}eval\ d{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ L{\isacharunderscore}{\kern0pt}{\isasymnu}{\isacharunderscore}{\kern0pt}fix\ policy{\isacharunderscore}{\kern0pt}eval{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymnu}\isactrlsub b\ {\isacharquery}{\kern0pt}dp\ {\isasymle}\ r{\isacharunderscore}{\kern0pt}dec\isactrlsub b\ {\isacharquery}{\kern0pt}d{\isacharprime}{\kern0pt}\ {\isacharplus}{\kern0pt}\ l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isasymP}\isactrlsub {\isadigit{1}}\ {\isacharquery}{\kern0pt}d{\isacharprime}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymnu}\isactrlsub b\ {\isacharquery}{\kern0pt}dp{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ policy{\isacharunderscore}{\kern0pt}eval{\isacharunderscore}{\kern0pt}def\ L{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}id{\isacharunderscore}{\kern0pt}blinfun\ {\isacharminus}{\kern0pt}\ l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isasymP}\isactrlsub {\isadigit{1}}\ {\isacharquery}{\kern0pt}d{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymnu}\isactrlsub b\ {\isacharquery}{\kern0pt}dp{\isacharparenright}{\kern0pt}\ {\isasymle}\ r{\isacharunderscore}{\kern0pt}dec\isactrlsub b\ {\isacharquery}{\kern0pt}d{\isacharprime}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ blinfun{\isachardot}{\kern0pt}diff{\isacharunderscore}{\kern0pt}left\ diff{\isacharunderscore}{\kern0pt}le{\isacharunderscore}{\kern0pt}eq\ scaleR{\isacharunderscore}{\kern0pt}blinfun{\isachardot}{\kern0pt}rep{\isacharunderscore}{\kern0pt}eq{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharquery}{\kern0pt}P\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}id{\isacharunderscore}{\kern0pt}blinfun\ {\isacharminus}{\kern0pt}\ l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isasymP}\isactrlsub {\isadigit{1}}\ {\isacharquery}{\kern0pt}d{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymnu}\isactrlsub b\ {\isacharquery}{\kern0pt}dp{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymle}\ {\isacharquery}{\kern0pt}P\ {\isacharparenleft}{\kern0pt}r{\isacharunderscore}{\kern0pt}dec\isactrlsub b\ {\isacharquery}{\kern0pt}d{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ lemma{\isacharunderscore}{\kern0pt}{\isadigit{6}}{\isacharunderscore}{\kern0pt}{\isadigit{1}}{\isacharunderscore}{\kern0pt}{\isadigit{2}}{\isacharunderscore}{\kern0pt}b\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymnu}\isactrlsub b\ {\isacharquery}{\kern0pt}dp\ {\isasymle}\ {\isacharquery}{\kern0pt}P\ {\isacharparenleft}{\kern0pt}r{\isacharunderscore}{\kern0pt}dec\isactrlsub b\ {\isacharquery}{\kern0pt}d{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ inv{\isacharunderscore}{\kern0pt}norm{\isacharunderscore}{\kern0pt}le{\isacharprime}{\kern0pt}{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isacharbrackleft}{\kern0pt}OF\ norm{\isacharunderscore}{\kern0pt}{\isasymP}\isactrlsub {\isadigit{1}}{\isacharunderscore}{\kern0pt}l{\isacharunderscore}{\kern0pt}less{\isacharbrackright}{\kern0pt}\ blincomp{\isacharunderscore}{\kern0pt}scaleR{\isacharunderscore}{\kern0pt}right\ suminf{\isacharunderscore}{\kern0pt}cong\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}mono{\isacharunderscore}{\kern0pt}tags{\isacharcomma}{\kern0pt}\ lifting{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ policy{\isacharunderscore}{\kern0pt}eval{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ {\isasymnu}{\isacharunderscore}{\kern0pt}stationary{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\begin{isamarkuptext}%
-If policy iteration terminates, i.e. \isa{d\ {\isacharequal}{\kern0pt}\ policy{\isacharunderscore}{\kern0pt}step\ d}, then it does so with optimal value.%
-\end{isamarkuptext}\isamarkuptrue%
-\isacommand{lemma}\isamarkupfalse%
-\ policy{\isacharunderscore}{\kern0pt}step{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}opt{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}is{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\ d{\isachardoublequoteclose}\ {\isachardoublequoteopen}d\ {\isacharequal}{\kern0pt}\ policy{\isacharunderscore}{\kern0pt}step\ d{\isachardoublequoteclose}\ \isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}{\isasymnu}\isactrlsub b\ {\isacharparenleft}{\kern0pt}mk{\isacharunderscore}{\kern0pt}stationary\ {\isacharparenleft}{\kern0pt}mk{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\ d{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}policy{\isacharunderscore}{\kern0pt}eval\ d\ {\isacharequal}{\kern0pt}\ {\isasymL}\isactrlsub b\ {\isacharparenleft}{\kern0pt}policy{\isacharunderscore}{\kern0pt}eval\ d{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ policy{\isacharunderscore}{\kern0pt}eval{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ L{\isacharunderscore}{\kern0pt}{\isasymnu}{\isacharunderscore}{\kern0pt}fix\ assms\ eval{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}step{\isacharunderscore}{\kern0pt}L{\isacharbrackleft}{\kern0pt}unfolded\ policy{\isacharunderscore}{\kern0pt}eval{\isacharunderscore}{\kern0pt}def{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ fastforce\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ policy{\isacharunderscore}{\kern0pt}eval{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isasymL}{\isacharunderscore}{\kern0pt}fix{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}opt\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{end}\isamarkupfalse%
-%
-\begin{isamarkuptext}%
-We prove termination of policy iteration only if both the state and action sets are finite.%
-\end{isamarkuptext}\isamarkuptrue%
-\isacommand{locale}\isamarkupfalse%
-\ MDP{\isacharunderscore}{\kern0pt}PI{\isacharunderscore}{\kern0pt}finite\ {\isacharequal}{\kern0pt}\ MDP{\isacharunderscore}{\kern0pt}act\ A\ K\ r\ l\ arb{\isacharunderscore}{\kern0pt}act\isanewline
-\ \ \isakeyword{for}\isanewline
-\ \ \ \ A\ \isakeyword{and}\isanewline
-\ \ \ \ K\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}s\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}countable\ {\isasymtimes}\ {\isacharprime}{\kern0pt}a\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}countable\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}s\ pmf{\isachardoublequoteclose}\ \isakeyword{and}\ r\ l\ arb{\isacharunderscore}{\kern0pt}act\ {\isacharplus}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ fin{\isacharunderscore}{\kern0pt}states{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite\ {\isacharparenleft}{\kern0pt}UNIV\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isacharprime}{\kern0pt}s\ set{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{and}\ fin{\isacharunderscore}{\kern0pt}actions{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymAnd}s{\isachardot}{\kern0pt}\ finite\ {\isacharparenleft}{\kern0pt}A\ s{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isakeyword{begin}%
-\begin{isamarkuptext}%
-If the state and action sets are both finite, 
-  then so is the set of deterministic decision rules \isa{D\isactrlsub D}%
-\end{isamarkuptext}\isamarkuptrue%
-\isacommand{lemma}\isamarkupfalse%
-\ finite{\isacharunderscore}{\kern0pt}D\isactrlsub D{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite\ D\isactrlsub D{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}set\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}d{\isachardot}{\kern0pt}\ {\isasymforall}x\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isacharprime}{\kern0pt}s{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}x\ {\isasymin}\ UNIV\ {\isasymlongrightarrow}\ d\ x\ {\isasymin}\ {\isacharparenleft}{\kern0pt}{\isasymUnion}s{\isachardot}{\kern0pt}\ A\ s{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymand}\ {\isacharparenleft}{\kern0pt}x\ {\isasymnotin}\ UNIV\ {\isasymlongrightarrow}\ d\ x\ {\isacharequal}{\kern0pt}\ undefined{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}finite\ {\isacharparenleft}{\kern0pt}{\isasymUnion}s{\isachardot}{\kern0pt}\ A\ s{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ fin{\isacharunderscore}{\kern0pt}actions\ fin{\isacharunderscore}{\kern0pt}states\ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}finite\ {\isacharquery}{\kern0pt}set{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ fin{\isacharunderscore}{\kern0pt}states\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}fastforce\ intro{\isacharcolon}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}set{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}finite{\isacharunderscore}{\kern0pt}funs{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}D\isactrlsub D\ {\isasymsubseteq}\ {\isacharquery}{\kern0pt}set{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ is{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det{\isacharunderscore}{\kern0pt}def\ \isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ finite{\isacharunderscore}{\kern0pt}subset\ \isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ finite{\isacharunderscore}{\kern0pt}rel{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite\ {\isacharbraceleft}{\kern0pt}{\isacharparenleft}{\kern0pt}u{\isacharcomma}{\kern0pt}\ v{\isacharparenright}{\kern0pt}{\isachardot}{\kern0pt}\ is{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\ u\ {\isasymand}\ is{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\ v\ {\isasymand}\ {\isasymnu}\isactrlsub b\ {\isacharparenleft}{\kern0pt}mk{\isacharunderscore}{\kern0pt}stationary{\isacharunderscore}{\kern0pt}det\ u{\isacharparenright}{\kern0pt}\ {\isachargreater}{\kern0pt}\ \isanewline
-\ \ {\isasymnu}\isactrlsub b\ {\isacharparenleft}{\kern0pt}mk{\isacharunderscore}{\kern0pt}stationary{\isacharunderscore}{\kern0pt}det\ v{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-{\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ aux{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite\ {\isacharbraceleft}{\kern0pt}{\isacharparenleft}{\kern0pt}u{\isacharcomma}{\kern0pt}\ v{\isacharparenright}{\kern0pt}{\isachardot}{\kern0pt}\ is{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\ u\ {\isasymand}\ is{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\ v{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ intro{\isacharcolon}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}subset{\isacharbrackleft}{\kern0pt}OF\ {\isacharunderscore}{\kern0pt}\ aux{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\begin{isamarkuptext}%
-This auxiliary lemma shows that policy iteration terminates if no improvement to the value of 
-the policy could be made, as then the policy remains unchanged.%
-\end{isamarkuptext}\isamarkuptrue%
-\isacommand{lemma}\isamarkupfalse%
-\ eval{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}eq{\isacharcolon}{\kern0pt}\ \isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}policy{\isacharunderscore}{\kern0pt}eval\ d\ {\isacharequal}{\kern0pt}\ policy{\isacharunderscore}{\kern0pt}eval\ {\isacharparenleft}{\kern0pt}policy{\isacharunderscore}{\kern0pt}step\ d{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ {\isachardoublequoteopen}is{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\ d{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}d\ {\isacharequal}{\kern0pt}\ policy{\isacharunderscore}{\kern0pt}step\ d{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}policy{\isacharunderscore}{\kern0pt}eval\ d\ s\ {\isacharequal}{\kern0pt}\ policy{\isacharunderscore}{\kern0pt}eval\ {\isacharparenleft}{\kern0pt}policy{\isacharunderscore}{\kern0pt}step\ d{\isacharparenright}{\kern0pt}\ s{\isachardoublequoteclose}\ \isakeyword{for}\ s\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assms\ \isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}policy{\isacharunderscore}{\kern0pt}eval\ d\ {\isacharequal}{\kern0pt}\ L\ {\isacharparenleft}{\kern0pt}mk{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\ d{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}policy{\isacharunderscore}{\kern0pt}eval\ {\isacharparenleft}{\kern0pt}policy{\isacharunderscore}{\kern0pt}step\ d{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ policy{\isacharunderscore}{\kern0pt}eval{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ L{\isacharunderscore}{\kern0pt}{\isasymnu}{\isacharunderscore}{\kern0pt}fix\ \isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ assms{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}{\isacharbrackleft}{\kern0pt}symmetric{\isacharcomma}{\kern0pt}\ unfolded\ policy{\isacharunderscore}{\kern0pt}eval{\isacharunderscore}{\kern0pt}def{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}policy{\isacharunderscore}{\kern0pt}eval\ d\ {\isacharequal}{\kern0pt}\ {\isasymL}\isactrlsub b\ {\isacharparenleft}{\kern0pt}policy{\isacharunderscore}{\kern0pt}eval\ d{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ L{\isacharunderscore}{\kern0pt}{\isasymnu}{\isacharunderscore}{\kern0pt}fix\ policy{\isacharunderscore}{\kern0pt}eval{\isacharunderscore}{\kern0pt}def\ assms\ eval{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}step{\isacharunderscore}{\kern0pt}L{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}L\ {\isacharparenleft}{\kern0pt}mk{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\ d{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}policy{\isacharunderscore}{\kern0pt}eval\ d{\isacharparenright}{\kern0pt}\ s\ {\isacharequal}{\kern0pt}\ {\isasymL}\isactrlsub b\ {\isacharparenleft}{\kern0pt}policy{\isacharunderscore}{\kern0pt}eval\ d{\isacharparenright}{\kern0pt}\ s{\isachardoublequoteclose}\ \isakeyword{for}\ s\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isacartoucheopen}policy{\isacharunderscore}{\kern0pt}eval\ d\ {\isacharequal}{\kern0pt}\ L\ {\isacharparenleft}{\kern0pt}mk{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\ d{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}policy{\isacharunderscore}{\kern0pt}eval\ {\isacharparenleft}{\kern0pt}policy{\isacharunderscore}{\kern0pt}step\ d{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacartoucheclose}\ assms{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}\ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max\ {\isacharparenleft}{\kern0pt}{\isasymlambda}a{\isachardot}{\kern0pt}\ L\isactrlsub a\ a\ {\isacharparenleft}{\kern0pt}{\isasymnu}\isactrlsub b\ {\isacharparenleft}{\kern0pt}mk{\isacharunderscore}{\kern0pt}stationary\ {\isacharparenleft}{\kern0pt}mk{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\ d{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ s{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymlambda}a{\isachardot}{\kern0pt}\ a\ {\isasymin}\ A\ s{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}d\ s{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{for}\ s\isanewline
-\ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ L{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}L\isactrlsub a{\isacharunderscore}{\kern0pt}det\isanewline
-\ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ policy{\isacharunderscore}{\kern0pt}eval{\isacharunderscore}{\kern0pt}def\ {\isasymL}\isactrlsub b{\isachardot}{\kern0pt}rep{\isacharunderscore}{\kern0pt}eq\ {\isasymL}{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}SUP{\isacharunderscore}{\kern0pt}det\ SUP{\isacharunderscore}{\kern0pt}step{\isacharunderscore}{\kern0pt}det{\isacharunderscore}{\kern0pt}eq\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assms{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}\ is{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det{\isacharunderscore}{\kern0pt}def\ L\isactrlsub a{\isacharunderscore}{\kern0pt}le\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp\ del{\isacharcolon}{\kern0pt}\ {\isasymnu}\isactrlsub b{\isachardot}{\kern0pt}rep{\isacharunderscore}{\kern0pt}eq\ simp{\isacharcolon}{\kern0pt}\ {\isasymnu}\isactrlsub b{\isachardot}{\kern0pt}rep{\isacharunderscore}{\kern0pt}eq{\isacharbrackleft}{\kern0pt}symmetric{\isacharbrackright}{\kern0pt}\ \isanewline
-\ \ \ \ \ \ \ \ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ SUP{\isacharunderscore}{\kern0pt}is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max\ boundedI{\isacharbrackleft}{\kern0pt}of\ {\isacharunderscore}{\kern0pt}\ {\isachardoublequoteopen}r\isactrlsub M\ {\isacharplus}{\kern0pt}\ l\ {\isacharasterisk}{\kern0pt}\ norm\ {\isacharunderscore}{\kern0pt}{\isachardoublequoteclose}{\isacharbrackright}{\kern0pt}\ bounded{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}bdd{\isacharunderscore}{\kern0pt}above{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ policy{\isacharunderscore}{\kern0pt}eval{\isacharunderscore}{\kern0pt}def\ policy{\isacharunderscore}{\kern0pt}step{\isacharunderscore}{\kern0pt}def\ policy{\isacharunderscore}{\kern0pt}improvement{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\begin{isamarkuptext}%
-We are now ready to prove termination in the context of finite state-action spaces.
-Intuitively, the algorithm terminates as there are only finitely many decision rules,
-and in each recursive call the value of the decision rule increases.%
-\end{isamarkuptext}\isamarkuptrue%
-\isacommand{termination}\isamarkupfalse%
-\ policy{\isacharunderscore}{\kern0pt}iteration\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}relation\ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}{\isacharparenleft}{\kern0pt}u{\isacharcomma}{\kern0pt}\ v{\isacharparenright}{\kern0pt}{\isachardot}{\kern0pt}\ u\ {\isasymin}\ D\isactrlsub D\ {\isasymand}\ v\ {\isasymin}\ D\isactrlsub D\ {\isasymand}\ {\isasymnu}\isactrlsub b\ {\isacharparenleft}{\kern0pt}mk{\isacharunderscore}{\kern0pt}stationary{\isacharunderscore}{\kern0pt}det\ u{\isacharparenright}{\kern0pt}\ {\isachargreater}{\kern0pt}\ {\isasymnu}\isactrlsub b\ {\isacharparenleft}{\kern0pt}mk{\isacharunderscore}{\kern0pt}stationary{\isacharunderscore}{\kern0pt}det\ v{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}wf\ {\isacharbraceleft}{\kern0pt}{\isacharparenleft}{\kern0pt}u{\isacharcomma}{\kern0pt}\ v{\isacharparenright}{\kern0pt}{\isachardot}{\kern0pt}\ u\ {\isasymin}\ D\isactrlsub D\ {\isasymand}\ v\ {\isasymin}\ D\isactrlsub D\ {\isasymand}\ {\isasymnu}\isactrlsub b\ {\isacharparenleft}{\kern0pt}mk{\isacharunderscore}{\kern0pt}stationary{\isacharunderscore}{\kern0pt}det\ v{\isacharparenright}{\kern0pt}\ {\isacharless}{\kern0pt}\ {\isasymnu}\isactrlsub b\ {\isacharparenleft}{\kern0pt}mk{\isacharunderscore}{\kern0pt}stationary{\isacharunderscore}{\kern0pt}det\ u{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ finite{\isacharunderscore}{\kern0pt}rel\ \isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}acyclic{\isacharunderscore}{\kern0pt}wf\ acyclicI{\isacharunderscore}{\kern0pt}order{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\ d\ x\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\ h{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isacharequal}{\kern0pt}\ policy{\isacharunderscore}{\kern0pt}step\ d{\isachardoublequoteclose}\ {\isachardoublequoteopen}{\isasymnot}\ {\isacharparenleft}{\kern0pt}d\ {\isacharequal}{\kern0pt}\ x\ {\isasymor}\ {\isasymnot}\ is{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\ d{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}is{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\ d\ {\isasymLongrightarrow}\ {\isasymnu}\isactrlsub b\ {\isacharparenleft}{\kern0pt}mk{\isacharunderscore}{\kern0pt}stationary{\isacharunderscore}{\kern0pt}det\ d{\isacharparenright}{\kern0pt}\ {\isasymle}\ {\isasymnu}\isactrlsub b\ {\isacharparenleft}{\kern0pt}mk{\isacharunderscore}{\kern0pt}stationary{\isacharunderscore}{\kern0pt}det\ {\isacharparenleft}{\kern0pt}policy{\isacharunderscore}{\kern0pt}step\ d{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ policy{\isacharunderscore}{\kern0pt}eval{\isacharunderscore}{\kern0pt}mon\ \ \isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ policy{\isacharunderscore}{\kern0pt}eval{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}is{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\ d\ {\isasymLongrightarrow}\ d\ {\isasymnoteq}\ policy{\isacharunderscore}{\kern0pt}step\ d\ {\isasymLongrightarrow}\isanewline
-\ \ \ \ {\isasymnu}\isactrlsub b\ {\isacharparenleft}{\kern0pt}mk{\isacharunderscore}{\kern0pt}stationary{\isacharunderscore}{\kern0pt}det\ d{\isacharparenright}{\kern0pt}\ {\isacharless}{\kern0pt}\ {\isasymnu}\isactrlsub b\ {\isacharparenleft}{\kern0pt}mk{\isacharunderscore}{\kern0pt}stationary{\isacharunderscore}{\kern0pt}det\ {\isacharparenleft}{\kern0pt}policy{\isacharunderscore}{\kern0pt}step\ d{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ eval{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}eq\ policy{\isacharunderscore}{\kern0pt}eval{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}force\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ order{\isachardot}{\kern0pt}not{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}order{\isacharunderscore}{\kern0pt}implies{\isacharunderscore}{\kern0pt}strict{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ d{\isacharparenright}{\kern0pt}\ {\isasymin}\ {\isacharbraceleft}{\kern0pt}{\isacharparenleft}{\kern0pt}u{\isacharcomma}{\kern0pt}\ v{\isacharparenright}{\kern0pt}{\isachardot}{\kern0pt}\ u\ {\isasymin}\ D\isactrlsub D\ {\isasymand}\ v\ {\isasymin}\ D\isactrlsub D\ {\isasymand}\ {\isasymnu}\isactrlsub b\ {\isacharparenleft}{\kern0pt}mk{\isacharunderscore}{\kern0pt}stationary{\isacharunderscore}{\kern0pt}det\ v{\isacharparenright}{\kern0pt}\ {\isacharless}{\kern0pt}\ {\isasymnu}\isactrlsub b\ {\isacharparenleft}{\kern0pt}mk{\isacharunderscore}{\kern0pt}stationary{\isacharunderscore}{\kern0pt}det\ u{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ is{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det{\isacharunderscore}{\kern0pt}pi\ policy{\isacharunderscore}{\kern0pt}step{\isacharunderscore}{\kern0pt}def\ h\ \isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\begin{isamarkuptext}%
-The termination proof gives us access to the induction rule/simplification lemmas associated 
-with the \isa{policy{\isacharunderscore}{\kern0pt}iteration} definition.
-Thus we can prove that the algorithm finds an optimal policy.%
-\end{isamarkuptext}\isamarkuptrue%
-\isacommand{lemma}\isamarkupfalse%
-\ is{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det{\isacharunderscore}{\kern0pt}pi{\isacharprime}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}d\ {\isasymin}\ D\isactrlsub D\ {\isasymLongrightarrow}\ is{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\ {\isacharparenleft}{\kern0pt}policy{\isacharunderscore}{\kern0pt}iteration\ d{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ is{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det{\isacharunderscore}{\kern0pt}pi\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}induction\ d\ rule{\isacharcolon}{\kern0pt}\ policy{\isacharunderscore}{\kern0pt}iteration{\isachardot}{\kern0pt}induct{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ Let{\isacharunderscore}{\kern0pt}def\ policy{\isacharunderscore}{\kern0pt}step{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ pi{\isacharunderscore}{\kern0pt}pi{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}d\ {\isasymin}\ D\isactrlsub D\ {\isasymLongrightarrow}\ policy{\isacharunderscore}{\kern0pt}step\ {\isacharparenleft}{\kern0pt}policy{\isacharunderscore}{\kern0pt}iteration\ d{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ policy{\isacharunderscore}{\kern0pt}iteration\ d{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ is{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det{\isacharunderscore}{\kern0pt}pi\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}induction\ d\ rule{\isacharcolon}{\kern0pt}\ policy{\isacharunderscore}{\kern0pt}iteration{\isachardot}{\kern0pt}induct{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ policy{\isacharunderscore}{\kern0pt}step{\isacharunderscore}{\kern0pt}def\ Let{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ policy{\isacharunderscore}{\kern0pt}iteration{\isacharunderscore}{\kern0pt}correct{\isacharcolon}{\kern0pt}\ \isanewline
-\ \ {\isachardoublequoteopen}d\ {\isasymin}\ D\isactrlsub D\ {\isasymLongrightarrow}\ {\isasymnu}\isactrlsub b\ {\isacharparenleft}{\kern0pt}mk{\isacharunderscore}{\kern0pt}stationary{\isacharunderscore}{\kern0pt}det\ {\isacharparenleft}{\kern0pt}policy{\isacharunderscore}{\kern0pt}iteration\ d{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt{\isachardoublequoteclose}\ \isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}induction\ d\ rule{\isacharcolon}{\kern0pt}\ policy{\isacharunderscore}{\kern0pt}iteration{\isachardot}{\kern0pt}induct{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ {\isacharparenleft}{\kern0pt}fastforce\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ policy{\isacharunderscore}{\kern0pt}step{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}opt\ is{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det{\isacharunderscore}{\kern0pt}pi{\isacharprime}{\kern0pt}\ simp\ del{\isacharcolon}{\kern0pt}\ policy{\isacharunderscore}{\kern0pt}iteration{\isachardot}{\kern0pt}simps{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isacommand{end}\isamarkupfalse%
-\isanewline
-\isanewline
-\isacommand{context}\isamarkupfalse%
-\ MDP{\isacharunderscore}{\kern0pt}finite{\isacharunderscore}{\kern0pt}type\ \isakeyword{begin}%
-\begin{isamarkuptext}%
-The following proofs concern code generation, i.e. how to represent \isa{{\isasymP}\isactrlsub {\isadigit{1}}} as a matrix.%
-\end{isamarkuptext}\isamarkuptrue%
-\isacommand{sublocale}\isamarkupfalse%
-\ MDP{\isacharunderscore}{\kern0pt}att{\isacharunderscore}{\kern0pt}{\isasymL}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ A{\isacharunderscore}{\kern0pt}ne\ finite{\isacharunderscore}{\kern0pt}is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max\ MDP{\isacharunderscore}{\kern0pt}att{\isacharunderscore}{\kern0pt}{\isasymL}{\isacharunderscore}{\kern0pt}def\ MDP{\isacharunderscore}{\kern0pt}att{\isacharunderscore}{\kern0pt}{\isasymL}{\isacharunderscore}{\kern0pt}axioms{\isacharunderscore}{\kern0pt}def\ max{\isacharunderscore}{\kern0pt}L{\isacharunderscore}{\kern0pt}ex{\isacharunderscore}{\kern0pt}def\ \isanewline
-\ \ \ \ \ \ has{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}def\ MDP{\isacharunderscore}{\kern0pt}reward{\isacharunderscore}{\kern0pt}axioms{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\ \isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ {\isachardoublequoteopen}fun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ f\ {\isacharequal}{\kern0pt}\ matrix\ {\isacharparenleft}{\kern0pt}{\isasymlambda}v{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymchi}\ j{\isachardot}{\kern0pt}\ f\ {\isacharparenleft}{\kern0pt}vec{\isacharunderscore}{\kern0pt}nth\ v{\isacharparenright}{\kern0pt}\ j{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ {\isachardoublequoteopen}Ek{\isacharunderscore}{\kern0pt}mat\ d\ {\isacharequal}{\kern0pt}\ fun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ {\isacharparenleft}{\kern0pt}{\isasymlambda}v{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}{\isasymP}\isactrlsub {\isadigit{1}}\ d{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}Bfun\ v{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ {\isachardoublequoteopen}nu{\isacharunderscore}{\kern0pt}inv{\isacharunderscore}{\kern0pt}mat\ d\ {\isacharequal}{\kern0pt}\ fun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}{\isasymlambda}v{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}id{\isacharunderscore}{\kern0pt}blinfun\ {\isacharminus}{\kern0pt}\ l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isasymP}\isactrlsub {\isadigit{1}}\ d{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}Bfun\ v{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ {\isachardoublequoteopen}nu{\isacharunderscore}{\kern0pt}mat\ d\ {\isacharequal}{\kern0pt}\ fun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ {\isacharparenleft}{\kern0pt}{\isasymlambda}v{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}{\isasymSum}i{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isasymP}\isactrlsub {\isadigit{1}}\ d{\isacharparenright}{\kern0pt}\ {\isacharcircum}{\kern0pt}{\isacharcircum}{\kern0pt}\ i{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}Bfun\ v{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ apply{\isacharunderscore}{\kern0pt}nu{\isacharunderscore}{\kern0pt}inv{\isacharunderscore}{\kern0pt}mat{\isacharcolon}{\kern0pt}\ \isanewline
-\ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}id{\isacharunderscore}{\kern0pt}blinfun\ {\isacharminus}{\kern0pt}\ l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isasymP}\isactrlsub {\isadigit{1}}\ d{\isacharparenright}{\kern0pt}\ v\ {\isacharequal}{\kern0pt}\ Bfun\ {\isacharparenleft}{\kern0pt}{\isasymlambda}i{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}nu{\isacharunderscore}{\kern0pt}inv{\isacharunderscore}{\kern0pt}mat\ d{\isacharparenright}{\kern0pt}\ {\isacharasterisk}{\kern0pt}v\ {\isacharparenleft}{\kern0pt}vec{\isacharunderscore}{\kern0pt}lambda\ v{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isachardollar}{\kern0pt}\ i{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ eq{\isacharunderscore}{\kern0pt}onpI{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}P\ x\ {\isasymLongrightarrow}\ eq{\isacharunderscore}{\kern0pt}onp\ P\ x\ x{\isachardoublequoteclose}\ \isakeyword{for}\ P\ x\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ eq{\isacharunderscore}{\kern0pt}onp{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}Real{\isacharunderscore}{\kern0pt}Vector{\isacharunderscore}{\kern0pt}Spaces{\isachardot}{\kern0pt}linear\ {\isacharparenleft}{\kern0pt}{\isasymlambda}v{\isachardot}{\kern0pt}\ vec{\isacharunderscore}{\kern0pt}lambda\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}id{\isacharunderscore}{\kern0pt}blinfun\ {\isacharminus}{\kern0pt}\ l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isasymP}\isactrlsub {\isadigit{1}}\ d{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}bfun{\isachardot}{\kern0pt}Bfun\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}{\isachardollar}{\kern0pt}{\isacharparenright}{\kern0pt}\ v{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp\ del{\isacharcolon}{\kern0pt}\ real{\isacharunderscore}{\kern0pt}scaleR{\isacharunderscore}{\kern0pt}def\ intro{\isacharcolon}{\kern0pt}\ linearI\isanewline
-\ \ \ \ \ \ \ \ simp{\isacharcolon}{\kern0pt}\ scaleR{\isacharunderscore}{\kern0pt}vec{\isacharunderscore}{\kern0pt}def\ eq{\isacharunderscore}{\kern0pt}onpI\ plus{\isacharunderscore}{\kern0pt}vec{\isacharunderscore}{\kern0pt}def\ vec{\isacharunderscore}{\kern0pt}lambda{\isacharunderscore}{\kern0pt}inverse\ plus{\isacharunderscore}{\kern0pt}bfun{\isachardot}{\kern0pt}abs{\isacharunderscore}{\kern0pt}eq{\isacharbrackleft}{\kern0pt}symmetric{\isacharbrackright}{\kern0pt}\ \isanewline
-\ \ \ \ \ \ \ \ scaleR{\isacharunderscore}{\kern0pt}bfun{\isachardot}{\kern0pt}abs{\isacharunderscore}{\kern0pt}eq{\isacharbrackleft}{\kern0pt}symmetric{\isacharbrackright}{\kern0pt}\ blinfun{\isachardot}{\kern0pt}scaleR{\isacharunderscore}{\kern0pt}right\ blinfun{\isachardot}{\kern0pt}add{\isacharunderscore}{\kern0pt}right{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ Ek{\isacharunderscore}{\kern0pt}mat{\isacharunderscore}{\kern0pt}def\ fun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}def\ nu{\isacharunderscore}{\kern0pt}inv{\isacharunderscore}{\kern0pt}mat{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ apply{\isacharunderscore}{\kern0pt}bfun{\isacharunderscore}{\kern0pt}inverse\ vec{\isacharunderscore}{\kern0pt}lambda{\isacharunderscore}{\kern0pt}inverse{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ bounded{\isacharunderscore}{\kern0pt}linear{\isacharunderscore}{\kern0pt}vec{\isacharunderscore}{\kern0pt}lambda{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}bounded{\isacharunderscore}{\kern0pt}linear\ {\isacharparenleft}{\kern0pt}{\isasymlambda}x{\isachardot}{\kern0pt}\ vec{\isacharunderscore}{\kern0pt}lambda\ {\isacharparenleft}{\kern0pt}x\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isacharprime}{\kern0pt}s\ {\isasymRightarrow}\isactrlsub b\ real{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}intro\ bounded{\isacharunderscore}{\kern0pt}linear{\isacharunderscore}{\kern0pt}intro{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\ x\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}s\ {\isasymRightarrow}\isactrlsub b\ real{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}sqrt\ {\isacharparenleft}{\kern0pt}{\isasymSum}\ i\ {\isasymin}\ UNIV\ {\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}apply{\isacharunderscore}{\kern0pt}bfun\ x\ i{\isacharparenright}{\kern0pt}\isactrlsup {\isadigit{2}}{\isacharparenright}{\kern0pt}\ {\isasymle}\ {\isacharparenleft}{\kern0pt}{\isasymSum}\ i\ {\isasymin}\ UNIV\ {\isachardot}{\kern0pt}\ {\isasymbar}{\isacharparenleft}{\kern0pt}apply{\isacharunderscore}{\kern0pt}bfun\ x\ i{\isacharparenright}{\kern0pt}{\isasymbar}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ L{\isadigit{2}}{\isacharunderscore}{\kern0pt}set{\isacharunderscore}{\kern0pt}le{\isacharunderscore}{\kern0pt}sum{\isacharunderscore}{\kern0pt}abs\ \isanewline
-\ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ L{\isadigit{2}}{\isacharunderscore}{\kern0pt}set{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isasymSum}\ i\ {\isasymin}\ UNIV\ {\isachardot}{\kern0pt}\ {\isasymbar}{\isacharparenleft}{\kern0pt}apply{\isacharunderscore}{\kern0pt}bfun\ x\ i{\isacharparenright}{\kern0pt}{\isasymbar}{\isacharparenright}{\kern0pt}\ {\isasymle}\ {\isacharparenleft}{\kern0pt}card\ {\isacharparenleft}{\kern0pt}UNIV\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isacharprime}{\kern0pt}s\ set{\isacharparenright}{\kern0pt}\ {\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSqunion}xa{\isachardot}{\kern0pt}\ {\isasymbar}apply{\isacharunderscore}{\kern0pt}bfun\ x\ xa{\isasymbar}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ cSup{\isacharunderscore}{\kern0pt}upper\ sum{\isacharunderscore}{\kern0pt}bounded{\isacharunderscore}{\kern0pt}above{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{finally}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}norm\ {\isacharparenleft}{\kern0pt}vec{\isacharunderscore}{\kern0pt}lambda\ {\isacharparenleft}{\kern0pt}apply{\isacharunderscore}{\kern0pt}bfun\ x{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymle}\ norm\ x\ {\isacharasterisk}{\kern0pt}\ CARD{\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}s{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ norm{\isacharunderscore}{\kern0pt}vec{\isacharunderscore}{\kern0pt}def\ norm{\isacharunderscore}{\kern0pt}bfun{\isacharunderscore}{\kern0pt}def\ dist{\isacharunderscore}{\kern0pt}bfun{\isacharunderscore}{\kern0pt}def\ L{\isadigit{2}}{\isacharunderscore}{\kern0pt}set{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp\ add{\isacharcolon}{\kern0pt}\ mult{\isachardot}{\kern0pt}commute{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ plus{\isacharunderscore}{\kern0pt}vec{\isacharunderscore}{\kern0pt}def\ scaleR{\isacharunderscore}{\kern0pt}vec{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ bounded{\isacharunderscore}{\kern0pt}linear{\isacharunderscore}{\kern0pt}vec{\isacharunderscore}{\kern0pt}lambda{\isacharunderscore}{\kern0pt}blinfun{\isacharcolon}{\kern0pt}\ \isanewline
-\ \ \isakeyword{fixes}\ f\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}s\ {\isasymRightarrow}\isactrlsub b\ real{\isacharparenright}{\kern0pt}\ {\isasymRightarrow}\isactrlsub L\ {\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}s\ {\isasymRightarrow}\isactrlsub b\ real{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}bounded{\isacharunderscore}{\kern0pt}linear\ {\isacharparenleft}{\kern0pt}{\isasymlambda}v{\isachardot}{\kern0pt}\ vec{\isacharunderscore}{\kern0pt}lambda\ {\isacharparenleft}{\kern0pt}apply{\isacharunderscore}{\kern0pt}bfun\ {\isacharparenleft}{\kern0pt}blinfun{\isacharunderscore}{\kern0pt}apply\ f\ {\isacharparenleft}{\kern0pt}bfun{\isachardot}{\kern0pt}Bfun\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}{\isachardollar}{\kern0pt}{\isacharparenright}{\kern0pt}\ v{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ blinfun{\isachardot}{\kern0pt}bounded{\isacharunderscore}{\kern0pt}linear{\isacharunderscore}{\kern0pt}right\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}fastforce\ intro{\isacharcolon}{\kern0pt}\ bounded{\isacharunderscore}{\kern0pt}linear{\isacharunderscore}{\kern0pt}compose{\isacharbrackleft}{\kern0pt}OF\ bounded{\isacharunderscore}{\kern0pt}linear{\isacharunderscore}{\kern0pt}vec{\isacharunderscore}{\kern0pt}lambda{\isacharbrackright}{\kern0pt}\ \isanewline
-\ \ \ \ \ \ bounded{\isacharunderscore}{\kern0pt}linear{\isacharunderscore}{\kern0pt}bfun{\isacharunderscore}{\kern0pt}nth\ bounded{\isacharunderscore}{\kern0pt}linear{\isacharunderscore}{\kern0pt}compose{\isacharbrackleft}{\kern0pt}of\ f{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ invertible{\isacharunderscore}{\kern0pt}nu{\isacharunderscore}{\kern0pt}inv{\isacharunderscore}{\kern0pt}max{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}invertible\ {\isacharparenleft}{\kern0pt}nu{\isacharunderscore}{\kern0pt}inv{\isacharunderscore}{\kern0pt}mat\ d{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ nu{\isacharunderscore}{\kern0pt}inv{\isacharunderscore}{\kern0pt}mat{\isacharunderscore}{\kern0pt}def\ fun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ matrix{\isacharunderscore}{\kern0pt}invertible\ inv{\isacharunderscore}{\kern0pt}norm{\isacharunderscore}{\kern0pt}le{\isacharprime}{\kern0pt}\ vec{\isacharunderscore}{\kern0pt}lambda{\isacharunderscore}{\kern0pt}inverse\ apply{\isacharunderscore}{\kern0pt}bfun{\isacharunderscore}{\kern0pt}inverse\ \isanewline
-\ \ \ \ \ \ bounded{\isacharunderscore}{\kern0pt}linear{\isachardot}{\kern0pt}linear{\isacharbrackleft}{\kern0pt}OF\ bounded{\isacharunderscore}{\kern0pt}linear{\isacharunderscore}{\kern0pt}vec{\isacharunderscore}{\kern0pt}lambda{\isacharunderscore}{\kern0pt}blinfun{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \ \ \ \ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ exI{\isacharbrackleft}{\kern0pt}of\ {\isacharunderscore}{\kern0pt}\ {\isachardoublequoteopen}{\isasymlambda}v{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymchi}\ j{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymlambda}v{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSum}i{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isasymP}\isactrlsub {\isadigit{1}}\ d{\isacharparenright}{\kern0pt}\ {\isacharcircum}{\kern0pt}{\isacharcircum}{\kern0pt}\ i{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}Bfun\ v{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}vec{\isacharunderscore}{\kern0pt}nth\ v{\isacharparenright}{\kern0pt}\ j{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{end}\isamarkupfalse%
-\isanewline
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ {\isachardoublequoteopen}least{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max\ f\ P\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}LEAST\ x{\isachardot}{\kern0pt}\ is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max\ f\ P\ x{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{locale}\isamarkupfalse%
-\ MDP{\isacharunderscore}{\kern0pt}ord\ {\isacharequal}{\kern0pt}\ MDP{\isacharunderscore}{\kern0pt}finite{\isacharunderscore}{\kern0pt}type\ A\ K\ r\ l\isanewline
-\ \ \isakeyword{for}\ A\ \isakeyword{and}\isanewline
-\ \ \ \ K\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}s\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}finite{\isacharcomma}{\kern0pt}\ wellorder{\isacharbraceright}{\kern0pt}\ {\isasymtimes}\ {\isacharprime}{\kern0pt}a\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}finite{\isacharcomma}{\kern0pt}\ wellorder{\isacharbraceright}{\kern0pt}\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}s\ pmf{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isakeyword{and}\ r\ l\isanewline
-\isakeyword{begin}\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ {\isasymL}{\isacharunderscore}{\kern0pt}fin{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}det{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymL}\ v\ s\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSqunion}a\ {\isasymin}\ A\ s{\isachardot}{\kern0pt}\ L\isactrlsub a\ a\ v\ s{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ SUP{\isacharunderscore}{\kern0pt}step{\isacharunderscore}{\kern0pt}det{\isacharunderscore}{\kern0pt}eq\ {\isasymL}{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}SUP{\isacharunderscore}{\kern0pt}det{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ {\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}fin{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}det{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymL}\isactrlsub b\ v\ s\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSqunion}a\ {\isasymin}\ A\ s{\isachardot}{\kern0pt}\ L\isactrlsub a\ a\ v\ s{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ SUP{\isacharunderscore}{\kern0pt}step{\isacharunderscore}{\kern0pt}det{\isacharunderscore}{\kern0pt}eq\ {\isasymL}\isactrlsub b{\isachardot}{\kern0pt}rep{\isacharunderscore}{\kern0pt}eq\ {\isasymL}{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}SUP{\isacharunderscore}{\kern0pt}det{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{sublocale}\isamarkupfalse%
-\ MDP{\isacharunderscore}{\kern0pt}PI{\isacharunderscore}{\kern0pt}finite\ A\ K\ r\ l\ {\isachardoublequoteopen}{\isasymlambda}X{\isachardot}{\kern0pt}\ Least\ {\isacharparenleft}{\kern0pt}{\isasymlambda}x{\isachardot}{\kern0pt}\ x\ {\isasymin}\ X{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ unfold{\isacharunderscore}{\kern0pt}locales\ {\isacharparenleft}{\kern0pt}auto\ intro{\isacharcolon}{\kern0pt}\ LeastI{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{end}\isamarkupfalse%
-\isanewline
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{end}\isamarkupfalse%
-%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\end{isabellebody}%
-\endinput
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diff --git a/thys/MDP-Algorithms/output/outline/Splitting_Methods.tex b/thys/MDP-Algorithms/output/outline/Splitting_Methods.tex
deleted file mode 100644
--- a/thys/MDP-Algorithms/output/outline/Splitting_Methods.tex
+++ /dev/null
@@ -1,9722 +0,0 @@
-%
-\begin{isabellebody}%
-\setisabellecontext{Splitting{\isacharunderscore}{\kern0pt}Methods}%
-%
-\isadelimtheory
-\isanewline
-\isanewline
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{theory}\isamarkupfalse%
-\ Splitting{\isacharunderscore}{\kern0pt}Methods\isanewline
-\ \ \isakeyword{imports}\ \isanewline
-\ \ \ \ Blinfun{\isacharunderscore}{\kern0pt}Matrix\isanewline
-\ \ \ \ Value{\isacharunderscore}{\kern0pt}Iteration\isanewline
-\ \ \ \ Policy{\isacharunderscore}{\kern0pt}Iteration\isanewline
-\isakeyword{begin}%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsection{Value Iteration using Splitting Methods%
-}
-\isamarkuptrue%
-%
-\isamarkupsubsection{Regular Splittings for Matrices and Bounded Linear Functions%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{definition}\isamarkupfalse%
-\ {\isachardoublequoteopen}is{\isacharunderscore}{\kern0pt}splitting{\isacharunderscore}{\kern0pt}mat\ X\ Q\ R\ {\isasymlongleftrightarrow}\isanewline
-\ \ X\ {\isacharequal}{\kern0pt}\ Q\ {\isacharminus}{\kern0pt}\ R\ {\isasymand}\ invertible\ Q\ {\isasymand}\ {\isadigit{0}}\ {\isasymle}\ matrix{\isacharunderscore}{\kern0pt}inv\ Q\ {\isasymand}\ {\isadigit{0}}\ {\isasymle}\ R{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ {\isachardoublequoteopen}is{\isacharunderscore}{\kern0pt}splitting{\isacharunderscore}{\kern0pt}blin\ X\ Q\ R\ {\isasymlongleftrightarrow}\ is{\isacharunderscore}{\kern0pt}splitting{\isacharunderscore}{\kern0pt}mat\ {\isacharparenleft}{\kern0pt}blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ X{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ Q{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ R{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ is{\isacharunderscore}{\kern0pt}splitting{\isacharunderscore}{\kern0pt}blin{\isacharunderscore}{\kern0pt}def{\isacharprime}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}is{\isacharunderscore}{\kern0pt}splitting{\isacharunderscore}{\kern0pt}blin\ X\ Q\ R\ {\isasymlongleftrightarrow}\isanewline
-\ \ X\ {\isacharequal}{\kern0pt}\ Q\ {\isacharminus}{\kern0pt}\ R\ {\isasymand}\ invertible\isactrlsub L\ Q\ {\isasymand}\ nonneg{\isacharunderscore}{\kern0pt}blinfun\ {\isacharparenleft}{\kern0pt}inv\isactrlsub L\ Q{\isacharparenright}{\kern0pt}\ {\isasymand}\ nonneg{\isacharunderscore}{\kern0pt}blinfun\ R{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ X\ {\isacharequal}{\kern0pt}\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ Q\ {\isacharminus}{\kern0pt}\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ R\ {\isasymlongleftrightarrow}\ X\ {\isacharequal}{\kern0pt}\ Q\ {\isacharminus}{\kern0pt}\ R{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ \ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}diff\ matrix{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}blinfun{\isacharunderscore}{\kern0pt}inv\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ is{\isacharunderscore}{\kern0pt}splitting{\isacharunderscore}{\kern0pt}blin{\isacharunderscore}{\kern0pt}def\ is{\isacharunderscore}{\kern0pt}splitting{\isacharunderscore}{\kern0pt}mat{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}inverse{\isacharbrackleft}{\kern0pt}of\ Q{\isacharbrackright}{\kern0pt}\ matrix{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}blinfun{\isacharunderscore}{\kern0pt}inv\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}fastforce\ simp{\isacharcolon}{\kern0pt}\ invertible{\isacharunderscore}{\kern0pt}invertible\isactrlsub L{\isacharunderscore}{\kern0pt}I{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ is{\isacharunderscore}{\kern0pt}splitting{\isacharunderscore}{\kern0pt}blinD{\isacharbrackleft}{\kern0pt}dest{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\ \isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}is{\isacharunderscore}{\kern0pt}splitting{\isacharunderscore}{\kern0pt}blin\ X\ Q\ R{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}X\ {\isacharequal}{\kern0pt}\ Q\ {\isacharminus}{\kern0pt}\ R{\isachardoublequoteclose}\ {\isachardoublequoteopen}invertible\isactrlsub L\ Q{\isachardoublequoteclose}\ {\isachardoublequoteopen}nonneg{\isacharunderscore}{\kern0pt}blinfun\ {\isacharparenleft}{\kern0pt}inv\isactrlsub L\ Q{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ {\isachardoublequoteopen}nonneg{\isacharunderscore}{\kern0pt}blinfun\ R{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ is{\isacharunderscore}{\kern0pt}splitting{\isacharunderscore}{\kern0pt}blin{\isacharunderscore}{\kern0pt}def{\isacharprime}{\kern0pt}\ assms\ \isacommand{by}\isamarkupfalse%
-\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Splitting Methods for MDPs%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{locale}\isamarkupfalse%
-\ MDP{\isacharunderscore}{\kern0pt}QR\ {\isacharequal}{\kern0pt}\ MDP{\isacharunderscore}{\kern0pt}finite{\isacharunderscore}{\kern0pt}type\ A\ K\ r\ l\ \isanewline
-\ \ \isakeyword{for}\ A\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}s\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ finite\ {\isasymRightarrow}\ {\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}a\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ finite{\isacharparenright}{\kern0pt}\ set{\isachardoublequoteclose}\ \isanewline
-\ \ \ \ \isakeyword{and}\ K\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}s\ {\isasymtimes}\ {\isacharprime}{\kern0pt}a{\isacharparenright}{\kern0pt}\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}s\ pmf{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isakeyword{and}\ \ r\ l\ {\isacharplus}{\kern0pt}\isanewline
-\ \ \isakeyword{fixes}\ Q\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}s\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a{\isacharparenright}{\kern0pt}\ {\isasymRightarrow}\ {\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}s\ {\isasymRightarrow}\isactrlsub b\ real{\isacharparenright}{\kern0pt}\ {\isasymRightarrow}\isactrlsub L\ {\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}s\ {\isasymRightarrow}\isactrlsub b\ real{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{fixes}\ R\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}s\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a{\isacharparenright}{\kern0pt}\ {\isasymRightarrow}\ {\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}s\ {\isasymRightarrow}\isactrlsub b\ real{\isacharparenright}{\kern0pt}\ {\isasymRightarrow}\isactrlsub L\ {\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}s\ {\isasymRightarrow}\isactrlsub b\ real{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{assumes}\ is{\isacharunderscore}{\kern0pt}splitting{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymAnd}d{\isachardot}{\kern0pt}\ d\ {\isasymin}\ D\isactrlsub D\ {\isasymLongrightarrow}\ is{\isacharunderscore}{\kern0pt}splitting{\isacharunderscore}{\kern0pt}blin\ {\isacharparenleft}{\kern0pt}id{\isacharunderscore}{\kern0pt}blinfun\ {\isacharminus}{\kern0pt}\ l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isasymP}\isactrlsub {\isadigit{1}}\ {\isacharparenleft}{\kern0pt}mk{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\ d{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}Q\ d{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}R\ d{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{assumes}\ QR{\isacharunderscore}{\kern0pt}contraction{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isasymSqunion}d{\isasymin}D\isactrlsub D{\isachardot}{\kern0pt}\ norm\ {\isacharparenleft}{\kern0pt}inv\isactrlsub L\ {\isacharparenleft}{\kern0pt}Q\ d{\isacharparenright}{\kern0pt}\ o\isactrlsub L\ R\ d{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharless}{\kern0pt}\ {\isadigit{1}}{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{assumes}\ arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}ex{\isacharunderscore}{\kern0pt}split{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymexists}d{\isachardot}{\kern0pt}\ {\isasymforall}s{\isachardot}{\kern0pt}\ is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max\ {\isacharparenleft}{\kern0pt}{\isasymlambda}d{\isachardot}{\kern0pt}\ inv\isactrlsub L\ {\isacharparenleft}{\kern0pt}Q\ d{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}r{\isacharunderscore}{\kern0pt}dec\isactrlsub b\ {\isacharparenleft}{\kern0pt}mk{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\ d{\isacharparenright}{\kern0pt}\ {\isacharplus}{\kern0pt}\ R\ d\ v{\isacharparenright}{\kern0pt}\ s{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymlambda}d{\isachardot}{\kern0pt}\ d\ {\isasymin}\ D\isactrlsub D{\isacharparenright}{\kern0pt}\ d{\isachardoublequoteclose}\isanewline
-\isakeyword{begin}\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ inv{\isacharunderscore}{\kern0pt}Q{\isacharunderscore}{\kern0pt}mono{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}d\ {\isasymin}\ D\isactrlsub D\ {\isasymLongrightarrow}\ u\ {\isasymle}\ v\ {\isasymLongrightarrow}\ {\isacharparenleft}{\kern0pt}inv\isactrlsub L\ {\isacharparenleft}{\kern0pt}Q\ d{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ u\ {\isasymle}\ {\isacharparenleft}{\kern0pt}inv\isactrlsub L\ {\isacharparenleft}{\kern0pt}Q\ d{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ v{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ is{\isacharunderscore}{\kern0pt}splitting\ \isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ nonneg{\isacharunderscore}{\kern0pt}blinfun{\isacharunderscore}{\kern0pt}mono{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ splitting{\isacharunderscore}{\kern0pt}eq{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}d\ {\isasymin}\ D\isactrlsub D\ {\isasymLongrightarrow}\ Q\ d\ {\isacharminus}{\kern0pt}\ R\ d\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}id{\isacharunderscore}{\kern0pt}blinfun\ {\isacharminus}{\kern0pt}\ l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isasymP}\isactrlsub {\isadigit{1}}\ {\isacharparenleft}{\kern0pt}mk{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\ d{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ is{\isacharunderscore}{\kern0pt}splitting\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ fastforce%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ Q{\isacharunderscore}{\kern0pt}nonneg{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}d\ {\isasymin}\ D\isactrlsub D\ {\isasymLongrightarrow}\ {\isadigit{0}}\ {\isasymle}\ v\ {\isasymLongrightarrow}\ {\isadigit{0}}\ {\isasymle}\ inv\isactrlsub L\ {\isacharparenleft}{\kern0pt}Q\ d{\isacharparenright}{\kern0pt}\ v{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ is{\isacharunderscore}{\kern0pt}splitting\ nonneg{\isacharunderscore}{\kern0pt}blinfun{\isacharunderscore}{\kern0pt}nonneg\ \isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ Q{\isacharunderscore}{\kern0pt}invertible{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}d\ {\isasymin}\ D\isactrlsub D\ {\isasymLongrightarrow}\ invertible\isactrlsub L\ {\isacharparenleft}{\kern0pt}Q\ d{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ is{\isacharunderscore}{\kern0pt}splitting\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ R{\isacharunderscore}{\kern0pt}nonneg{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}d\ {\isasymin}\ D\isactrlsub D\ {\isasymLongrightarrow}\ {\isadigit{0}}\ {\isasymle}\ v\ {\isasymLongrightarrow}\ {\isadigit{0}}\ {\isasymle}\ R\ d\ v{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ is{\isacharunderscore}{\kern0pt}splitting{\isacharunderscore}{\kern0pt}blinD{\isacharbrackleft}{\kern0pt}OF\ is{\isacharunderscore}{\kern0pt}splitting{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}fastforce\ simp{\isacharcolon}{\kern0pt}\ nonneg{\isacharunderscore}{\kern0pt}blinfun{\isacharunderscore}{\kern0pt}nonneg\ intro{\isacharcolon}{\kern0pt}\ nonneg{\isacharunderscore}{\kern0pt}blinfun{\isacharunderscore}{\kern0pt}mono{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ R{\isacharunderscore}{\kern0pt}mono{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}d\ {\isasymin}\ D\isactrlsub D\ {\isasymLongrightarrow}\ u\ {\isasymle}\ v\ {\isasymLongrightarrow}\ {\isacharparenleft}{\kern0pt}R\ d{\isacharparenright}{\kern0pt}\ u\ {\isasymle}\ {\isacharparenleft}{\kern0pt}R\ d{\isacharparenright}{\kern0pt}\ v{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ R{\isacharunderscore}{\kern0pt}nonneg{\isacharbrackleft}{\kern0pt}of\ d\ {\isachardoublequoteopen}v\ {\isacharminus}{\kern0pt}\ u{\isachardoublequoteclose}{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ blinfun{\isachardot}{\kern0pt}bilinear{\isacharunderscore}{\kern0pt}simps{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ QR{\isacharunderscore}{\kern0pt}nonneg{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}d\ {\isasymin}\ D\isactrlsub D\ {\isasymLongrightarrow}\ {\isadigit{0}}\ {\isasymle}\ v\ {\isasymLongrightarrow}\ {\isadigit{0}}\ {\isasymle}\ {\isacharparenleft}{\kern0pt}inv\isactrlsub L\ {\isacharparenleft}{\kern0pt}Q\ d{\isacharparenright}{\kern0pt}\ o\isactrlsub L\ R\ d{\isacharparenright}{\kern0pt}\ v{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ Q{\isacharunderscore}{\kern0pt}nonneg\ R{\isacharunderscore}{\kern0pt}nonneg{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ QR{\isacharunderscore}{\kern0pt}mono{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}d\ {\isasymin}\ D\isactrlsub D\ {\isasymLongrightarrow}\ u\ {\isasymle}\ v\ {\isasymLongrightarrow}\ {\isacharparenleft}{\kern0pt}inv\isactrlsub L\ {\isacharparenleft}{\kern0pt}Q\ d{\isacharparenright}{\kern0pt}\ o\isactrlsub L\ R\ d{\isacharparenright}{\kern0pt}\ u\ {\isasymle}\ {\isacharparenleft}{\kern0pt}inv\isactrlsub L\ {\isacharparenleft}{\kern0pt}Q\ d{\isacharparenright}{\kern0pt}\ o\isactrlsub L\ R\ d{\isacharparenright}{\kern0pt}\ v{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ QR{\isacharunderscore}{\kern0pt}nonneg{\isacharbrackleft}{\kern0pt}of\ d\ {\isachardoublequoteopen}v\ {\isacharminus}{\kern0pt}\ u{\isachardoublequoteclose}{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ blinfun{\isachardot}{\kern0pt}bilinear{\isacharunderscore}{\kern0pt}simps{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ norm{\isacharunderscore}{\kern0pt}QR{\isacharunderscore}{\kern0pt}less{\isacharunderscore}{\kern0pt}one{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}d\ {\isasymin}\ D\isactrlsub D\ {\isasymLongrightarrow}\ norm\ {\isacharparenleft}{\kern0pt}inv\isactrlsub L\ {\isacharparenleft}{\kern0pt}Q\ d{\isacharparenright}{\kern0pt}\ o\isactrlsub L\ R\ d{\isacharparenright}{\kern0pt}\ {\isacharless}{\kern0pt}\ {\isadigit{1}}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ QR{\isacharunderscore}{\kern0pt}contraction\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ cSUP{\isacharunderscore}{\kern0pt}lessD{\isacharbrackleft}{\kern0pt}of\ {\isachardoublequoteopen}{\isasymlambda}d{\isachardot}{\kern0pt}\ norm\ {\isacharparenleft}{\kern0pt}inv\isactrlsub L\ {\isacharparenleft}{\kern0pt}Q\ d{\isacharparenright}{\kern0pt}\ o\isactrlsub L\ R\ d{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ splitting{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}d\ {\isasymin}\ D\isactrlsub D\ {\isasymLongrightarrow}\ id{\isacharunderscore}{\kern0pt}blinfun\ {\isacharminus}{\kern0pt}\ l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isasymP}\isactrlsub {\isadigit{1}}\ {\isacharparenleft}{\kern0pt}mk{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\ d{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ Q\ d\ {\isacharminus}{\kern0pt}\ R\ d{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ is{\isacharunderscore}{\kern0pt}splitting\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Discount Factor \isa{QR{\isacharunderscore}{\kern0pt}disc}%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{abbreviation}\isamarkupfalse%
-\ {\isachardoublequoteopen}QR{\isacharunderscore}{\kern0pt}disc\ {\isasymequiv}\ {\isacharparenleft}{\kern0pt}{\isasymSqunion}d\ {\isasymin}\ D\isactrlsub D{\isachardot}{\kern0pt}\ norm\ {\isacharparenleft}{\kern0pt}inv\isactrlsub L\ {\isacharparenleft}{\kern0pt}Q\ d{\isacharparenright}{\kern0pt}\ o\isactrlsub L\ R\ d{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ QR{\isacharunderscore}{\kern0pt}le{\isacharunderscore}{\kern0pt}QR{\isacharunderscore}{\kern0pt}disc{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}d\ {\isasymin}\ D\isactrlsub D\ {\isasymLongrightarrow}\ norm\ {\isacharparenleft}{\kern0pt}inv\isactrlsub L\ {\isacharparenleft}{\kern0pt}Q\ d{\isacharparenright}{\kern0pt}\ o\isactrlsub L\ {\isacharparenleft}{\kern0pt}R\ d{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymle}\ QR{\isacharunderscore}{\kern0pt}disc{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ intro{\isacharcolon}{\kern0pt}\ cSUP{\isacharunderscore}{\kern0pt}upper{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ a{\isacharunderscore}{\kern0pt}nonneg{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isadigit{0}}\ {\isasymle}\ QR{\isacharunderscore}{\kern0pt}disc{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ QR{\isacharunderscore}{\kern0pt}contraction\ norm{\isacharunderscore}{\kern0pt}ge{\isacharunderscore}{\kern0pt}zero\ ex{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}fastforce\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ cSUP{\isacharunderscore}{\kern0pt}upper{\isadigit{2}}{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Bellman-Operator%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{abbreviation}\isamarkupfalse%
-\ {\isachardoublequoteopen}L{\isacharunderscore}{\kern0pt}split\ d\ v\ {\isasymequiv}\ inv\isactrlsub L\ {\isacharparenleft}{\kern0pt}Q\ d{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}r{\isacharunderscore}{\kern0pt}dec\isactrlsub b\ {\isacharparenleft}{\kern0pt}mk{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\ d{\isacharparenright}{\kern0pt}\ {\isacharplus}{\kern0pt}\ R\ d\ v{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymL}{\isacharunderscore}{\kern0pt}split\ v\ s\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSqunion}d\ {\isasymin}\ D\isactrlsub D{\isachardot}{\kern0pt}\ L{\isacharunderscore}{\kern0pt}split\ d\ v\ s{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ {\isasymL}{\isacharunderscore}{\kern0pt}split{\isacharunderscore}{\kern0pt}bfun{\isacharunderscore}{\kern0pt}aux{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}d\ {\isasymin}\ D\isactrlsub D{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}norm\ {\isacharparenleft}{\kern0pt}L{\isacharunderscore}{\kern0pt}split\ d\ v{\isacharparenright}{\kern0pt}\ {\isasymle}\ {\isacharparenleft}{\kern0pt}{\isasymSqunion}d\ {\isasymin}\ D\isactrlsub D{\isachardot}{\kern0pt}\ norm\ {\isacharparenleft}{\kern0pt}inv\isactrlsub L\ {\isacharparenleft}{\kern0pt}Q\ d{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharasterisk}{\kern0pt}\ r\isactrlsub M\ {\isacharplus}{\kern0pt}\ norm\ v{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}norm\ {\isacharparenleft}{\kern0pt}L{\isacharunderscore}{\kern0pt}split\ d\ v{\isacharparenright}{\kern0pt}\ {\isasymle}\ norm\ {\isacharparenleft}{\kern0pt}inv\isactrlsub L\ {\isacharparenleft}{\kern0pt}Q\ d{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}r{\isacharunderscore}{\kern0pt}dec\isactrlsub b\ {\isacharparenleft}{\kern0pt}mk{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\ d{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharplus}{\kern0pt}\ norm\ {\isacharparenleft}{\kern0pt}inv\isactrlsub L\ {\isacharparenleft}{\kern0pt}Q\ d{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}R\ d\ v{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ blinfun{\isachardot}{\kern0pt}add{\isacharunderscore}{\kern0pt}right\ norm{\isacharunderscore}{\kern0pt}triangle{\isacharunderscore}{\kern0pt}ineq{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isasymle}\ norm\ {\isacharparenleft}{\kern0pt}inv\isactrlsub L\ {\isacharparenleft}{\kern0pt}Q\ d{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}r{\isacharunderscore}{\kern0pt}dec\isactrlsub b\ {\isacharparenleft}{\kern0pt}mk{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\ d{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharplus}{\kern0pt}\ norm\ {\isacharparenleft}{\kern0pt}inv\isactrlsub L\ {\isacharparenleft}{\kern0pt}Q\ d{\isacharparenright}{\kern0pt}\ o\isactrlsub L\ R\ d{\isacharparenright}{\kern0pt}\ {\isacharasterisk}{\kern0pt}\ norm\ v{\isachardoublequoteclose}\ \ \ \ \isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ blinfun{\isacharunderscore}{\kern0pt}apply{\isacharunderscore}{\kern0pt}blinfun{\isacharunderscore}{\kern0pt}compose{\isacharbrackleft}{\kern0pt}symmetric{\isacharbrackright}{\kern0pt}\ norm{\isacharunderscore}{\kern0pt}blinfun\ simp\ del{\isacharcolon}{\kern0pt}\ blinfun{\isacharunderscore}{\kern0pt}apply{\isacharunderscore}{\kern0pt}blinfun{\isacharunderscore}{\kern0pt}compose{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isasymle}\ norm\ {\isacharparenleft}{\kern0pt}inv\isactrlsub L\ {\isacharparenleft}{\kern0pt}Q\ d{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}r{\isacharunderscore}{\kern0pt}dec\isactrlsub b\ {\isacharparenleft}{\kern0pt}mk{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\ d{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharplus}{\kern0pt}\ norm\ v{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ norm{\isacharunderscore}{\kern0pt}QR{\isacharunderscore}{\kern0pt}less{\isacharunderscore}{\kern0pt}one\ assms\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}fastforce\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ mult{\isacharunderscore}{\kern0pt}left{\isacharunderscore}{\kern0pt}le{\isacharunderscore}{\kern0pt}one{\isacharunderscore}{\kern0pt}le{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isasymle}\ norm\ {\isacharparenleft}{\kern0pt}inv\isactrlsub L\ {\isacharparenleft}{\kern0pt}Q\ d{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharasterisk}{\kern0pt}\ r\isactrlsub M\ {\isacharplus}{\kern0pt}\ norm\ v{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ order{\isachardot}{\kern0pt}trans{\isacharbrackleft}{\kern0pt}OF\ norm{\isacharunderscore}{\kern0pt}blinfun{\isacharbrackright}{\kern0pt}\ mult{\isacharunderscore}{\kern0pt}left{\isacharunderscore}{\kern0pt}mono\ simp{\isacharcolon}{\kern0pt}\ \ norm{\isacharunderscore}{\kern0pt}r{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}le{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isasymle}\ {\isacharparenleft}{\kern0pt}{\isasymSqunion}d\ {\isasymin}\ D\isactrlsub D{\isachardot}{\kern0pt}\ norm\ {\isacharparenleft}{\kern0pt}inv\isactrlsub L\ {\isacharparenleft}{\kern0pt}Q\ d{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharasterisk}{\kern0pt}\ r\isactrlsub M\ {\isacharplus}{\kern0pt}\ norm\ v{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ mult{\isacharunderscore}{\kern0pt}right{\isacharunderscore}{\kern0pt}mono\ cSUP{\isacharunderscore}{\kern0pt}upper\ assms\ simp{\isacharcolon}{\kern0pt}\ r\isactrlsub M{\isacharunderscore}{\kern0pt}nonneg{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{finally}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isacommand{{\isachardot}{\kern0pt}}\isamarkupfalse%
-\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lift{\isacharunderscore}{\kern0pt}definition}\isamarkupfalse%
-\ {\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}split\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}s\ {\isasymRightarrow}\isactrlsub b\ real{\isacharparenright}{\kern0pt}\ {\isasymRightarrow}\ {\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}s\ {\isasymRightarrow}\isactrlsub b\ real{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{is}\ {\isasymL}{\isacharunderscore}{\kern0pt}split\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ fastforce%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ {\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}split{\isacharunderscore}{\kern0pt}def{\isacharprime}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}split\ v\ s\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSqunion}d{\isasymin}D\isactrlsub D{\isachardot}{\kern0pt}\ L{\isacharunderscore}{\kern0pt}split\ d\ v\ s{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ {\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}split{\isachardot}{\kern0pt}rep{\isacharunderscore}{\kern0pt}eq\ {\isasymL}{\isacharunderscore}{\kern0pt}split{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ {\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}split{\isacharunderscore}{\kern0pt}contraction{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}dist\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}split\ v{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}split\ u{\isacharparenright}{\kern0pt}\ {\isasymle}\ QR{\isacharunderscore}{\kern0pt}disc\ {\isacharasterisk}{\kern0pt}\ dist\ v\ u{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ aux{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}split\ v\ s\ {\isacharminus}{\kern0pt}\ {\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}split\ u\ s\ {\isasymle}\ QR{\isacharunderscore}{\kern0pt}disc\ {\isacharasterisk}{\kern0pt}\ norm\ {\isacharparenleft}{\kern0pt}v\ {\isacharminus}{\kern0pt}\ u{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{if}\ h{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}split\ u\ s\ {\isasymle}\ {\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}split\ v\ s{\isachardoublequoteclose}\ \isakeyword{for}\ u\ v\ s\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{obtain}\isamarkupfalse%
-\ d\ \isakeyword{where}\ d{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max\ {\isacharparenleft}{\kern0pt}{\isasymlambda}d{\isachardot}{\kern0pt}\ inv\isactrlsub L\ {\isacharparenleft}{\kern0pt}Q\ d{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}r{\isacharunderscore}{\kern0pt}dec\isactrlsub b\ {\isacharparenleft}{\kern0pt}mk{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\ d{\isacharparenright}{\kern0pt}\ {\isacharplus}{\kern0pt}\ R\ d\ v{\isacharparenright}{\kern0pt}\ s{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymlambda}d{\isachardot}{\kern0pt}\ d\ {\isasymin}\ D\isactrlsub D{\isacharparenright}{\kern0pt}\ d{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ finite{\isacharunderscore}{\kern0pt}is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharbrackleft}{\kern0pt}of\ {\isachardoublequoteopen}D\isactrlsub D{\isachardoublequoteclose}{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isacharasterisk}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}inv\isactrlsub L\ {\isacharparenleft}{\kern0pt}Q\ d{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}r{\isacharunderscore}{\kern0pt}dec\isactrlsub b\ {\isacharparenleft}{\kern0pt}mk{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\ d{\isacharparenright}{\kern0pt}\ {\isacharplus}{\kern0pt}\ R\ d\ u{\isacharparenright}{\kern0pt}\ s\ {\isasymle}\ {\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}split\ u\ s{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ d\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ {\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}split{\isacharunderscore}{\kern0pt}def{\isacharprime}{\kern0pt}\ is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}linorder\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ cSUP{\isacharunderscore}{\kern0pt}upper{\isadigit{2}}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}inv\isactrlsub L\ {\isacharparenleft}{\kern0pt}Q\ d{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}r{\isacharunderscore}{\kern0pt}dec\isactrlsub b\ {\isacharparenleft}{\kern0pt}mk{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\ d{\isacharparenright}{\kern0pt}\ {\isacharplus}{\kern0pt}\ R\ d\ v{\isacharparenright}{\kern0pt}\ s\ {\isacharequal}{\kern0pt}\ {\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}split\ v\ s{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ {\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}split{\isacharunderscore}{\kern0pt}def{\isacharprime}{\kern0pt}\ arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}SUP{\isacharbrackleft}{\kern0pt}OF\ d{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}split\ v\ s\ {\isacharminus}{\kern0pt}\ {\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}split\ u\ s\ {\isacharequal}{\kern0pt}\ inv\isactrlsub L\ {\isacharparenleft}{\kern0pt}Q\ d{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}r{\isacharunderscore}{\kern0pt}dec\isactrlsub b\ {\isacharparenleft}{\kern0pt}mk{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\ d{\isacharparenright}{\kern0pt}\ {\isacharplus}{\kern0pt}\ R\ d\ v{\isacharparenright}{\kern0pt}\ s\ {\isacharminus}{\kern0pt}\ {\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}split\ u\ s{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isasymle}\ {\isacharparenleft}{\kern0pt}inv\isactrlsub L\ {\isacharparenleft}{\kern0pt}Q\ d{\isacharparenright}{\kern0pt}\ o\isactrlsub L\ R\ d{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}v\ {\isacharminus}{\kern0pt}\ u{\isacharparenright}{\kern0pt}\ s{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isacharasterisk}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ blinfun{\isachardot}{\kern0pt}bilinear{\isacharunderscore}{\kern0pt}simps{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isasymle}\ norm\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}inv\isactrlsub L\ {\isacharparenleft}{\kern0pt}Q\ d{\isacharparenright}{\kern0pt}\ o\isactrlsub L\ R\ d{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharasterisk}{\kern0pt}\ norm\ {\isacharparenleft}{\kern0pt}v\ {\isacharminus}{\kern0pt}\ u{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}fastforce\ intro{\isacharcolon}{\kern0pt}\ order{\isachardot}{\kern0pt}trans{\isacharbrackleft}{\kern0pt}OF\ le{\isacharunderscore}{\kern0pt}norm{\isacharunderscore}{\kern0pt}bfun\ norm{\isacharunderscore}{\kern0pt}blinfun{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isasymle}\ QR{\isacharunderscore}{\kern0pt}disc\ {\isacharasterisk}{\kern0pt}\ norm\ {\isacharparenleft}{\kern0pt}v\ {\isacharminus}{\kern0pt}\ u{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ QR{\isacharunderscore}{\kern0pt}contraction\ d\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}linorder\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ mult{\isacharunderscore}{\kern0pt}right{\isacharunderscore}{\kern0pt}mono\ cSUP{\isacharunderscore}{\kern0pt}upper{\isadigit{2}}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{finally}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isacommand{{\isachardot}{\kern0pt}}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymbar}{\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}split\ v\ {\isacharminus}{\kern0pt}\ {\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}split\ u{\isacharparenright}{\kern0pt}\ s{\isasymbar}\ {\isasymle}\ QR{\isacharunderscore}{\kern0pt}disc\ {\isacharasterisk}{\kern0pt}\ dist\ v\ u{\isachardoublequoteclose}\ \isakeyword{for}\ s\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ aux\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}cases\ {\isachardoublequoteopen}{\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}split\ v\ s\ {\isasymle}\ {\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}split\ u\ s{\isachardoublequoteclose}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}fastforce\ simp{\isacharcolon}{\kern0pt}\ dist{\isacharunderscore}{\kern0pt}norm\ norm{\isacharunderscore}{\kern0pt}minus{\isacharunderscore}{\kern0pt}commute{\isacharparenright}{\kern0pt}{\isacharplus}{\kern0pt}\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ cSUP{\isacharunderscore}{\kern0pt}least\ simp{\isacharcolon}{\kern0pt}\ dist{\isacharunderscore}{\kern0pt}bfun{\isachardot}{\kern0pt}rep{\isacharunderscore}{\kern0pt}eq\ dist{\isacharunderscore}{\kern0pt}real{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ {\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}lim{\isacharcolon}{\kern0pt}\isanewline
-\ \ {\isachardoublequoteopen}{\isasymexists}{\isacharbang}{\kern0pt}v{\isachardot}{\kern0pt}\ {\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}split\ v\ {\isacharequal}{\kern0pt}\ v{\isachardoublequoteclose}\isanewline
-\ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isasymlambda}n{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}split\ {\isacharcircum}{\kern0pt}{\isacharcircum}{\kern0pt}\ n{\isacharparenright}{\kern0pt}\ v{\isacharparenright}{\kern0pt}\ {\isasymlonglonglongrightarrow}\ {\isacharparenleft}{\kern0pt}THE\ v{\isachardot}{\kern0pt}\ {\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}split\ v\ {\isacharequal}{\kern0pt}\ v{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ banach{\isacharprime}{\kern0pt}{\isacharbrackleft}{\kern0pt}of\ {\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}split{\isacharbrackright}{\kern0pt}\ a{\isacharunderscore}{\kern0pt}nonneg\ QR{\isacharunderscore}{\kern0pt}contraction\ {\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}split{\isacharunderscore}{\kern0pt}contraction\isanewline
-\ \ \isacommand{unfolding}\isamarkupfalse%
-\ is{\isacharunderscore}{\kern0pt}contraction{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ {\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}split{\isacharunderscore}{\kern0pt}tendsto{\isacharunderscore}{\kern0pt}opt{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isasymlambda}n{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}split\ {\isacharcircum}{\kern0pt}{\isacharcircum}{\kern0pt}\ n{\isacharparenright}{\kern0pt}\ v{\isacharparenright}{\kern0pt}\ {\isasymlonglonglongrightarrow}\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{obtain}\isamarkupfalse%
-\ L\ \isakeyword{where}\ l{\isacharunderscore}{\kern0pt}fix{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}split\ L\ {\isacharequal}{\kern0pt}\ L{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}lim{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymnu}\isactrlsub b\ {\isacharparenleft}{\kern0pt}mk{\isacharunderscore}{\kern0pt}stationary{\isacharunderscore}{\kern0pt}det\ d{\isacharparenright}{\kern0pt}\ {\isasymle}\ L{\isachardoublequoteclose}\ \isakeyword{if}\ d{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}d\ {\isasymin}\ D\isactrlsub D{\isachardoublequoteclose}\ \isakeyword{for}\ d\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}QR\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}inv\isactrlsub L\ {\isacharparenleft}{\kern0pt}Q\ d{\isacharparenright}{\kern0pt}\ o\isactrlsub L\ R\ d{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}inv\isactrlsub L\ {\isacharparenleft}{\kern0pt}Q\ d{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}r{\isacharunderscore}{\kern0pt}dec\isactrlsub b\ {\isacharparenleft}{\kern0pt}mk{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\ d{\isacharparenright}{\kern0pt}\ {\isacharplus}{\kern0pt}\ R\ d\ L{\isacharparenright}{\kern0pt}\ {\isasymle}\ {\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}split\ L{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ d\ l{\isacharunderscore}{\kern0pt}fix\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}fastforce\ simp{\isacharcolon}{\kern0pt}\ {\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}split{\isacharunderscore}{\kern0pt}def{\isacharprime}{\kern0pt}\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ cSUP{\isacharunderscore}{\kern0pt}upper{\isadigit{2}}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}inv\isactrlsub L\ {\isacharparenleft}{\kern0pt}Q\ d{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}r{\isacharunderscore}{\kern0pt}dec\isactrlsub b\ {\isacharparenleft}{\kern0pt}mk{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\ d{\isacharparenright}{\kern0pt}\ {\isacharplus}{\kern0pt}\ R\ d\ L{\isacharparenright}{\kern0pt}\ {\isasymle}\ L{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ l{\isacharunderscore}{\kern0pt}fix\ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ aux{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}inv\isactrlsub L\ {\isacharparenleft}{\kern0pt}Q\ d{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}r{\isacharunderscore}{\kern0pt}dec\isactrlsub b\ {\isacharparenleft}{\kern0pt}mk{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\ d{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymle}\ {\isacharparenleft}{\kern0pt}id{\isacharunderscore}{\kern0pt}blinfun\ {\isacharminus}{\kern0pt}\ {\isacharquery}{\kern0pt}QR{\isacharparenright}{\kern0pt}\ L{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ that\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ blinfun{\isachardot}{\kern0pt}bilinear{\isacharunderscore}{\kern0pt}simps\ le{\isacharunderscore}{\kern0pt}diff{\isacharunderscore}{\kern0pt}eq{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ inv{\isacharunderscore}{\kern0pt}eq{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}inv\isactrlsub L\ {\isacharparenleft}{\kern0pt}id{\isacharunderscore}{\kern0pt}blinfun\ {\isacharminus}{\kern0pt}\ {\isacharquery}{\kern0pt}QR{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSum}i{\isachardot}{\kern0pt}\ {\isacharquery}{\kern0pt}QR\ {\isacharcircum}{\kern0pt}{\isacharcircum}{\kern0pt}\ i{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ QR{\isacharunderscore}{\kern0pt}contraction\ d\ norm{\isacharunderscore}{\kern0pt}QR{\isacharunderscore}{\kern0pt}less{\isacharunderscore}{\kern0pt}one\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ inv\isactrlsub L{\isacharunderscore}{\kern0pt}inf{\isacharunderscore}{\kern0pt}sum{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ summable{\isacharunderscore}{\kern0pt}QR{\isacharcolon}{\kern0pt}{\isachardoublequoteopen}summable\ {\isacharparenleft}{\kern0pt}{\isasymlambda}i{\isachardot}{\kern0pt}\ norm\ {\isacharparenleft}{\kern0pt}{\isacharquery}{\kern0pt}QR\ {\isacharcircum}{\kern0pt}{\isacharcircum}{\kern0pt}\ i{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ QR{\isacharunderscore}{\kern0pt}contraction\ d\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}fastforce\ simp{\isacharcolon}{\kern0pt}\ a{\isacharunderscore}{\kern0pt}nonneg\isanewline
-\ \ \ \ \ \ \ \ \ \ intro{\isacharcolon}{\kern0pt}\ summable{\isacharunderscore}{\kern0pt}comparison{\isacharunderscore}{\kern0pt}test{\isacharprime}{\kern0pt}{\isacharbrackleft}{\kern0pt}\isakeyword{where}\ g\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}{\isasymlambda}i{\isachardot}{\kern0pt}\ QR{\isacharunderscore}{\kern0pt}disc{\isacharcircum}{\kern0pt}i{\isachardoublequoteclose}{\isacharbrackright}{\kern0pt}\ \isanewline
-\ \ \ \ \ \ \ \ \ \ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ cSUP{\isacharunderscore}{\kern0pt}upper{\isadigit{2}}\ power{\isacharunderscore}{\kern0pt}mono\ order{\isachardot}{\kern0pt}trans{\isacharbrackleft}{\kern0pt}OF\ norm{\isacharunderscore}{\kern0pt}blinfunpow{\isacharunderscore}{\kern0pt}le{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}summable\ {\isacharparenleft}{\kern0pt}{\isasymlambda}i{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharquery}{\kern0pt}QR\ {\isacharcircum}{\kern0pt}{\isacharcircum}{\kern0pt}\ i{\isacharparenright}{\kern0pt}\ v\ s{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{for}\ v\ s\ \isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}rule\ summable{\isacharunderscore}{\kern0pt}comparison{\isacharunderscore}{\kern0pt}test{\isacharprime}{\kern0pt}{\isacharbrackleft}{\kern0pt}\isakeyword{where}\ g\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}{\isasymlambda}i{\isachardot}{\kern0pt}\ norm\ {\isacharparenleft}{\kern0pt}{\isacharquery}{\kern0pt}QR\ {\isacharcircum}{\kern0pt}{\isacharcircum}{\kern0pt}\ i{\isacharparenright}{\kern0pt}\ {\isacharasterisk}{\kern0pt}\ norm\ v{\isachardoublequoteclose}{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}auto\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ summable{\isacharunderscore}{\kern0pt}QR\ summable{\isacharunderscore}{\kern0pt}norm{\isacharunderscore}{\kern0pt}cancel\ order{\isachardot}{\kern0pt}trans{\isacharbrackleft}{\kern0pt}OF\ abs{\isacharunderscore}{\kern0pt}le{\isacharunderscore}{\kern0pt}norm{\isacharunderscore}{\kern0pt}bfun{\isacharbrackright}{\kern0pt}\ order{\isachardot}{\kern0pt}trans{\isacharbrackleft}{\kern0pt}OF\ norm{\isacharunderscore}{\kern0pt}blinfun{\isacharbrackright}{\kern0pt}\ summable{\isacharunderscore}{\kern0pt}mult{\isadigit{2}}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isadigit{0}}\ {\isasymle}\ v\ {\isasymLongrightarrow}\ {\isadigit{0}}\ {\isasymle}\ {\isacharparenleft}{\kern0pt}{\isasymSum}i{\isacharless}{\kern0pt}n{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharquery}{\kern0pt}QR\ {\isacharcircum}{\kern0pt}{\isacharcircum}{\kern0pt}\ i{\isacharparenright}{\kern0pt}\ v\ s{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{for}\ n\ v\ s\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ blinfunpow{\isacharunderscore}{\kern0pt}nonneg{\isacharbrackleft}{\kern0pt}OF\ QR{\isacharunderscore}{\kern0pt}nonneg{\isacharbrackleft}{\kern0pt}OF\ d{\isacharbrackright}{\kern0pt}{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}induction\ n{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}auto\ simp\ add{\isacharcolon}{\kern0pt}\ less{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}bfun{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isadigit{0}}\ {\isasymle}\ v\ {\isasymLongrightarrow}\ {\isadigit{0}}\ {\isasymle}\ {\isacharparenleft}{\kern0pt}{\isasymSum}i{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}{\isacharquery}{\kern0pt}QR\ {\isacharcircum}{\kern0pt}{\isacharcircum}{\kern0pt}\ i{\isacharparenright}{\kern0pt}\ v\ s{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isachardoublequoteclose}\ \isakeyword{for}\ v\ s\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ summable{\isacharunderscore}{\kern0pt}LIMSEQ\ LIMSEQ{\isacharunderscore}{\kern0pt}le{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isadigit{0}}\ {\isasymle}\ v\ {\isasymLongrightarrow}\ {\isadigit{0}}\ {\isasymle}\ {\isacharparenleft}{\kern0pt}{\isasymSum}i{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}{\isacharquery}{\kern0pt}QR\ {\isacharcircum}{\kern0pt}{\isacharcircum}{\kern0pt}\ i{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ v\ s{\isachardoublequoteclose}\ \isakeyword{for}\ v\ s\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ bounded{\isacharunderscore}{\kern0pt}linear{\isacharunderscore}{\kern0pt}apply{\isacharunderscore}{\kern0pt}bfun\ summable{\isacharunderscore}{\kern0pt}QR\ summable{\isacharunderscore}{\kern0pt}comparison{\isacharunderscore}{\kern0pt}test{\isacharprime}{\kern0pt}\ \isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}subst\ bounded{\isacharunderscore}{\kern0pt}linear{\isachardot}{\kern0pt}suminf{\isacharbrackleft}{\kern0pt}\isakeyword{where}\ f\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isasymlambda}i{\isachardot}{\kern0pt}\ apply{\isacharunderscore}{\kern0pt}bfun\ {\isacharparenleft}{\kern0pt}blinfun{\isacharunderscore}{\kern0pt}apply\ i\ v{\isacharparenright}{\kern0pt}\ s{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}fastforce\ intro{\isacharcolon}{\kern0pt}\ bounded{\isacharunderscore}{\kern0pt}linear{\isacharunderscore}{\kern0pt}compose{\isacharbrackleft}{\kern0pt}of\ {\isachardoublequoteopen}{\isasymlambda}s{\isachardot}{\kern0pt}\ apply{\isacharunderscore}{\kern0pt}bfun\ s\ {\isacharunderscore}{\kern0pt}{\isachardoublequoteclose}{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharplus}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isadigit{0}}\ {\isasymle}\ v\ {\isasymLongrightarrow}\ {\isadigit{0}}\ {\isasymle}\ inv\isactrlsub L\ {\isacharparenleft}{\kern0pt}id{\isacharunderscore}{\kern0pt}blinfun\ {\isacharminus}{\kern0pt}\ {\isacharquery}{\kern0pt}QR{\isacharparenright}{\kern0pt}\ v{\isachardoublequoteclose}\ \isakeyword{for}\ v\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ inv{\isacharunderscore}{\kern0pt}eq\ less{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}bfun{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}inv\isactrlsub L\ {\isacharparenleft}{\kern0pt}id{\isacharunderscore}{\kern0pt}blinfun\ {\isacharminus}{\kern0pt}\ {\isacharquery}{\kern0pt}QR{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}inv\isactrlsub L\ {\isacharparenleft}{\kern0pt}Q\ d{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}r{\isacharunderscore}{\kern0pt}dec\isactrlsub b\ {\isacharparenleft}{\kern0pt}mk{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\ d{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ {\isasymle}\ {\isacharparenleft}{\kern0pt}inv\isactrlsub L\ {\isacharparenleft}{\kern0pt}id{\isacharunderscore}{\kern0pt}blinfun\ {\isacharminus}{\kern0pt}\ {\isacharquery}{\kern0pt}QR{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}id{\isacharunderscore}{\kern0pt}blinfun\ {\isacharminus}{\kern0pt}\ {\isacharquery}{\kern0pt}QR{\isacharparenright}{\kern0pt}\ L{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ aux\ blinfun{\isachardot}{\kern0pt}diff{\isacharunderscore}{\kern0pt}right\ diff{\isacharunderscore}{\kern0pt}ge{\isacharunderscore}{\kern0pt}{\isadigit{0}}{\isacharunderscore}{\kern0pt}iff{\isacharunderscore}{\kern0pt}ge{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}inv\isactrlsub L\ {\isacharparenleft}{\kern0pt}id{\isacharunderscore}{\kern0pt}blinfun\ {\isacharminus}{\kern0pt}\ {\isacharquery}{\kern0pt}QR{\isacharparenright}{\kern0pt}\ o\isactrlsub L\ inv\isactrlsub L\ {\isacharparenleft}{\kern0pt}Q\ d{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}r{\isacharunderscore}{\kern0pt}dec\isactrlsub b\ {\isacharparenleft}{\kern0pt}mk{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\ d{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymle}\ L{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ invertible\isactrlsub L{\isacharunderscore}{\kern0pt}inf{\isacharunderscore}{\kern0pt}sum{\isacharbrackleft}{\kern0pt}OF\ norm{\isacharunderscore}{\kern0pt}QR{\isacharunderscore}{\kern0pt}less{\isacharunderscore}{\kern0pt}one{\isacharbrackleft}{\kern0pt}OF\ that{\isacharbrackright}{\kern0pt}{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}inv\isactrlsub L\ {\isacharparenleft}{\kern0pt}Q\ d\ o\isactrlsub L\ {\isacharparenleft}{\kern0pt}id{\isacharunderscore}{\kern0pt}blinfun\ {\isacharminus}{\kern0pt}\ {\isacharquery}{\kern0pt}QR{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}r{\isacharunderscore}{\kern0pt}dec\isactrlsub b\ {\isacharparenleft}{\kern0pt}mk{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\ d{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymle}\ L{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ d\ norm{\isacharunderscore}{\kern0pt}QR{\isacharunderscore}{\kern0pt}less{\isacharunderscore}{\kern0pt}one\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ inv\isactrlsub L{\isacharunderscore}{\kern0pt}compose{\isacharbrackleft}{\kern0pt}OF\ Q{\isacharunderscore}{\kern0pt}invertible\ invertible\isactrlsub L{\isacharunderscore}{\kern0pt}inf{\isacharunderscore}{\kern0pt}sum{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}inv\isactrlsub L\ {\isacharparenleft}{\kern0pt}Q\ d\ {\isacharminus}{\kern0pt}\ R\ d{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}r{\isacharunderscore}{\kern0pt}dec\isactrlsub b\ {\isacharparenleft}{\kern0pt}mk{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\ d{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymle}\ L{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ Q{\isacharunderscore}{\kern0pt}invertible\ d\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ blinfun{\isacharunderscore}{\kern0pt}compose{\isacharunderscore}{\kern0pt}diff{\isacharunderscore}{\kern0pt}right\ blinfun{\isacharunderscore}{\kern0pt}compose{\isacharunderscore}{\kern0pt}assoc{\isacharbrackleft}{\kern0pt}symmetric{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymnu}\isactrlsub b\ {\isacharparenleft}{\kern0pt}mk{\isacharunderscore}{\kern0pt}stationary{\isacharunderscore}{\kern0pt}det\ d{\isacharparenright}{\kern0pt}\ {\isasymle}\ L{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ {\isasymnu}{\isacharunderscore}{\kern0pt}stationary\ splitting{\isacharbrackleft}{\kern0pt}OF\ that{\isacharcomma}{\kern0pt}\ symmetric{\isacharbrackright}{\kern0pt}\ inv\isactrlsub L{\isacharunderscore}{\kern0pt}inf{\isacharunderscore}{\kern0pt}sum\ blincomp{\isacharunderscore}{\kern0pt}scaleR{\isacharunderscore}{\kern0pt}right{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ opt{\isacharunderscore}{\kern0pt}le{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt\ {\isasymle}\ L{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ thm{\isacharunderscore}{\kern0pt}{\isadigit{6}}{\isacharunderscore}{\kern0pt}{\isadigit{2}}{\isacharunderscore}{\kern0pt}{\isadigit{1}}{\isadigit{0}}\ finite\ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\isanewline
-\ \ \isacommand{obtain}\isamarkupfalse%
-\ d\ \isakeyword{where}\ d{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max\ {\isacharparenleft}{\kern0pt}{\isasymlambda}d{\isachardot}{\kern0pt}\ inv\isactrlsub L\ {\isacharparenleft}{\kern0pt}Q\ d{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}r{\isacharunderscore}{\kern0pt}dec\isactrlsub b\ {\isacharparenleft}{\kern0pt}mk{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\ d{\isacharparenright}{\kern0pt}\ {\isacharplus}{\kern0pt}\ R\ d\ L{\isacharparenright}{\kern0pt}\ s{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymlambda}d{\isachardot}{\kern0pt}\ d\ {\isasymin}\ D\isactrlsub D{\isacharparenright}{\kern0pt}\ d{\isachardoublequoteclose}\ \isakeyword{for}\ s\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}ex{\isacharunderscore}{\kern0pt}split\ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}d\ {\isasymin}\ D\isactrlsub D{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}linorder\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}L\ {\isacharequal}{\kern0pt}\ inv\isactrlsub L\ {\isacharparenleft}{\kern0pt}Q\ d{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}r{\isacharunderscore}{\kern0pt}dec\isactrlsub b\ {\isacharparenleft}{\kern0pt}mk{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\ d{\isacharparenright}{\kern0pt}\ {\isacharplus}{\kern0pt}\ R\ d\ L{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}subst\ l{\isacharunderscore}{\kern0pt}fix{\isacharbrackleft}{\kern0pt}symmetric{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}fastforce\ simp{\isacharcolon}{\kern0pt}\ {\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}split{\isacharunderscore}{\kern0pt}def{\isacharprime}{\kern0pt}\ arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}SUP{\isacharbrackleft}{\kern0pt}OF\ d{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}Q\ d\ L\ {\isacharequal}{\kern0pt}\ r{\isacharunderscore}{\kern0pt}dec\isactrlsub b\ {\isacharparenleft}{\kern0pt}mk{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\ d{\isacharparenright}{\kern0pt}\ {\isacharplus}{\kern0pt}\ R\ d\ L{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ Q{\isacharunderscore}{\kern0pt}invertible\ {\isacartoucheopen}d\ {\isasymin}\ D\isactrlsub D{\isacartoucheclose}\ inv{\isacharunderscore}{\kern0pt}app{\isadigit{2}}{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}id{\isacharunderscore}{\kern0pt}blinfun\ {\isacharminus}{\kern0pt}\ l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isasymP}\isactrlsub {\isadigit{1}}\ {\isacharparenleft}{\kern0pt}mk{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\ d{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ L\ {\isacharequal}{\kern0pt}\ r{\isacharunderscore}{\kern0pt}dec\isactrlsub b\ {\isacharparenleft}{\kern0pt}mk{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\ d{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ splitting{\isacharbrackleft}{\kern0pt}OF\ {\isacartoucheopen}d\ {\isasymin}\ D\isactrlsub D{\isacartoucheclose}{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ blinfun{\isachardot}{\kern0pt}diff{\isacharunderscore}{\kern0pt}left{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}L\ {\isacharequal}{\kern0pt}\ inv\isactrlsub L\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}id{\isacharunderscore}{\kern0pt}blinfun\ {\isacharminus}{\kern0pt}\ l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isasymP}\isactrlsub {\isadigit{1}}\ {\isacharparenleft}{\kern0pt}mk{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\ d{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}r{\isacharunderscore}{\kern0pt}dec\isactrlsub b\ {\isacharparenleft}{\kern0pt}mk{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\ d{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ invertible\isactrlsub L{\isacharunderscore}{\kern0pt}inf{\isacharunderscore}{\kern0pt}sum{\isacharbrackleft}{\kern0pt}OF\ norm{\isacharunderscore}{\kern0pt}{\isasymP}\isactrlsub {\isadigit{1}}{\isacharunderscore}{\kern0pt}l{\isacharunderscore}{\kern0pt}less{\isacharbrackright}{\kern0pt}\ inv{\isacharunderscore}{\kern0pt}app{\isadigit{1}}{\isacharprime}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}L\ {\isacharequal}{\kern0pt}\ {\isasymnu}\isactrlsub b\ {\isacharparenleft}{\kern0pt}mk{\isacharunderscore}{\kern0pt}stationary{\isacharunderscore}{\kern0pt}det\ d{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ inv\isactrlsub L{\isacharunderscore}{\kern0pt}inf{\isacharunderscore}{\kern0pt}sum\ {\isasymnu}{\isacharunderscore}{\kern0pt}stationary\ blincomp{\isacharunderscore}{\kern0pt}scaleR{\isacharunderscore}{\kern0pt}right{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt\ {\isacharequal}{\kern0pt}\ L{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ opt{\isacharunderscore}{\kern0pt}le\ {\isacartoucheopen}d\ {\isasymin}\ D\isactrlsub D{\isacartoucheclose}\ is{\isacharunderscore}{\kern0pt}markovian{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ intro{\isacharcolon}{\kern0pt}\ order{\isachardot}{\kern0pt}antisym{\isacharbrackleft}{\kern0pt}OF\ {\isacharunderscore}{\kern0pt}\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}le{\isacharunderscore}{\kern0pt}opt{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}lim\ l{\isacharunderscore}{\kern0pt}fix\ the{\isadigit{1}}{\isacharunderscore}{\kern0pt}equality{\isacharbrackleft}{\kern0pt}OF\ {\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}lim{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ {\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}split{\isacharunderscore}{\kern0pt}fix{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}split\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt\ {\isacharequal}{\kern0pt}\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ {\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}lim\ {\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}split{\isacharunderscore}{\kern0pt}tendsto{\isacharunderscore}{\kern0pt}opt\ the{\isacharunderscore}{\kern0pt}equality\ limI\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}mono{\isacharunderscore}{\kern0pt}tags{\isacharcomma}{\kern0pt}\ lifting{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ dist{\isacharunderscore}{\kern0pt}{\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}split{\isacharunderscore}{\kern0pt}opt{\isacharunderscore}{\kern0pt}eps{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}eps\ {\isachargreater}{\kern0pt}\ {\isadigit{0}}{\isachardoublequoteclose}\ {\isachardoublequoteopen}{\isadigit{2}}\ {\isacharasterisk}{\kern0pt}\ QR{\isacharunderscore}{\kern0pt}disc\ {\isacharasterisk}{\kern0pt}\ dist\ v\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}split\ v{\isacharparenright}{\kern0pt}\ {\isacharless}{\kern0pt}\ eps\ {\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharminus}{\kern0pt}QR{\isacharunderscore}{\kern0pt}disc{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}dist\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}split\ v{\isacharparenright}{\kern0pt}\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt\ {\isacharless}{\kern0pt}\ eps\ {\isacharslash}{\kern0pt}\ {\isadigit{2}}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isadigit{1}}\ {\isacharminus}{\kern0pt}\ QR{\isacharunderscore}{\kern0pt}disc{\isacharparenright}{\kern0pt}\ {\isacharasterisk}{\kern0pt}\ dist\ v\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt\ {\isasymle}\ dist\ v\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}split\ v{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ dist{\isacharunderscore}{\kern0pt}triangle\ {\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}split{\isacharunderscore}{\kern0pt}contraction{\isacharbrackleft}{\kern0pt}of\ v\ {\isachardoublequoteopen}{\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt{\isachardoublequoteclose}{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}fastforce\ simp{\isacharcolon}{\kern0pt}\ algebra{\isacharunderscore}{\kern0pt}simps\ intro{\isacharcolon}{\kern0pt}\ order{\isachardot}{\kern0pt}trans{\isacharbrackleft}{\kern0pt}OF\ {\isacharunderscore}{\kern0pt}\ add{\isacharunderscore}{\kern0pt}left{\isacharunderscore}{\kern0pt}mono{\isacharbrackleft}{\kern0pt}of\ {\isachardoublequoteopen}dist\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}split\ v{\isacharparenright}{\kern0pt}\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt{\isachardoublequoteclose}{\isacharbrackright}{\kern0pt}{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}dist\ v\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt\ {\isasymle}\ dist\ v\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}split\ v{\isacharparenright}{\kern0pt}\ {\isacharslash}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}\ {\isacharminus}{\kern0pt}\ QR{\isacharunderscore}{\kern0pt}disc{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ QR{\isacharunderscore}{\kern0pt}contraction\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ mult{\isachardot}{\kern0pt}commute\ pos{\isacharunderscore}{\kern0pt}le{\isacharunderscore}{\kern0pt}divide{\isacharunderscore}{\kern0pt}eq{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isadigit{2}}\ {\isacharasterisk}{\kern0pt}\ QR{\isacharunderscore}{\kern0pt}disc\ {\isacharasterisk}{\kern0pt}\ dist\ v\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt\ {\isasymle}\ {\isadigit{2}}\ {\isacharasterisk}{\kern0pt}\ QR{\isacharunderscore}{\kern0pt}disc\ {\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}dist\ v\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}split\ v{\isacharparenright}{\kern0pt}\ {\isacharslash}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}\ {\isacharminus}{\kern0pt}\ QR{\isacharunderscore}{\kern0pt}disc{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}split{\isacharunderscore}{\kern0pt}contraction\ assms\ mult{\isacharunderscore}{\kern0pt}le{\isacharunderscore}{\kern0pt}cancel{\isacharunderscore}{\kern0pt}left{\isacharunderscore}{\kern0pt}pos{\isacharbrackleft}{\kern0pt}of\ {\isachardoublequoteopen}{\isadigit{2}}\ {\isacharasterisk}{\kern0pt}\ QR{\isacharunderscore}{\kern0pt}disc{\isachardoublequoteclose}{\isacharbrackright}{\kern0pt}\ a{\isacharunderscore}{\kern0pt}nonneg\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}fastforce\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ mult{\isacharunderscore}{\kern0pt}left{\isacharunderscore}{\kern0pt}mono{\isacharbrackleft}{\kern0pt}of\ {\isacharunderscore}{\kern0pt}\ {\isacharunderscore}{\kern0pt}\ {\isachardoublequoteopen}{\isadigit{2}}\ {\isacharasterisk}{\kern0pt}\ QR{\isacharunderscore}{\kern0pt}disc{\isachardoublequoteclose}{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isadigit{2}}\ {\isacharasterisk}{\kern0pt}\ QR{\isacharunderscore}{\kern0pt}disc\ {\isacharasterisk}{\kern0pt}\ dist\ v\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt\ {\isacharless}{\kern0pt}\ eps{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ a{\isacharunderscore}{\kern0pt}nonneg\ QR{\isacharunderscore}{\kern0pt}contraction\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ assms{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}\ pos{\isacharunderscore}{\kern0pt}divide{\isacharunderscore}{\kern0pt}less{\isacharunderscore}{\kern0pt}eq\ intro{\isacharcolon}{\kern0pt}\ order{\isachardot}{\kern0pt}strict{\isacharunderscore}{\kern0pt}trans{\isadigit{1}}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}dist\ v\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt\ {\isacharasterisk}{\kern0pt}\ QR{\isacharunderscore}{\kern0pt}disc\ {\isacharless}{\kern0pt}\ eps\ {\isacharslash}{\kern0pt}\ {\isadigit{2}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ argo\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}dist\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}split\ v{\isacharparenright}{\kern0pt}\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt\ {\isacharless}{\kern0pt}\ eps\ {\isacharslash}{\kern0pt}\ {\isadigit{2}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}split{\isacharunderscore}{\kern0pt}contraction{\isacharbrackleft}{\kern0pt}of\ v\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt{\isacharbrackright}{\kern0pt}\ \isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ algebra{\isacharunderscore}{\kern0pt}simps{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ L{\isacharunderscore}{\kern0pt}split{\isacharunderscore}{\kern0pt}fix{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}d\ {\isasymin}\ D\isactrlsub D{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}L{\isacharunderscore}{\kern0pt}split\ d\ {\isacharparenleft}{\kern0pt}{\isasymnu}\isactrlsub b\ {\isacharparenleft}{\kern0pt}mk{\isacharunderscore}{\kern0pt}stationary{\isacharunderscore}{\kern0pt}det\ d{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isasymnu}\isactrlsub b\ {\isacharparenleft}{\kern0pt}mk{\isacharunderscore}{\kern0pt}stationary{\isacharunderscore}{\kern0pt}det\ d{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}d\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}mk{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\ d{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}p\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}mk{\isacharunderscore}{\kern0pt}stationary{\isacharunderscore}{\kern0pt}det\ d{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}Q\ d\ {\isacharminus}{\kern0pt}\ R\ d{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymnu}\isactrlsub b\ {\isacharquery}{\kern0pt}p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ r{\isacharunderscore}{\kern0pt}dec\isactrlsub b\ {\isacharquery}{\kern0pt}d{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ L{\isacharunderscore}{\kern0pt}{\isasymnu}{\isacharunderscore}{\kern0pt}fix{\isacharbrackleft}{\kern0pt}of\ {\isachardoublequoteopen}mk{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\ d{\isachardoublequoteclose}{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ L{\isacharunderscore}{\kern0pt}def\ splitting{\isacharbrackleft}{\kern0pt}OF\ assms{\isacharcomma}{\kern0pt}\ symmetric{\isacharbrackright}{\kern0pt}\ blinfun{\isachardot}{\kern0pt}bilinear{\isacharunderscore}{\kern0pt}simps\ diff{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}eq{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assms\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ blinfun{\isachardot}{\kern0pt}bilinear{\isacharunderscore}{\kern0pt}simps\ diff{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}eq\ inv\isactrlsub L{\isacharunderscore}{\kern0pt}cancel{\isacharunderscore}{\kern0pt}iff{\isacharbrackleft}{\kern0pt}OF\ Q{\isacharunderscore}{\kern0pt}invertible{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ L{\isacharunderscore}{\kern0pt}split{\isacharunderscore}{\kern0pt}contraction{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}d\ {\isasymin}\ D\isactrlsub D{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}dist\ {\isacharparenleft}{\kern0pt}L{\isacharunderscore}{\kern0pt}split\ d\ v{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}L{\isacharunderscore}{\kern0pt}split\ d\ u{\isacharparenright}{\kern0pt}\ {\isasymle}\ QR{\isacharunderscore}{\kern0pt}disc\ {\isacharasterisk}{\kern0pt}\ dist\ v\ u{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ aux{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}L{\isacharunderscore}{\kern0pt}split\ d\ v\ s\ {\isacharminus}{\kern0pt}\ L{\isacharunderscore}{\kern0pt}split\ d\ u\ s\ {\isasymle}\ QR{\isacharunderscore}{\kern0pt}disc\ {\isacharasterisk}{\kern0pt}\ dist\ v\ u{\isachardoublequoteclose}\ \isakeyword{if}\ lea{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}L{\isacharunderscore}{\kern0pt}split\ d\ u\ s{\isacharparenright}{\kern0pt}\ {\isasymle}\ {\isacharparenleft}{\kern0pt}L{\isacharunderscore}{\kern0pt}split\ d\ v\ s{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{for}\ v\ s\ u\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}L{\isacharunderscore}{\kern0pt}split\ d\ v\ s\ {\isacharminus}{\kern0pt}\ L{\isacharunderscore}{\kern0pt}split\ d\ u\ s\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}inv\isactrlsub L\ {\isacharparenleft}{\kern0pt}Q\ d{\isacharparenright}{\kern0pt}\ o\isactrlsub L\ {\isacharparenleft}{\kern0pt}R\ d{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}v\ {\isacharminus}{\kern0pt}\ u{\isacharparenright}{\kern0pt}\ s{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ blinfun{\isachardot}{\kern0pt}bilinear{\isacharunderscore}{\kern0pt}simps{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isasymle}\ norm\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}inv\isactrlsub L\ {\isacharparenleft}{\kern0pt}Q\ d{\isacharparenright}{\kern0pt}\ o\isactrlsub L\ {\isacharparenleft}{\kern0pt}R\ d{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}v\ {\isacharminus}{\kern0pt}\ u{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ le{\isacharunderscore}{\kern0pt}norm{\isacharunderscore}{\kern0pt}bfun{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isasymle}\ norm\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}inv\isactrlsub L\ {\isacharparenleft}{\kern0pt}Q\ d{\isacharparenright}{\kern0pt}\ o\isactrlsub L\ {\isacharparenleft}{\kern0pt}R\ d{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharasterisk}{\kern0pt}\ dist\ v\ u{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp\ only{\isacharcolon}{\kern0pt}\ dist{\isacharunderscore}{\kern0pt}norm\ norm{\isacharunderscore}{\kern0pt}blinfun{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isasymle}\ QR{\isacharunderscore}{\kern0pt}disc\ {\isacharasterisk}{\kern0pt}\ dist\ v\ u{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assms\ QR{\isacharunderscore}{\kern0pt}le{\isacharunderscore}{\kern0pt}QR{\isacharunderscore}{\kern0pt}disc\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ mult{\isacharunderscore}{\kern0pt}right{\isacharunderscore}{\kern0pt}mono{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{finally}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}dist\ {\isacharparenleft}{\kern0pt}L{\isacharunderscore}{\kern0pt}split\ d\ v\ s{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}L{\isacharunderscore}{\kern0pt}split\ d\ u\ s{\isacharparenright}{\kern0pt}\ {\isasymle}\ QR{\isacharunderscore}{\kern0pt}disc\ {\isacharasterisk}{\kern0pt}\ dist\ v\ u{\isachardoublequoteclose}\ \isakeyword{for}\ v\ s\ u\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ aux\ aux{\isacharbrackleft}{\kern0pt}of\ v\ {\isacharunderscore}{\kern0pt}\ u{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}cases\ {\isachardoublequoteopen}L{\isacharunderscore}{\kern0pt}split\ d\ v\ s\ {\isasymge}\ L{\isacharunderscore}{\kern0pt}split\ d\ u\ s{\isachardoublequoteclose}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ dist{\isacharunderscore}{\kern0pt}real{\isacharunderscore}{\kern0pt}def\ dist{\isacharunderscore}{\kern0pt}commute{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}dist\ {\isacharparenleft}{\kern0pt}L{\isacharunderscore}{\kern0pt}split\ d\ v{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}L{\isacharunderscore}{\kern0pt}split\ d\ u{\isacharparenright}{\kern0pt}\ {\isasymle}\ QR{\isacharunderscore}{\kern0pt}disc\ {\isacharasterisk}{\kern0pt}\ dist\ v\ u{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ dist{\isacharunderscore}{\kern0pt}bound{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ find{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}QR{\isacharunderscore}{\kern0pt}error{\isacharunderscore}{\kern0pt}bound{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}eps\ {\isachargreater}{\kern0pt}\ {\isadigit{0}}{\isachardoublequoteclose}\ {\isachardoublequoteopen}{\isadigit{2}}\ {\isacharasterisk}{\kern0pt}\ QR{\isacharunderscore}{\kern0pt}disc\ {\isacharasterisk}{\kern0pt}\ dist\ v\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}split\ v{\isacharparenright}{\kern0pt}\ {\isacharless}{\kern0pt}\ eps\ {\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharminus}{\kern0pt}QR{\isacharunderscore}{\kern0pt}disc{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{assumes}\ am{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymAnd}s{\isachardot}{\kern0pt}\ is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max\ {\isacharparenleft}{\kern0pt}{\isasymlambda}d{\isachardot}{\kern0pt}\ L{\isacharunderscore}{\kern0pt}split\ d\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}split\ v{\isacharparenright}{\kern0pt}\ s{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymlambda}d{\isachardot}{\kern0pt}\ d\ {\isasymin}\ D\isactrlsub D{\isacharparenright}{\kern0pt}\ d{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}dist\ {\isacharparenleft}{\kern0pt}{\isasymnu}\isactrlsub b\ {\isacharparenleft}{\kern0pt}mk{\isacharunderscore}{\kern0pt}stationary{\isacharunderscore}{\kern0pt}det\ d{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt\ {\isacharless}{\kern0pt}\ eps{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}p\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}mk{\isacharunderscore}{\kern0pt}stationary{\isacharunderscore}{\kern0pt}det\ d{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ L{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}{\isasymL}\isactrlsub b{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}L{\isacharunderscore}{\kern0pt}split\ d\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}split\ v{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}split\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}split\ v{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ {\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}split{\isacharunderscore}{\kern0pt}def{\isacharprime}{\kern0pt}\ arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}SUP{\isacharbrackleft}{\kern0pt}OF\ am{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}dist\ {\isacharparenleft}{\kern0pt}{\isasymnu}\isactrlsub b\ {\isacharquery}{\kern0pt}p{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}split\ v{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ dist\ {\isacharparenleft}{\kern0pt}L{\isacharunderscore}{\kern0pt}split\ d\ {\isacharparenleft}{\kern0pt}{\isasymnu}\isactrlsub b\ {\isacharquery}{\kern0pt}p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}split\ v{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ am\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}linorder\ L{\isacharunderscore}{\kern0pt}split{\isacharunderscore}{\kern0pt}fix{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isasymle}\ dist\ {\isacharparenleft}{\kern0pt}L{\isacharunderscore}{\kern0pt}split\ d\ {\isacharparenleft}{\kern0pt}{\isasymnu}\isactrlsub b\ {\isacharquery}{\kern0pt}p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}split\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}split\ v{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharplus}{\kern0pt}\ dist\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}split\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}split\ v{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}split\ v{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ intro{\isacharcolon}{\kern0pt}\ dist{\isacharunderscore}{\kern0pt}triangle{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isacharequal}{\kern0pt}\ dist\ {\isacharparenleft}{\kern0pt}L{\isacharunderscore}{\kern0pt}split\ d\ {\isacharparenleft}{\kern0pt}{\isasymnu}\isactrlsub b\ {\isacharquery}{\kern0pt}p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}L{\isacharunderscore}{\kern0pt}split\ d\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}split\ v{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharplus}{\kern0pt}\ dist\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}split\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}split\ v{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}split\ v{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ L{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}{\isasymL}\isactrlsub b{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isasymle}\ QR{\isacharunderscore}{\kern0pt}disc\ {\isacharasterisk}{\kern0pt}\ dist\ {\isacharparenleft}{\kern0pt}{\isasymnu}\isactrlsub b\ {\isacharquery}{\kern0pt}p{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}split\ v{\isacharparenright}{\kern0pt}\ {\isacharplus}{\kern0pt}\ QR{\isacharunderscore}{\kern0pt}disc\ {\isacharasterisk}{\kern0pt}\ dist\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}split\ v{\isacharparenright}{\kern0pt}\ v{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}split{\isacharunderscore}{\kern0pt}contraction\ L{\isacharunderscore}{\kern0pt}split{\isacharunderscore}{\kern0pt}contraction\ am\ \isacommand{unfolding}\isamarkupfalse%
-\ is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ add{\isacharunderscore}{\kern0pt}mono{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{finally}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ aux{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}dist\ {\isacharparenleft}{\kern0pt}{\isasymnu}\isactrlsub b\ {\isacharquery}{\kern0pt}p{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}split\ v{\isacharparenright}{\kern0pt}\ {\isasymle}\ QR{\isacharunderscore}{\kern0pt}disc\ {\isacharasterisk}{\kern0pt}\ dist\ {\isacharparenleft}{\kern0pt}{\isasymnu}\isactrlsub b\ {\isacharquery}{\kern0pt}p{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}split\ v{\isacharparenright}{\kern0pt}\ {\isacharplus}{\kern0pt}\ QR{\isacharunderscore}{\kern0pt}disc\ {\isacharasterisk}{\kern0pt}\ dist\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}split\ v{\isacharparenright}{\kern0pt}\ v{\isachardoublequoteclose}\ \isacommand{{\isachardot}{\kern0pt}}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}dist\ {\isacharparenleft}{\kern0pt}{\isasymnu}\isactrlsub b\ {\isacharquery}{\kern0pt}p{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}split\ v{\isacharparenright}{\kern0pt}\ {\isacharminus}{\kern0pt}\ QR{\isacharunderscore}{\kern0pt}disc\ {\isacharasterisk}{\kern0pt}\ dist\ {\isacharparenleft}{\kern0pt}{\isasymnu}\isactrlsub b\ {\isacharquery}{\kern0pt}p{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}split\ v{\isacharparenright}{\kern0pt}\ {\isasymle}\ QR{\isacharunderscore}{\kern0pt}disc\ {\isacharasterisk}{\kern0pt}\ dist\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}split\ v{\isacharparenright}{\kern0pt}\ v{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}dist\ {\isacharparenleft}{\kern0pt}{\isasymnu}\isactrlsub b\ {\isacharquery}{\kern0pt}p{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}split\ v{\isacharparenright}{\kern0pt}\ {\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}\ {\isacharminus}{\kern0pt}\ QR{\isacharunderscore}{\kern0pt}disc{\isacharparenright}{\kern0pt}\ {\isasymle}\ QR{\isacharunderscore}{\kern0pt}disc\ {\isacharasterisk}{\kern0pt}\ dist\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}split\ v{\isacharparenright}{\kern0pt}\ v{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ argo\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ \ {\isachardoublequoteopen}{\isadigit{2}}\ {\isacharasterisk}{\kern0pt}\ dist\ {\isacharparenleft}{\kern0pt}{\isasymnu}\isactrlsub b\ {\isacharquery}{\kern0pt}p{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}split\ v{\isacharparenright}{\kern0pt}\ {\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharminus}{\kern0pt}QR{\isacharunderscore}{\kern0pt}disc{\isacharparenright}{\kern0pt}\ {\isasymle}\ {\isadigit{2}}\ {\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}QR{\isacharunderscore}{\kern0pt}disc\ {\isacharasterisk}{\kern0pt}\ dist\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}split\ v{\isacharparenright}{\kern0pt}\ v{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ mult{\isacharunderscore}{\kern0pt}left{\isacharunderscore}{\kern0pt}mono\ \isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isadigit{2}}\ {\isacharasterisk}{\kern0pt}\ dist\ {\isacharparenleft}{\kern0pt}{\isasymnu}\isactrlsub b\ {\isacharquery}{\kern0pt}p{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}split\ v{\isacharparenright}{\kern0pt}\ {\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}\ {\isacharminus}{\kern0pt}\ QR{\isacharunderscore}{\kern0pt}disc{\isacharparenright}{\kern0pt}\ {\isasymle}\ eps\ {\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}\ {\isacharminus}{\kern0pt}\ QR{\isacharunderscore}{\kern0pt}disc{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assms\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ mult{\isacharunderscore}{\kern0pt}left{\isacharunderscore}{\kern0pt}mono\ simp{\isacharcolon}{\kern0pt}\ dist{\isacharunderscore}{\kern0pt}commute\ pos{\isacharunderscore}{\kern0pt}divide{\isacharunderscore}{\kern0pt}le{\isacharunderscore}{\kern0pt}eq{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isadigit{2}}\ {\isacharasterisk}{\kern0pt}\ dist\ {\isacharparenleft}{\kern0pt}{\isasymnu}\isactrlsub b\ {\isacharquery}{\kern0pt}p{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}split\ v{\isacharparenright}{\kern0pt}\ {\isasymle}\ eps{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ QR{\isacharunderscore}{\kern0pt}contraction\ mult{\isacharunderscore}{\kern0pt}right{\isacharunderscore}{\kern0pt}le{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}le\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isadigit{2}}\ {\isacharasterisk}{\kern0pt}\ dist\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}split\ v{\isacharparenright}{\kern0pt}\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt\ {\isacharless}{\kern0pt}\ eps{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ dist{\isacharunderscore}{\kern0pt}{\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}split{\isacharunderscore}{\kern0pt}opt{\isacharunderscore}{\kern0pt}eps\ assms\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ fastforce\isanewline
-\ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\ \isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ dist{\isacharunderscore}{\kern0pt}triangle{\isacharbrackleft}{\kern0pt}of\ {\isachardoublequoteopen}{\isasymnu}\isactrlsub b\ {\isacharquery}{\kern0pt}p{\isachardoublequoteclose}\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt\ {\isachardoublequoteopen}{\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}split\ v{\isachardoublequoteclose}{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isacommand{end}\isamarkupfalse%
-\isanewline
-\isanewline
-\isacommand{context}\isamarkupfalse%
-\ MDP{\isacharunderscore}{\kern0pt}ord\ \isakeyword{begin}\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ inv{\isacharunderscore}{\kern0pt}one{\isacharunderscore}{\kern0pt}sub{\isacharunderscore}{\kern0pt}Q{\isacharprime}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{fixes}\ Q\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}c\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ banach\ {\isasymRightarrow}\isactrlsub L\ {\isacharprime}{\kern0pt}c{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{assumes}\ onorm{\isacharunderscore}{\kern0pt}le{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}norm\ {\isacharparenleft}{\kern0pt}id{\isacharunderscore}{\kern0pt}blinfun\ {\isacharminus}{\kern0pt}\ Q{\isacharparenright}{\kern0pt}\ {\isacharless}{\kern0pt}\ {\isadigit{1}}{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}inv\isactrlsub L\ Q\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSum}i{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}id{\isacharunderscore}{\kern0pt}blinfun\ {\isacharminus}{\kern0pt}\ Q{\isacharparenright}{\kern0pt}{\isacharcircum}{\kern0pt}{\isacharcircum}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ inv\isactrlsub L{\isacharunderscore}{\kern0pt}I\ inv{\isacharunderscore}{\kern0pt}one{\isacharunderscore}{\kern0pt}sub{\isacharunderscore}{\kern0pt}Q\ assms{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\begin{isamarkuptext}%
-An important theorem: allows to compare the rate of convergence for different splittings%
-\end{isamarkuptext}\isamarkuptrue%
-\isacommand{lemma}\isamarkupfalse%
-\ norm{\isacharunderscore}{\kern0pt}splitting{\isacharunderscore}{\kern0pt}le{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}is{\isacharunderscore}{\kern0pt}splitting{\isacharunderscore}{\kern0pt}blin\ {\isacharparenleft}{\kern0pt}id{\isacharunderscore}{\kern0pt}blinfun\ {\isacharminus}{\kern0pt}\ l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isasymP}\isactrlsub {\isadigit{1}}\ d{\isacharparenright}{\kern0pt}\ Q{\isadigit{1}}\ R{\isadigit{1}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isakeyword{and}\ {\isachardoublequoteopen}is{\isacharunderscore}{\kern0pt}splitting{\isacharunderscore}{\kern0pt}blin\ {\isacharparenleft}{\kern0pt}id{\isacharunderscore}{\kern0pt}blinfun\ {\isacharminus}{\kern0pt}\ l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isasymP}\isactrlsub {\isadigit{1}}\ d{\isacharparenright}{\kern0pt}\ Q{\isadigit{2}}\ R{\isadigit{2}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isakeyword{and}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ R{\isadigit{2}}{\isacharparenright}{\kern0pt}\ {\isasymle}\ {\isacharparenleft}{\kern0pt}blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ R{\isadigit{1}}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isakeyword{and}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ R{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isasymle}\ {\isacharparenleft}{\kern0pt}blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ {\isacharparenleft}{\kern0pt}l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isasymP}\isactrlsub {\isadigit{1}}\ d{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}norm\ {\isacharparenleft}{\kern0pt}inv\isactrlsub L\ Q{\isadigit{2}}\ o\isactrlsub L\ R{\isadigit{2}}{\isacharparenright}{\kern0pt}\ {\isasymle}\ norm\ {\isacharparenleft}{\kern0pt}inv\isactrlsub L\ Q{\isadigit{1}}\ o\isactrlsub L\ R{\isadigit{1}}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}R{\isadigit{1}}\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ R{\isadigit{1}}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}R{\isadigit{2}}\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ R{\isadigit{2}}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}Q{\isadigit{1}}\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ Q{\isadigit{1}}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}Q{\isadigit{2}}\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ Q{\isadigit{2}}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ \isanewline
-\ \ \ \ inv{\isacharunderscore}{\kern0pt}Q{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}inv\isactrlsub L\ Q\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSum}i{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}id{\isacharunderscore}{\kern0pt}blinfun\ {\isacharminus}{\kern0pt}\ Q{\isacharparenright}{\kern0pt}{\isacharcircum}{\kern0pt}{\isacharcircum}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ {\isachardoublequoteopen}norm\ {\isacharparenleft}{\kern0pt}id{\isacharunderscore}{\kern0pt}blinfun\ {\isacharminus}{\kern0pt}\ Q{\isacharparenright}{\kern0pt}\ {\isacharless}{\kern0pt}\ {\isadigit{1}}{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ splitting{\isacharunderscore}{\kern0pt}eq{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}id{\isacharunderscore}{\kern0pt}blinfun\ {\isacharminus}{\kern0pt}\ Q\ {\isacharequal}{\kern0pt}\ l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isasymP}\isactrlsub {\isadigit{1}}\ d\ {\isacharminus}{\kern0pt}\ R{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ nonneg{\isacharunderscore}{\kern0pt}Q{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isadigit{0}}\ {\isasymle}\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ {\isacharparenleft}{\kern0pt}id{\isacharunderscore}{\kern0pt}blinfun\ {\isacharminus}{\kern0pt}\ Q{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isakeyword{if}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ R{\isacharparenright}{\kern0pt}\ {\isasymle}\ {\isacharparenleft}{\kern0pt}blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ {\isacharparenleft}{\kern0pt}l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isasymP}\isactrlsub {\isadigit{1}}\ d{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isakeyword{and}\ {\isachardoublequoteopen}is{\isacharunderscore}{\kern0pt}splitting{\isacharunderscore}{\kern0pt}blin\ {\isacharparenleft}{\kern0pt}id{\isacharunderscore}{\kern0pt}blinfun\ {\isacharminus}{\kern0pt}\ l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isasymP}\isactrlsub {\isadigit{1}}\ d{\isacharparenright}{\kern0pt}\ Q\ R{\isachardoublequoteclose}\ \isakeyword{for}\ Q\ R\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}R\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ R{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{show}\isamarkupfalse%
-\ splitting{\isacharunderscore}{\kern0pt}eq{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}id{\isacharunderscore}{\kern0pt}blinfun\ {\isacharminus}{\kern0pt}\ Q\ {\isacharequal}{\kern0pt}\ l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isasymP}\isactrlsub {\isadigit{1}}\ d\ {\isacharminus}{\kern0pt}\ R{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ that\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ eq{\isacharunderscore}{\kern0pt}diff{\isacharunderscore}{\kern0pt}eq\ is{\isacharunderscore}{\kern0pt}splitting{\isacharunderscore}{\kern0pt}blin{\isacharunderscore}{\kern0pt}def{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ R{\isacharunderscore}{\kern0pt}nonneg{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isadigit{0}}\ {\isasymle}\ {\isacharquery}{\kern0pt}R{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ that\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \ \ \isacommand{show}\isamarkupfalse%
-\ nonneg{\isacharunderscore}{\kern0pt}Q{\isacharcolon}{\kern0pt}\ \ {\isachardoublequoteopen}{\isadigit{0}}\ {\isasymle}\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ {\isacharparenleft}{\kern0pt}id{\isacharunderscore}{\kern0pt}blinfun\ {\isacharminus}{\kern0pt}\ Q{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ that\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ splitting{\isacharunderscore}{\kern0pt}eq\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}diff{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ {\isacharparenleft}{\kern0pt}id{\isacharunderscore}{\kern0pt}blinfun\ {\isacharminus}{\kern0pt}\ Q{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymle}\ {\isacharparenleft}{\kern0pt}blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ {\isacharparenleft}{\kern0pt}l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isasymP}\isactrlsub {\isadigit{1}}\ d{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ R{\isacharunderscore}{\kern0pt}nonneg\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ splitting{\isacharunderscore}{\kern0pt}eq\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}diff{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}norm\ {\isacharparenleft}{\kern0pt}id{\isacharunderscore}{\kern0pt}blinfun\ {\isacharminus}{\kern0pt}\ Q{\isacharparenright}{\kern0pt}\ {\isasymle}\ norm\ {\isacharparenleft}{\kern0pt}l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isasymP}\isactrlsub {\isadigit{1}}\ d{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ matrix{\isacharunderscore}{\kern0pt}le{\isacharunderscore}{\kern0pt}norm{\isacharunderscore}{\kern0pt}mono\ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}norm\ {\isacharparenleft}{\kern0pt}id{\isacharunderscore}{\kern0pt}blinfun\ {\isacharminus}{\kern0pt}\ Q{\isacharparenright}{\kern0pt}\ {\isacharless}{\kern0pt}\ {\isadigit{1}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ norm{\isacharunderscore}{\kern0pt}{\isasymP}\isactrlsub {\isadigit{1}}{\isacharunderscore}{\kern0pt}l{\isacharunderscore}{\kern0pt}less\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ order{\isachardot}{\kern0pt}strict{\isacharunderscore}{\kern0pt}trans{\isadigit{1}}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}inv\isactrlsub L\ Q\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSum}i{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}id{\isacharunderscore}{\kern0pt}blinfun\ {\isacharminus}{\kern0pt}\ Q{\isacharparenright}{\kern0pt}\ {\isacharcircum}{\kern0pt}{\isacharcircum}{\kern0pt}\ i{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ inv{\isacharunderscore}{\kern0pt}one{\isacharunderscore}{\kern0pt}sub{\isacharunderscore}{\kern0pt}Q{\isacharprime}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ i{\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}inv\isactrlsub L\ Q{\isadigit{1}}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSum}i{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}id{\isacharunderscore}{\kern0pt}blinfun\ {\isacharminus}{\kern0pt}\ Q{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isacharcircum}{\kern0pt}{\isacharcircum}{\kern0pt}\ i{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ {\isachardoublequoteopen}norm\ {\isacharparenleft}{\kern0pt}id{\isacharunderscore}{\kern0pt}blinfun\ {\isacharminus}{\kern0pt}\ Q{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isacharless}{\kern0pt}\ {\isadigit{1}}{\isachardoublequoteclose}\ \isanewline
-\ \ \ \ \isakeyword{and}\ i{\isadigit{2}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}inv\isactrlsub L\ Q{\isadigit{2}}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSum}i{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}id{\isacharunderscore}{\kern0pt}blinfun\ {\isacharminus}{\kern0pt}\ Q{\isadigit{2}}{\isacharparenright}{\kern0pt}\ {\isacharcircum}{\kern0pt}{\isacharcircum}{\kern0pt}\ i{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ {\isachardoublequoteopen}norm\ {\isacharparenleft}{\kern0pt}id{\isacharunderscore}{\kern0pt}blinfun\ {\isacharminus}{\kern0pt}\ Q{\isadigit{2}}{\isacharparenright}{\kern0pt}\ {\isacharless}{\kern0pt}\ {\isadigit{1}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assms\ \ \isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ intro{\isacharcolon}{\kern0pt}\ inv{\isacharunderscore}{\kern0pt}Q{\isacharbrackleft}{\kern0pt}of\ R{\isadigit{2}}\ Q{\isadigit{2}}{\isacharbrackright}{\kern0pt}\ inv{\isacharunderscore}{\kern0pt}Q{\isacharbrackleft}{\kern0pt}of\ R{\isadigit{1}}\ Q{\isadigit{1}}{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ Q{\isadigit{1}}{\isacharunderscore}{\kern0pt}le{\isacharunderscore}{\kern0pt}Q{\isadigit{2}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ {\isacharparenleft}{\kern0pt}id{\isacharunderscore}{\kern0pt}blinfun\ {\isacharminus}{\kern0pt}\ Q{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isasymle}\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ {\isacharparenleft}{\kern0pt}id{\isacharunderscore}{\kern0pt}blinfun\ {\isacharminus}{\kern0pt}\ Q{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assms\ \isacommand{unfolding}\isamarkupfalse%
-\ is{\isacharunderscore}{\kern0pt}splitting{\isacharunderscore}{\kern0pt}blin{\isacharunderscore}{\kern0pt}def{\isacharprime}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}diff\ eq{\isacharunderscore}{\kern0pt}diff{\isacharunderscore}{\kern0pt}eq\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}add{\isacharparenright}{\kern0pt}\isanewline
-\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ {\isacharparenleft}{\kern0pt}inv\isactrlsub L\ Q{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}{\isasymSum}i{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}id{\isacharunderscore}{\kern0pt}blinfun\ {\isacharminus}{\kern0pt}\ Q{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isacharcircum}{\kern0pt}{\isacharcircum}{\kern0pt}\ i{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ i{\isadigit{1}}\ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isacharequal}{\kern0pt}\ \ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}{\isasymSum}i{\isachardot}{\kern0pt}\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}id{\isacharunderscore}{\kern0pt}blinfun\ {\isacharminus}{\kern0pt}\ Q{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isacharcircum}{\kern0pt}{\isacharcircum}{\kern0pt}\ i{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ bounded{\isacharunderscore}{\kern0pt}linear{\isachardot}{\kern0pt}suminf{\isacharbrackleft}{\kern0pt}OF\ bounded{\isacharunderscore}{\kern0pt}linear{\isacharunderscore}{\kern0pt}blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix{\isacharbrackright}{\kern0pt}\ summable{\isacharunderscore}{\kern0pt}inv{\isacharunderscore}{\kern0pt}Q\ i{\isadigit{1}}{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}\ \isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isasymle}\ {\isacharparenleft}{\kern0pt}{\isasymSum}i{\isachardot}{\kern0pt}\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}id{\isacharunderscore}{\kern0pt}blinfun\ {\isacharminus}{\kern0pt}\ Q{\isadigit{2}}{\isacharparenright}{\kern0pt}\ {\isacharcircum}{\kern0pt}{\isacharcircum}{\kern0pt}\ i{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ le{\isacharunderscore}{\kern0pt}n{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymAnd}n{\isachardot}{\kern0pt}\ {\isadigit{0}}\ {\isasymle}\ n\ {\isasymLongrightarrow}\ {\isacharparenleft}{\kern0pt}{\isasymSum}i{\isacharless}{\kern0pt}n{\isachardot}{\kern0pt}\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}id{\isacharunderscore}{\kern0pt}blinfun\ {\isacharminus}{\kern0pt}\ Q{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isacharcircum}{\kern0pt}{\isacharcircum}{\kern0pt}\ i{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymle}\ {\isacharparenleft}{\kern0pt}{\isasymSum}i{\isacharless}{\kern0pt}n{\isachardot}{\kern0pt}\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}id{\isacharunderscore}{\kern0pt}blinfun\ {\isacharminus}{\kern0pt}\ Q{\isadigit{2}}{\isacharparenright}{\kern0pt}\ {\isacharcircum}{\kern0pt}{\isacharcircum}{\kern0pt}\ i{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assms\ nonneg{\isacharunderscore}{\kern0pt}Q\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ sum{\isacharunderscore}{\kern0pt}mono\ matpow{\isacharunderscore}{\kern0pt}mono\ simp{\isacharcolon}{\kern0pt}\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}matpow\ Q{\isadigit{1}}{\isacharunderscore}{\kern0pt}le{\isacharunderscore}{\kern0pt}Q{\isadigit{2}}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ le{\isacharunderscore}{\kern0pt}n{\isacharunderscore}{\kern0pt}elem{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymAnd}n{\isachardot}{\kern0pt}\ {\isadigit{0}}\ {\isasymle}\ n\ {\isasymLongrightarrow}\ {\isacharparenleft}{\kern0pt}{\isasymSum}i{\isacharless}{\kern0pt}n{\isachardot}{\kern0pt}\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}id{\isacharunderscore}{\kern0pt}blinfun\ {\isacharminus}{\kern0pt}\ Q{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isacharcircum}{\kern0pt}{\isacharcircum}{\kern0pt}\ i{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isachardollar}{\kern0pt}\ i\ {\isachardollar}{\kern0pt}\ j\ {\isasymle}\ {\isacharparenleft}{\kern0pt}{\isasymSum}i{\isacharless}{\kern0pt}n{\isachardot}{\kern0pt}\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}id{\isacharunderscore}{\kern0pt}blinfun\ {\isacharminus}{\kern0pt}\ Q{\isadigit{2}}{\isacharparenright}{\kern0pt}\ {\isacharcircum}{\kern0pt}{\isacharcircum}{\kern0pt}\ i{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isachardollar}{\kern0pt}\ i\ {\isachardollar}{\kern0pt}\ j\ {\isachardoublequoteclose}\ \isakeyword{for}\ i\ j\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ less{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}vec{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isasymlambda}n{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSum}i{\isacharless}{\kern0pt}n{\isachardot}{\kern0pt}\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}id{\isacharunderscore}{\kern0pt}blinfun\ {\isacharminus}{\kern0pt}\ Q{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isacharcircum}{\kern0pt}{\isacharcircum}{\kern0pt}\ i{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymlonglonglongrightarrow}\ {\isacharparenleft}{\kern0pt}{\isasymSum}i{\isachardot}{\kern0pt}\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}id{\isacharunderscore}{\kern0pt}blinfun\ {\isacharminus}{\kern0pt}\ Q{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isacharcircum}{\kern0pt}{\isacharcircum}{\kern0pt}\ i{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ bounded{\isacharunderscore}{\kern0pt}linear{\isachardot}{\kern0pt}summable{\isacharbrackleft}{\kern0pt}of\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix{\isacharbrackright}{\kern0pt}\ summable{\isacharunderscore}{\kern0pt}LIMSEQ\ simp\ add{\isacharcolon}{\kern0pt}\ bounded{\isacharunderscore}{\kern0pt}linear{\isacharunderscore}{\kern0pt}blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ i{\isadigit{1}}{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}\ summable{\isacharunderscore}{\kern0pt}inv{\isacharunderscore}{\kern0pt}Q{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ le{\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isasymlambda}n{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSum}i{\isacharless}{\kern0pt}n{\isachardot}{\kern0pt}\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}id{\isacharunderscore}{\kern0pt}blinfun\ {\isacharminus}{\kern0pt}\ Q{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isacharcircum}{\kern0pt}{\isacharcircum}{\kern0pt}\ i{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isachardollar}{\kern0pt}\ j\ {\isachardollar}{\kern0pt}\ k{\isacharparenright}{\kern0pt}\ {\isasymlonglonglongrightarrow}\ {\isacharparenleft}{\kern0pt}{\isasymSum}i{\isachardot}{\kern0pt}\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}id{\isacharunderscore}{\kern0pt}blinfun\ {\isacharminus}{\kern0pt}\ Q{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isacharcircum}{\kern0pt}{\isacharcircum}{\kern0pt}\ i{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isachardollar}{\kern0pt}\ j\ {\isachardollar}{\kern0pt}\ k{\isachardoublequoteclose}\ \isakeyword{for}\ j\ k\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ tendsto{\isacharunderscore}{\kern0pt}vec{\isacharunderscore}{\kern0pt}nth\ \isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isasymlambda}n{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSum}i{\isacharless}{\kern0pt}n{\isachardot}{\kern0pt}\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}id{\isacharunderscore}{\kern0pt}blinfun\ {\isacharminus}{\kern0pt}\ Q{\isadigit{2}}{\isacharparenright}{\kern0pt}\ {\isacharcircum}{\kern0pt}{\isacharcircum}{\kern0pt}\ i{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymlonglonglongrightarrow}\ {\isacharparenleft}{\kern0pt}{\isasymSum}i{\isachardot}{\kern0pt}\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}id{\isacharunderscore}{\kern0pt}blinfun\ {\isacharminus}{\kern0pt}\ Q{\isadigit{2}}{\isacharparenright}{\kern0pt}\ {\isacharcircum}{\kern0pt}{\isacharcircum}{\kern0pt}\ i{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ bounded{\isacharunderscore}{\kern0pt}linear{\isachardot}{\kern0pt}summable{\isacharbrackleft}{\kern0pt}of\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix{\isacharbrackright}{\kern0pt}\ summable{\isacharunderscore}{\kern0pt}LIMSEQ\ simp\ add{\isacharcolon}{\kern0pt}\ bounded{\isacharunderscore}{\kern0pt}linear{\isacharunderscore}{\kern0pt}blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ i{\isadigit{2}}{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}\ summable{\isacharunderscore}{\kern0pt}inv{\isacharunderscore}{\kern0pt}Q{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ le{\isadigit{2}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isasymlambda}n{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSum}i{\isacharless}{\kern0pt}n{\isachardot}{\kern0pt}\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}id{\isacharunderscore}{\kern0pt}blinfun\ {\isacharminus}{\kern0pt}\ Q{\isadigit{2}}{\isacharparenright}{\kern0pt}\ {\isacharcircum}{\kern0pt}{\isacharcircum}{\kern0pt}\ i{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isachardollar}{\kern0pt}\ j\ {\isachardollar}{\kern0pt}\ k{\isacharparenright}{\kern0pt}\ {\isasymlonglonglongrightarrow}\ {\isacharparenleft}{\kern0pt}{\isasymSum}i{\isachardot}{\kern0pt}\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}id{\isacharunderscore}{\kern0pt}blinfun\ {\isacharminus}{\kern0pt}\ Q{\isadigit{2}}{\isacharparenright}{\kern0pt}\ {\isacharcircum}{\kern0pt}{\isacharcircum}{\kern0pt}\ i{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isachardollar}{\kern0pt}\ j\ {\isachardollar}{\kern0pt}\ k{\isachardoublequoteclose}\ \isakeyword{for}\ j\ k\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ tendsto{\isacharunderscore}{\kern0pt}vec{\isacharunderscore}{\kern0pt}nth\ \isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}{\isasymSum}i{\isachardot}{\kern0pt}\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}id{\isacharunderscore}{\kern0pt}blinfun\ {\isacharminus}{\kern0pt}\ Q{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isacharcircum}{\kern0pt}{\isacharcircum}{\kern0pt}\ i{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardollar}{\kern0pt}\ j\ {\isachardollar}{\kern0pt}\ k{\isacharparenright}{\kern0pt}\ {\isasymle}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}{\isasymSum}i{\isachardot}{\kern0pt}\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}id{\isacharunderscore}{\kern0pt}blinfun\ {\isacharminus}{\kern0pt}\ Q{\isadigit{2}}{\isacharparenright}{\kern0pt}\ {\isacharcircum}{\kern0pt}{\isacharcircum}{\kern0pt}\ i{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isachardollar}{\kern0pt}\ j\ {\isachardollar}{\kern0pt}\ k{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{for}\ j\ k\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}fastforce\ intro{\isacharcolon}{\kern0pt}\ Topological{\isacharunderscore}{\kern0pt}Spaces{\isachardot}{\kern0pt}lim{\isacharunderscore}{\kern0pt}mono{\isacharbrackleft}{\kern0pt}OF\ le{\isacharunderscore}{\kern0pt}n{\isacharunderscore}{\kern0pt}elem\ le{\isadigit{1}}\ le{\isadigit{2}}{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ less{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}vec{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isacharequal}{\kern0pt}\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ {\isacharparenleft}{\kern0pt}inv\isactrlsub L\ Q{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ summable{\isacharunderscore}{\kern0pt}inv{\isacharunderscore}{\kern0pt}Q\ i{\isadigit{2}}{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}\ i{\isadigit{2}}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ bounded{\isacharunderscore}{\kern0pt}linear{\isachardot}{\kern0pt}suminf{\isacharbrackleft}{\kern0pt}OF\ bounded{\isacharunderscore}{\kern0pt}linear{\isacharunderscore}{\kern0pt}blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix{\isacharcomma}{\kern0pt}\ symmetric{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{finally}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ Q{\isadigit{1}}{\isacharunderscore}{\kern0pt}le{\isacharunderscore}{\kern0pt}Q{\isadigit{2}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ {\isacharparenleft}{\kern0pt}inv\isactrlsub L\ Q{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isasymle}\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ {\isacharparenleft}{\kern0pt}inv\isactrlsub L\ Q{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isacommand{{\isachardot}{\kern0pt}}\isamarkupfalse%
-\isanewline
-\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isacharasterisk}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isadigit{0}}\ {\isasymle}\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}inv\isactrlsub L\ Q{\isadigit{1}}{\isacharparenright}{\kern0pt}\ o\isactrlsub L\ R{\isadigit{1}}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ {\isachardoublequoteopen}{\isadigit{0}}\ {\isasymle}\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}inv\isactrlsub L\ Q{\isadigit{2}}{\isacharparenright}{\kern0pt}\ o\isactrlsub L\ R{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assms\ is{\isacharunderscore}{\kern0pt}splitting{\isacharunderscore}{\kern0pt}blin{\isacharunderscore}{\kern0pt}def{\isacharprime}{\kern0pt}\ \isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}comp\ intro{\isacharcolon}{\kern0pt}\ nonneg{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}mult{\isacharparenright}{\kern0pt}\isanewline
-\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isadigit{0}}\ {\isasymle}\ {\isacharparenleft}{\kern0pt}id{\isacharunderscore}{\kern0pt}blinfun\ {\isacharminus}{\kern0pt}\ l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isasymP}\isactrlsub {\isadigit{1}}\ d{\isacharparenright}{\kern0pt}\ {\isadigit{1}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ less{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}le{\isacharbrackleft}{\kern0pt}OF\ disc{\isacharunderscore}{\kern0pt}lt{\isacharunderscore}{\kern0pt}one{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ blinfun{\isachardot}{\kern0pt}diff{\isacharunderscore}{\kern0pt}left\ less{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}bfun{\isacharunderscore}{\kern0pt}def\ blinfun{\isachardot}{\kern0pt}scaleR{\isacharunderscore}{\kern0pt}left{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}inv\isactrlsub L\ Q{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}id{\isacharunderscore}{\kern0pt}blinfun\ {\isacharminus}{\kern0pt}\ l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isasymP}\isactrlsub {\isadigit{1}}\ d{\isacharparenright}{\kern0pt}\ {\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isasymle}\ {\isacharparenleft}{\kern0pt}inv\isactrlsub L\ Q{\isadigit{2}}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}id{\isacharunderscore}{\kern0pt}blinfun\ {\isacharminus}{\kern0pt}\ l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isasymP}\isactrlsub {\isadigit{1}}\ d{\isacharparenright}{\kern0pt}\ {\isadigit{1}}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ Q{\isadigit{1}}{\isacharunderscore}{\kern0pt}le{\isacharunderscore}{\kern0pt}Q{\isadigit{2}}\ blinfun{\isachardot}{\kern0pt}diff{\isacharunderscore}{\kern0pt}left\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}diff\ diff{\isacharunderscore}{\kern0pt}ge{\isacharunderscore}{\kern0pt}{\isadigit{0}}{\isacharunderscore}{\kern0pt}iff{\isacharunderscore}{\kern0pt}ge\ nonneg{\isacharunderscore}{\kern0pt}blinfun{\isacharunderscore}{\kern0pt}nonneg{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}inv\isactrlsub L\ Q{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}Q{\isadigit{1}}\ {\isacharminus}{\kern0pt}\ R{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isasymle}\ {\isacharparenleft}{\kern0pt}inv\isactrlsub L\ Q{\isadigit{2}}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}Q{\isadigit{2}}\ {\isacharminus}{\kern0pt}\ R{\isadigit{2}}{\isacharparenright}{\kern0pt}\ {\isadigit{1}}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharcomma}{\kern0pt}\ opaque{\isacharunderscore}{\kern0pt}lifting{\isacharparenright}{\kern0pt}\ assms{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}\ assms{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}\ is{\isacharunderscore}{\kern0pt}splitting{\isacharunderscore}{\kern0pt}blin{\isacharunderscore}{\kern0pt}def{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}inv\isactrlsub L\ Q{\isadigit{1}}\ o\isactrlsub L\ Q{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isadigit{1}}\ {\isacharminus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}inv\isactrlsub L\ Q{\isadigit{1}}\ o\isactrlsub L\ R{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isadigit{1}}\ {\isasymle}\ {\isacharparenleft}{\kern0pt}inv\isactrlsub L\ Q{\isadigit{2}}\ o\isactrlsub L\ Q{\isadigit{2}}{\isacharparenright}{\kern0pt}\ {\isadigit{1}}\ {\isacharminus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}inv\isactrlsub L\ Q{\isadigit{2}}\ o\isactrlsub L\ R{\isadigit{2}}{\isacharparenright}{\kern0pt}\ {\isadigit{1}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ blinfun{\isachardot}{\kern0pt}add{\isacharunderscore}{\kern0pt}left\ blinfun{\isachardot}{\kern0pt}diff{\isacharunderscore}{\kern0pt}right\ blinfun{\isachardot}{\kern0pt}diff{\isacharunderscore}{\kern0pt}left{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}inv\isactrlsub L\ Q{\isadigit{2}}\ o\isactrlsub L\ R{\isadigit{2}}{\isacharparenright}{\kern0pt}\ {\isadigit{1}}\ {\isasymle}\ {\isacharparenleft}{\kern0pt}inv\isactrlsub L\ Q{\isadigit{1}}\ o\isactrlsub L\ R{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isadigit{1}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assms\ \isanewline
-\ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ is{\isacharunderscore}{\kern0pt}splitting{\isacharunderscore}{\kern0pt}blin{\isacharunderscore}{\kern0pt}def{\isacharprime}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isadigit{0}}\ {\isasymle}\ {\isacharparenleft}{\kern0pt}inv\isactrlsub L\ Q{\isadigit{2}}\ o\isactrlsub L\ R{\isadigit{2}}{\isacharparenright}{\kern0pt}\ {\isadigit{1}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isacharasterisk}{\kern0pt}\ \isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}fastforce\ simp{\isacharcolon}{\kern0pt}\ less{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}bfunI\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ nonneg{\isacharunderscore}{\kern0pt}blinfun{\isacharunderscore}{\kern0pt}nonneg{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}norm\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}inv\isactrlsub L\ Q{\isadigit{2}}\ o\isactrlsub L\ R{\isadigit{2}}{\isacharparenright}{\kern0pt}\ {\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isasymle}\ norm\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}inv\isactrlsub L\ Q{\isadigit{1}}\ o\isactrlsub L\ R{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isadigit{1}}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ less{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}bfun{\isacharunderscore}{\kern0pt}def\ norm{\isacharunderscore}{\kern0pt}bfun{\isacharunderscore}{\kern0pt}def{\isacharprime}{\kern0pt}\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ abs{\isacharunderscore}{\kern0pt}ge{\isacharunderscore}{\kern0pt}self\ cSUP{\isacharunderscore}{\kern0pt}mono\ intro{\isacharcolon}{\kern0pt}\ order{\isachardot}{\kern0pt}trans{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}norm\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}inv\isactrlsub L\ Q{\isadigit{2}}\ o\isactrlsub L\ R{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymle}\ norm\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}inv\isactrlsub L\ Q{\isadigit{1}}\ o\isactrlsub L\ R{\isadigit{1}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ norm{\isacharunderscore}{\kern0pt}nonneg{\isacharunderscore}{\kern0pt}blinfun{\isacharunderscore}{\kern0pt}one\ {\isacharasterisk}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Gauss Seidel Splitting%
-}
-\isamarkuptrue%
-%
-\isamarkupsubsubsection{Definition of Upper and Lower Triangular Matrices%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{definition}\isamarkupfalse%
-\ {\isachardoublequoteopen}P{\isacharunderscore}{\kern0pt}dec\ d\ {\isasymequiv}\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ {\isacharparenleft}{\kern0pt}{\isasymP}\isactrlsub {\isadigit{1}}\ {\isacharparenleft}{\kern0pt}mk{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\ d{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ {\isachardoublequoteopen}P{\isacharunderscore}{\kern0pt}upper\ d\ {\isasymequiv}\ {\isacharparenleft}{\kern0pt}{\isasymchi}\ i\ j{\isachardot}{\kern0pt}\ if\ i\ {\isasymle}\ j\ then\ P{\isacharunderscore}{\kern0pt}dec\ d\ {\isachardollar}{\kern0pt}\ i\ {\isachardollar}{\kern0pt}\ j\ else\ {\isadigit{0}}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ {\isachardoublequoteopen}P{\isacharunderscore}{\kern0pt}lower\ d\ {\isasymequiv}\ {\isacharparenleft}{\kern0pt}{\isasymchi}\ i\ j{\isachardot}{\kern0pt}\ if\ j\ {\isacharless}{\kern0pt}\ i\ then\ P{\isacharunderscore}{\kern0pt}dec\ d\ {\isachardollar}{\kern0pt}\ i\ {\isachardollar}{\kern0pt}\ j\ else\ {\isadigit{0}}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymP}\isactrlsub U\ d\ {\isacharequal}{\kern0pt}\ matrix{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}blinfun\ {\isacharparenleft}{\kern0pt}P{\isacharunderscore}{\kern0pt}upper\ d{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymP}\isactrlsub L\ d\ {\isacharequal}{\kern0pt}\ matrix{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}blinfun\ {\isacharparenleft}{\kern0pt}P{\isacharunderscore}{\kern0pt}lower\ d{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ P{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}elem{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}P{\isacharunderscore}{\kern0pt}dec\ d\ {\isachardollar}{\kern0pt}\ i\ {\isachardollar}{\kern0pt}\ j\ {\isacharequal}{\kern0pt}\ pmf\ {\isacharparenleft}{\kern0pt}K\ {\isacharparenleft}{\kern0pt}i{\isacharcomma}{\kern0pt}\ d\ i{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ j{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}def\ matrix{\isacharunderscore}{\kern0pt}def\ {\isasymP}\isactrlsub {\isadigit{1}}{\isachardot}{\kern0pt}rep{\isacharunderscore}{\kern0pt}eq\ K{\isacharunderscore}{\kern0pt}st{\isacharunderscore}{\kern0pt}def\ \ P{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}def\ push{\isacharunderscore}{\kern0pt}exp{\isachardot}{\kern0pt}rep{\isacharunderscore}{\kern0pt}eq\ vec{\isacharunderscore}{\kern0pt}lambda{\isacharunderscore}{\kern0pt}beta\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}subst\ pmf{\isacharunderscore}{\kern0pt}expectation{\isacharunderscore}{\kern0pt}bind{\isacharbrackleft}{\kern0pt}of\ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}d\ i{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\ \isanewline
-\ \ \ \ {\isacharparenleft}{\kern0pt}auto\ split{\isacharcolon}{\kern0pt}\ if{\isacharunderscore}{\kern0pt}splits\ simp{\isacharcolon}{\kern0pt}\ mk{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det{\isacharunderscore}{\kern0pt}def\ axis{\isacharunderscore}{\kern0pt}def\ vec{\isacharunderscore}{\kern0pt}lambda{\isacharunderscore}{\kern0pt}inverse\ integral{\isacharunderscore}{\kern0pt}measure{\isacharunderscore}{\kern0pt}pmf{\isacharbrackleft}{\kern0pt}of\ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}j{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\ \ \isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ nonneg{\isacharunderscore}{\kern0pt}{\isasymP}\isactrlsub U{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}nonneg{\isacharunderscore}{\kern0pt}blinfun\ {\isacharparenleft}{\kern0pt}{\isasymP}\isactrlsub U\ d{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ {\isasymP}\isactrlsub U{\isacharunderscore}{\kern0pt}def\ Finite{\isacharunderscore}{\kern0pt}Cartesian{\isacharunderscore}{\kern0pt}Product{\isachardot}{\kern0pt}less{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}vec{\isacharunderscore}{\kern0pt}def\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}inv\ P{\isacharunderscore}{\kern0pt}upper{\isacharunderscore}{\kern0pt}def\ P{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}elem\ \isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ nonneg{\isacharunderscore}{\kern0pt}P{\isacharunderscore}{\kern0pt}dec{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isadigit{0}}\ {\isasymle}\ P{\isacharunderscore}{\kern0pt}dec\ d{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ Finite{\isacharunderscore}{\kern0pt}Cartesian{\isacharunderscore}{\kern0pt}Product{\isachardot}{\kern0pt}less{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}vec{\isacharunderscore}{\kern0pt}def\ P{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}elem{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ nonneg{\isacharunderscore}{\kern0pt}P{\isacharunderscore}{\kern0pt}upper{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isadigit{0}}\ {\isasymle}\ P{\isacharunderscore}{\kern0pt}upper\ d{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ nonneg{\isacharunderscore}{\kern0pt}P{\isacharunderscore}{\kern0pt}dec\ \isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ Finite{\isacharunderscore}{\kern0pt}Cartesian{\isacharunderscore}{\kern0pt}Product{\isachardot}{\kern0pt}less{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}vec{\isacharunderscore}{\kern0pt}def\ P{\isacharunderscore}{\kern0pt}upper{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ nonneg{\isacharunderscore}{\kern0pt}P{\isacharunderscore}{\kern0pt}lower{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isadigit{0}}\ {\isasymle}\ P{\isacharunderscore}{\kern0pt}lower\ d{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ nonneg{\isacharunderscore}{\kern0pt}P{\isacharunderscore}{\kern0pt}dec\ \isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ Finite{\isacharunderscore}{\kern0pt}Cartesian{\isacharunderscore}{\kern0pt}Product{\isachardot}{\kern0pt}less{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}vec{\isacharunderscore}{\kern0pt}def\ P{\isacharunderscore}{\kern0pt}lower{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ nonneg{\isacharunderscore}{\kern0pt}{\isasymP}\isactrlsub L{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}nonneg{\isacharunderscore}{\kern0pt}blinfun\ {\isacharparenleft}{\kern0pt}{\isasymP}\isactrlsub L\ d{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ {\isasymP}\isactrlsub L{\isacharunderscore}{\kern0pt}def\ Finite{\isacharunderscore}{\kern0pt}Cartesian{\isacharunderscore}{\kern0pt}Product{\isachardot}{\kern0pt}less{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}vec{\isacharunderscore}{\kern0pt}def\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}inv\ P{\isacharunderscore}{\kern0pt}lower{\isacharunderscore}{\kern0pt}def\ P{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}elem\ \isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ nonneg{\isacharunderscore}{\kern0pt}{\isasymP}\isactrlsub {\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}nonneg{\isacharunderscore}{\kern0pt}blinfun\ {\isacharparenleft}{\kern0pt}{\isasymP}\isactrlsub {\isadigit{1}}\ d{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}def\ matrix{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ Finite{\isacharunderscore}{\kern0pt}Cartesian{\isacharunderscore}{\kern0pt}Product{\isachardot}{\kern0pt}less{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}vec{\isacharunderscore}{\kern0pt}def\ axis{\isacharunderscore}{\kern0pt}def\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isasymP}\isactrlsub {\isadigit{1}}{\isacharunderscore}{\kern0pt}pos\ less{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}bfunD{\isacharbrackleft}{\kern0pt}of\ {\isadigit{0}}{\isacharcomma}{\kern0pt}\ simplified{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ norm{\isacharunderscore}{\kern0pt}{\isasymP}\isactrlsub L{\isacharunderscore}{\kern0pt}le{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}norm\ {\isacharparenleft}{\kern0pt}{\isasymP}\isactrlsub L\ d{\isacharparenright}{\kern0pt}\ {\isasymle}\ norm\ {\isacharparenleft}{\kern0pt}{\isasymP}\isactrlsub {\isadigit{1}}\ {\isacharparenleft}{\kern0pt}mk{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\ d{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ nonneg{\isacharunderscore}{\kern0pt}{\isasymP}\isactrlsub {\isadigit{1}}\ \isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}fastforce\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ matrix{\isacharunderscore}{\kern0pt}le{\isacharunderscore}{\kern0pt}norm{\isacharunderscore}{\kern0pt}mono\ simp{\isacharcolon}{\kern0pt}\ Finite{\isacharunderscore}{\kern0pt}Cartesian{\isacharunderscore}{\kern0pt}Product{\isachardot}{\kern0pt}less{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}vec{\isacharunderscore}{\kern0pt}def\ P{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}def\ P{\isacharunderscore}{\kern0pt}lower{\isacharunderscore}{\kern0pt}def\ {\isasymP}\isactrlsub L{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ norm{\isacharunderscore}{\kern0pt}{\isasymP}\isactrlsub L{\isacharunderscore}{\kern0pt}le{\isacharunderscore}{\kern0pt}one{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}norm\ {\isacharparenleft}{\kern0pt}{\isasymP}\isactrlsub L\ d{\isacharparenright}{\kern0pt}\ {\isasymle}\ {\isadigit{1}}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ norm{\isacharunderscore}{\kern0pt}{\isasymP}\isactrlsub L{\isacharunderscore}{\kern0pt}le\ norm{\isacharunderscore}{\kern0pt}{\isasymP}\isactrlsub {\isadigit{1}}\ \isacommand{by}\isamarkupfalse%
-\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ norm{\isacharunderscore}{\kern0pt}{\isasymP}\isactrlsub L{\isacharunderscore}{\kern0pt}less{\isacharunderscore}{\kern0pt}one{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}norm\ {\isacharparenleft}{\kern0pt}l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isasymP}\isactrlsub L\ d{\isacharparenright}{\kern0pt}\ {\isacharless}{\kern0pt}\ {\isadigit{1}}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ order{\isachardot}{\kern0pt}strict{\isacharunderscore}{\kern0pt}trans{\isadigit{1}}{\isacharbrackleft}{\kern0pt}OF\ mult{\isacharunderscore}{\kern0pt}left{\isacharunderscore}{\kern0pt}le\ disc{\isacharunderscore}{\kern0pt}lt{\isacharunderscore}{\kern0pt}one{\isacharbrackright}{\kern0pt}\ zero{\isacharunderscore}{\kern0pt}le{\isacharunderscore}{\kern0pt}disc\ norm{\isacharunderscore}{\kern0pt}{\isasymP}\isactrlsub L{\isacharunderscore}{\kern0pt}le{\isacharunderscore}{\kern0pt}one\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ {\isasymP}\isactrlsub L{\isacharunderscore}{\kern0pt}le{\isacharunderscore}{\kern0pt}{\isasymP}\isactrlsub {\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isadigit{0}}\ {\isasymle}\ v\ {\isasymLongrightarrow}\ {\isasymP}\isactrlsub L\ d\ v\ {\isasymle}\ {\isasymP}\isactrlsub {\isadigit{1}}\ {\isacharparenleft}{\kern0pt}mk{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\ d{\isacharparenright}{\kern0pt}\ v{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isadigit{0}}\ {\isasymle}\ v{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}P{\isacharunderscore}{\kern0pt}lower\ d\ {\isasymle}\ P{\isacharunderscore}{\kern0pt}dec\ d{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ nonneg{\isacharunderscore}{\kern0pt}P{\isacharunderscore}{\kern0pt}dec\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ P{\isacharunderscore}{\kern0pt}lower{\isacharunderscore}{\kern0pt}def\ less{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}vec{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ P{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}def\ {\isasymP}\isactrlsub L{\isacharunderscore}{\kern0pt}def\ blinfun{\isacharunderscore}{\kern0pt}apply{\isacharunderscore}{\kern0pt}mono\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}inv\ nonneg{\isacharunderscore}{\kern0pt}{\isasymP}\isactrlsub L{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ {\isasymP}\isactrlsub U{\isacharunderscore}{\kern0pt}le{\isacharunderscore}{\kern0pt}{\isasymP}\isactrlsub {\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isadigit{0}}\ {\isasymle}\ v\ {\isasymLongrightarrow}\ {\isasymP}\isactrlsub U\ d\ v\ {\isasymle}\ {\isasymP}\isactrlsub {\isadigit{1}}\ {\isacharparenleft}{\kern0pt}mk{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\ d{\isacharparenright}{\kern0pt}\ v{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isadigit{0}}\ {\isasymle}\ v{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}P{\isacharunderscore}{\kern0pt}upper\ d\ {\isasymle}\ P{\isacharunderscore}{\kern0pt}dec\ d{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ nonneg{\isacharunderscore}{\kern0pt}P{\isacharunderscore}{\kern0pt}dec\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ P{\isacharunderscore}{\kern0pt}upper{\isacharunderscore}{\kern0pt}def\ less{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}vec{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ P{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}def\ {\isasymP}\isactrlsub U{\isacharunderscore}{\kern0pt}def\ blinfun{\isacharunderscore}{\kern0pt}apply{\isacharunderscore}{\kern0pt}mono\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}inv\ nonneg{\isacharunderscore}{\kern0pt}{\isasymP}\isactrlsub U{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ row{\isacharunderscore}{\kern0pt}P{\isacharunderscore}{\kern0pt}upper{\isacharunderscore}{\kern0pt}indep{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}d\ s\ {\isacharequal}{\kern0pt}\ d{\isacharprime}{\kern0pt}\ s\ {\isasymLongrightarrow}\ row\ s\ {\isacharparenleft}{\kern0pt}P{\isacharunderscore}{\kern0pt}upper\ d{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ row\ s\ {\isacharparenleft}{\kern0pt}P{\isacharunderscore}{\kern0pt}upper\ d{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ row{\isacharunderscore}{\kern0pt}def\ P{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}elem\ P{\isacharunderscore}{\kern0pt}upper{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ row{\isacharunderscore}{\kern0pt}P{\isacharunderscore}{\kern0pt}lower{\isacharunderscore}{\kern0pt}indep{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}d\ s\ {\isacharequal}{\kern0pt}\ d{\isacharprime}{\kern0pt}\ s\ {\isasymLongrightarrow}\ row\ s\ {\isacharparenleft}{\kern0pt}P{\isacharunderscore}{\kern0pt}lower\ d{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ row\ s\ {\isacharparenleft}{\kern0pt}P{\isacharunderscore}{\kern0pt}lower\ d{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ row{\isacharunderscore}{\kern0pt}def\ P{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}elem\ P{\isacharunderscore}{\kern0pt}lower{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ triangular{\isacharunderscore}{\kern0pt}mat{\isacharunderscore}{\kern0pt}P{\isacharunderscore}{\kern0pt}upper{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}upper{\isacharunderscore}{\kern0pt}triangular{\isacharunderscore}{\kern0pt}mat\ {\isacharparenleft}{\kern0pt}P{\isacharunderscore}{\kern0pt}upper\ d{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ upper{\isacharunderscore}{\kern0pt}triangular{\isacharunderscore}{\kern0pt}mat{\isacharunderscore}{\kern0pt}def\ P{\isacharunderscore}{\kern0pt}upper{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ slt{\isacharunderscore}{\kern0pt}P{\isacharunderscore}{\kern0pt}lower{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}strict{\isacharunderscore}{\kern0pt}lower{\isacharunderscore}{\kern0pt}triangular{\isacharunderscore}{\kern0pt}mat\ {\isacharparenleft}{\kern0pt}P{\isacharunderscore}{\kern0pt}lower\ d{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ strict{\isacharunderscore}{\kern0pt}lower{\isacharunderscore}{\kern0pt}triangular{\isacharunderscore}{\kern0pt}mat{\isacharunderscore}{\kern0pt}def\ P{\isacharunderscore}{\kern0pt}lower{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ lt{\isacharunderscore}{\kern0pt}P{\isacharunderscore}{\kern0pt}lower{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}lower{\isacharunderscore}{\kern0pt}triangular{\isacharunderscore}{\kern0pt}mat\ {\isacharparenleft}{\kern0pt}P{\isacharunderscore}{\kern0pt}lower\ d{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ lower{\isacharunderscore}{\kern0pt}triangular{\isacharunderscore}{\kern0pt}mat{\isacharunderscore}{\kern0pt}def\ P{\isacharunderscore}{\kern0pt}lower{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsubsection{Gauss Seidel is a Regular Splitting%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{definition}\isamarkupfalse%
-\ {\isachardoublequoteopen}Q{\isacharunderscore}{\kern0pt}GS\ d\ {\isacharequal}{\kern0pt}\ id{\isacharunderscore}{\kern0pt}blinfun\ {\isacharminus}{\kern0pt}\ l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isasymP}\isactrlsub L\ d{\isachardoublequoteclose}\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ {\isachardoublequoteopen}R{\isacharunderscore}{\kern0pt}GS\ d\ {\isacharequal}{\kern0pt}\ l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isasymP}\isactrlsub U\ d{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ splitting{\isacharunderscore}{\kern0pt}gauss{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}is{\isacharunderscore}{\kern0pt}splitting{\isacharunderscore}{\kern0pt}blin\ {\isacharparenleft}{\kern0pt}id{\isacharunderscore}{\kern0pt}blinfun\ {\isacharminus}{\kern0pt}\ l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isasymP}\isactrlsub {\isadigit{1}}\ {\isacharparenleft}{\kern0pt}mk{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\ d{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}Q{\isacharunderscore}{\kern0pt}GS\ d{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}R{\isacharunderscore}{\kern0pt}GS\ d{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ is{\isacharunderscore}{\kern0pt}splitting{\isacharunderscore}{\kern0pt}blin{\isacharunderscore}{\kern0pt}def{\isacharprime}{\kern0pt}\isanewline
-\isacommand{proof}\isamarkupfalse%
-\ safe\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}nonneg{\isacharunderscore}{\kern0pt}blinfun\ {\isacharparenleft}{\kern0pt}R{\isacharunderscore}{\kern0pt}GS\ d{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ R{\isacharunderscore}{\kern0pt}GS{\isacharunderscore}{\kern0pt}def\ {\isasymP}\isactrlsub U{\isacharunderscore}{\kern0pt}def\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}scaleR\ Finite{\isacharunderscore}{\kern0pt}Cartesian{\isacharunderscore}{\kern0pt}Product{\isachardot}{\kern0pt}less{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}vec{\isacharunderscore}{\kern0pt}def\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}inv\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ nonneg{\isacharunderscore}{\kern0pt}P{\isacharunderscore}{\kern0pt}upper\ \isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ mult{\isacharunderscore}{\kern0pt}nonneg{\isacharunderscore}{\kern0pt}nonneg{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymP}\isactrlsub L\ d\ {\isacharplus}{\kern0pt}\ {\isasymP}\isactrlsub U\ d\ {\isacharequal}{\kern0pt}\ {\isasymP}\isactrlsub {\isadigit{1}}\ {\isacharparenleft}{\kern0pt}mk{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\ d{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{for}\ d\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymP}\isactrlsub L\ d\ {\isacharplus}{\kern0pt}\ {\isasymP}\isactrlsub U\ d\ \ {\isacharequal}{\kern0pt}\ matrix{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}blinfun\ {\isacharparenleft}{\kern0pt}{\isasymchi}\ i\ j{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ {\isacharparenleft}{\kern0pt}{\isasymP}\isactrlsub {\isadigit{1}}\ {\isacharparenleft}{\kern0pt}mk{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\ d{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isachardollar}{\kern0pt}\ i\ {\isachardollar}{\kern0pt}\ j{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ {\isasymP}\isactrlsub L{\isacharunderscore}{\kern0pt}def\ {\isasymP}\isactrlsub U{\isacharunderscore}{\kern0pt}def\ P{\isacharunderscore}{\kern0pt}lower{\isacharunderscore}{\kern0pt}def\ P{\isacharunderscore}{\kern0pt}upper{\isacharunderscore}{\kern0pt}def\ P{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}def\ matrix{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}blinfun{\isacharunderscore}{\kern0pt}add{\isacharbrackleft}{\kern0pt}symmetric{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ vec{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}iff\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ arg{\isacharunderscore}{\kern0pt}cong{\isacharbrackleft}{\kern0pt}of\ {\isacharunderscore}{\kern0pt}\ {\isacharunderscore}{\kern0pt}\ matrix{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}blinfun{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymP}\isactrlsub {\isadigit{1}}\ {\isacharparenleft}{\kern0pt}mk{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\ d{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ matrix{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}blinfun{\isacharunderscore}{\kern0pt}inv{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{finally}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymP}\isactrlsub L\ d\ {\isacharplus}{\kern0pt}\ {\isasymP}\isactrlsub U\ d\ {\isacharequal}{\kern0pt}\ {\isasymP}\isactrlsub {\isadigit{1}}\ {\isacharparenleft}{\kern0pt}mk{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\ d{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isacommand{{\isachardot}{\kern0pt}}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}id{\isacharunderscore}{\kern0pt}blinfun\ {\isacharminus}{\kern0pt}\ l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isasymP}\isactrlsub {\isadigit{1}}\ {\isacharparenleft}{\kern0pt}mk{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\ d{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ Q{\isacharunderscore}{\kern0pt}GS\ d\ {\isacharminus}{\kern0pt}\ R{\isacharunderscore}{\kern0pt}GS\ d{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ Q{\isacharunderscore}{\kern0pt}GS{\isacharunderscore}{\kern0pt}def\ R{\isacharunderscore}{\kern0pt}GS{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ algebra{\isacharunderscore}{\kern0pt}simps\ scaleR{\isacharunderscore}{\kern0pt}add{\isacharunderscore}{\kern0pt}right{\isacharbrackleft}{\kern0pt}symmetric{\isacharbrackright}{\kern0pt}\ simp\ del{\isacharcolon}{\kern0pt}\ scaleR{\isacharunderscore}{\kern0pt}add{\isacharunderscore}{\kern0pt}right{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ n{\isacharunderscore}{\kern0pt}le{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}norm\ {\isacharparenleft}{\kern0pt}l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isasymP}\isactrlsub L\ d{\isacharparenright}{\kern0pt}\ {\isacharless}{\kern0pt}\ {\isadigit{1}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ mult{\isacharunderscore}{\kern0pt}left{\isacharunderscore}{\kern0pt}le{\isacharbrackleft}{\kern0pt}OF\ norm{\isacharunderscore}{\kern0pt}{\isasymP}\isactrlsub L{\isacharunderscore}{\kern0pt}le{\isacharunderscore}{\kern0pt}one{\isacharbrackleft}{\kern0pt}of\ d{\isacharbrackright}{\kern0pt}\ zero{\isacharunderscore}{\kern0pt}le{\isacharunderscore}{\kern0pt}disc{\isacharbrackright}{\kern0pt}\ order{\isachardot}{\kern0pt}strict{\isacharunderscore}{\kern0pt}trans{\isadigit{1}}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ intro{\isacharcolon}{\kern0pt}\ disc{\isacharunderscore}{\kern0pt}lt{\isacharunderscore}{\kern0pt}one{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}invertible\isactrlsub L\ {\isacharparenleft}{\kern0pt}Q{\isacharunderscore}{\kern0pt}GS\ d{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ Q{\isacharunderscore}{\kern0pt}GS{\isacharunderscore}{\kern0pt}def\ invertible\isactrlsub L{\isacharunderscore}{\kern0pt}inf{\isacharunderscore}{\kern0pt}sum{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}inv\isactrlsub L\ {\isacharparenleft}{\kern0pt}Q{\isacharunderscore}{\kern0pt}GS\ d{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSum}i{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isasymP}\isactrlsub L\ d{\isacharparenright}{\kern0pt}\ {\isacharcircum}{\kern0pt}{\isacharcircum}{\kern0pt}\ i{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ inv\isactrlsub L{\isacharunderscore}{\kern0pt}inf{\isacharunderscore}{\kern0pt}sum\ n{\isacharunderscore}{\kern0pt}le\ \isacommand{unfolding}\isamarkupfalse%
-\ Q{\isacharunderscore}{\kern0pt}GS{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isacharasterisk}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ {\isacharparenleft}{\kern0pt}inv\isactrlsub L\ {\isacharparenleft}{\kern0pt}Q{\isacharunderscore}{\kern0pt}GS\ d{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isachardollar}{\kern0pt}\ i\ {\isachardollar}{\kern0pt}\ j\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSum}k{\isachardot}{\kern0pt}\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isasymP}\isactrlsub L\ d{\isacharparenright}{\kern0pt}\ {\isacharcircum}{\kern0pt}{\isacharcircum}{\kern0pt}\ k{\isacharparenright}{\kern0pt}\ {\isachardollar}{\kern0pt}\ i\ {\isachardollar}{\kern0pt}\ j{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{for}\ i\ j\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ summable{\isacharunderscore}{\kern0pt}inv{\isacharunderscore}{\kern0pt}Q{\isacharbrackleft}{\kern0pt}of\ {\isachardoublequoteopen}Q{\isacharunderscore}{\kern0pt}GS\ d{\isachardoublequoteclose}{\isacharbrackright}{\kern0pt}\ norm{\isacharunderscore}{\kern0pt}{\isasymP}\isactrlsub L{\isacharunderscore}{\kern0pt}less{\isacharunderscore}{\kern0pt}one\ \isanewline
-\ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ Q{\isacharunderscore}{\kern0pt}GS{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}subst\ bounded{\isacharunderscore}{\kern0pt}linear{\isachardot}{\kern0pt}suminf{\isacharbrackleft}{\kern0pt}symmetric{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\ \isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}auto\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ bounded{\isacharunderscore}{\kern0pt}linear{\isacharunderscore}{\kern0pt}compose{\isacharbrackleft}{\kern0pt}OF\ bounded{\isacharunderscore}{\kern0pt}linear{\isacharunderscore}{\kern0pt}vec{\isacharunderscore}{\kern0pt}nth{\isacharbrackright}{\kern0pt}\ bounded{\isacharunderscore}{\kern0pt}linear{\isacharunderscore}{\kern0pt}compose{\isacharbrackleft}{\kern0pt}OF\ bounded{\isacharunderscore}{\kern0pt}linear{\isacharunderscore}{\kern0pt}blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isadigit{0}}\ {\isasymle}\ l{\isacharcircum}{\kern0pt}i\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ matpow\ {\isacharparenleft}{\kern0pt}P{\isacharunderscore}{\kern0pt}lower\ d{\isacharparenright}{\kern0pt}\ i{\isachardoublequoteclose}\ \isakeyword{for}\ i\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ nonneg{\isacharunderscore}{\kern0pt}matpow{\isacharbrackleft}{\kern0pt}OF\ nonneg{\isacharunderscore}{\kern0pt}P{\isacharunderscore}{\kern0pt}lower{\isacharbrackright}{\kern0pt}\ \isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}meson\ scaleR{\isacharunderscore}{\kern0pt}nonneg{\isacharunderscore}{\kern0pt}nonneg\ zero{\isacharunderscore}{\kern0pt}le{\isacharunderscore}{\kern0pt}disc\ zero{\isacharunderscore}{\kern0pt}le{\isacharunderscore}{\kern0pt}power{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isadigit{0}}\ {\isasymle}\ {\isacharparenleft}{\kern0pt}{\isasymSum}k{\isachardot}{\kern0pt}\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isasymP}\isactrlsub L\ d{\isacharparenright}{\kern0pt}\ {\isacharcircum}{\kern0pt}{\isacharcircum}{\kern0pt}\ k{\isacharparenright}{\kern0pt}\ {\isachardollar}{\kern0pt}\ i\ {\isachardollar}{\kern0pt}\ j{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{for}\ i\ j\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}intro\ suminf{\isacharunderscore}{\kern0pt}nonneg{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}summable\ {\isacharparenleft}{\kern0pt}{\isasymlambda}k{\isachardot}{\kern0pt}\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isasymP}\isactrlsub L\ d{\isacharparenright}{\kern0pt}\ {\isacharcircum}{\kern0pt}{\isacharcircum}{\kern0pt}\ k{\isacharparenright}{\kern0pt}\ {\isachardollar}{\kern0pt}\ i\ {\isachardollar}{\kern0pt}\ j{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ summable{\isacharunderscore}{\kern0pt}inv{\isacharunderscore}{\kern0pt}Q{\isacharbrackleft}{\kern0pt}of\ {\isachardoublequoteopen}Q{\isacharunderscore}{\kern0pt}GS\ d{\isachardoublequoteclose}{\isacharbrackright}{\kern0pt}\ norm{\isacharunderscore}{\kern0pt}{\isasymP}\isactrlsub L{\isacharunderscore}{\kern0pt}less{\isacharunderscore}{\kern0pt}one\ \isanewline
-\ \ \ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ Q{\isacharunderscore}{\kern0pt}GS{\isacharunderscore}{\kern0pt}def\ \isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}fastforce\isanewline
-\ \ \ \ \ \ \ \ \ \ simp{\isacharcolon}{\kern0pt}\ \isanewline
-\ \ \ \ \ \ \ \ \ \ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}matpow\ nonneg{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}nonneg\ blincomp{\isacharunderscore}{\kern0pt}scaleR{\isacharunderscore}{\kern0pt}right\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}scaleR\isanewline
-\ \ \ \ \ \ \ \ \ \ intro{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ bounded{\isacharunderscore}{\kern0pt}linear{\isachardot}{\kern0pt}summable{\isacharbrackleft}{\kern0pt}of\ {\isacharunderscore}{\kern0pt}\ {\isachardoublequoteopen}{\isasymlambda}i{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isasymP}\isactrlsub L\ d{\isacharparenright}{\kern0pt}\ {\isacharcircum}{\kern0pt}{\isacharcircum}{\kern0pt}\ i{\isachardoublequoteclose}{\isacharbrackright}{\kern0pt}\ \isanewline
-\ \ \ \ \ \ \ \ \ \ bounded{\isacharunderscore}{\kern0pt}linear{\isacharunderscore}{\kern0pt}compose{\isacharbrackleft}{\kern0pt}OF\ bounded{\isacharunderscore}{\kern0pt}linear{\isacharunderscore}{\kern0pt}vec{\isacharunderscore}{\kern0pt}nth{\isacharbrackright}{\kern0pt}\ \isanewline
-\ \ \ \ \ \ \ \ \ \ bounded{\isacharunderscore}{\kern0pt}linear{\isacharunderscore}{\kern0pt}compose{\isacharbrackleft}{\kern0pt}OF\ bounded{\isacharunderscore}{\kern0pt}linear{\isacharunderscore}{\kern0pt}blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymAnd}n{\isachardot}{\kern0pt}\ {\isadigit{0}}\ {\isasymle}\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isasymP}\isactrlsub L\ d{\isacharparenright}{\kern0pt}\ {\isacharcircum}{\kern0pt}{\isacharcircum}{\kern0pt}\ n{\isacharparenright}{\kern0pt}\ {\isachardollar}{\kern0pt}\ i\ {\isachardollar}{\kern0pt}\ j{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ nonneg{\isacharunderscore}{\kern0pt}matpow{\isacharbrackleft}{\kern0pt}OF\ nonneg{\isacharunderscore}{\kern0pt}P{\isacharunderscore}{\kern0pt}lower{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ {\isasymP}\isactrlsub L{\isacharunderscore}{\kern0pt}def\ nonneg{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}nonneg\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}scaleR\ matpow{\isacharunderscore}{\kern0pt}scaleR\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}matpow{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}nonneg{\isacharunderscore}{\kern0pt}blinfun\ {\isacharparenleft}{\kern0pt}inv\isactrlsub L\ {\isacharparenleft}{\kern0pt}Q{\isacharunderscore}{\kern0pt}GS\ d{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ {\isacharasterisk}{\kern0pt}\ Finite{\isacharunderscore}{\kern0pt}Cartesian{\isacharunderscore}{\kern0pt}Product{\isachardot}{\kern0pt}less{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}vec{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{abbreviation}\isamarkupfalse%
-\ {\isachardoublequoteopen}r{\isacharunderscore}{\kern0pt}det\isactrlsub b\ d\ {\isasymequiv}\ r{\isacharunderscore}{\kern0pt}dec\isactrlsub b\ {\isacharparenleft}{\kern0pt}mk{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\ d{\isacharparenright}{\kern0pt}\ {\isachardoublequoteclose}\isanewline
-\isacommand{abbreviation}\isamarkupfalse%
-\ {\isachardoublequoteopen}r{\isacharunderscore}{\kern0pt}vec\ d\ {\isasymequiv}\ {\isasymchi}\ i{\isachardot}{\kern0pt}\ r{\isacharunderscore}{\kern0pt}dec\isactrlsub b\ {\isacharparenleft}{\kern0pt}mk{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\ d{\isacharparenright}{\kern0pt}\ i{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{abbreviation}\isamarkupfalse%
-\ {\isachardoublequoteopen}Q{\isacharunderscore}{\kern0pt}mat\ d\ {\isasymequiv}\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ {\isacharparenleft}{\kern0pt}Q{\isacharunderscore}{\kern0pt}GS\ d{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isacommand{abbreviation}\isamarkupfalse%
-\ {\isachardoublequoteopen}R{\isacharunderscore}{\kern0pt}mat\ d\ {\isasymequiv}\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ {\isacharparenleft}{\kern0pt}R{\isacharunderscore}{\kern0pt}GS\ d{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ Q{\isacharunderscore}{\kern0pt}mat{\isacharunderscore}{\kern0pt}def{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}Q{\isacharunderscore}{\kern0pt}mat\ d\ {\isacharequal}{\kern0pt}\ mat\ {\isadigit{1}}\ {\isacharminus}{\kern0pt}\ l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ P{\isacharunderscore}{\kern0pt}lower\ d{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ Q{\isacharunderscore}{\kern0pt}GS{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ {\isasymP}\isactrlsub L{\isacharunderscore}{\kern0pt}def\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}diff\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}id\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}scaleR{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ R{\isacharunderscore}{\kern0pt}mat{\isacharunderscore}{\kern0pt}def{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}R{\isacharunderscore}{\kern0pt}mat\ d\ {\isacharequal}{\kern0pt}\ l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ P{\isacharunderscore}{\kern0pt}upper\ d{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ R{\isacharunderscore}{\kern0pt}GS{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ {\isasymP}\isactrlsub U{\isacharunderscore}{\kern0pt}def\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}scaleR{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ triangular{\isacharunderscore}{\kern0pt}mat{\isacharunderscore}{\kern0pt}R{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}upper{\isacharunderscore}{\kern0pt}triangular{\isacharunderscore}{\kern0pt}mat\ {\isacharparenleft}{\kern0pt}R{\isacharunderscore}{\kern0pt}mat\ d{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ triangular{\isacharunderscore}{\kern0pt}mat{\isacharunderscore}{\kern0pt}P{\isacharunderscore}{\kern0pt}upper\isanewline
-\ \ \isacommand{unfolding}\isamarkupfalse%
-\ upper{\isacharunderscore}{\kern0pt}triangular{\isacharunderscore}{\kern0pt}mat{\isacharunderscore}{\kern0pt}def\ R{\isacharunderscore}{\kern0pt}mat{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ {\isachardoublequoteopen}GS{\isacharunderscore}{\kern0pt}inv\ d\ v\ {\isasymequiv}\ matrix{\isacharunderscore}{\kern0pt}inv\ {\isacharparenleft}{\kern0pt}Q{\isacharunderscore}{\kern0pt}mat\ d{\isacharparenright}{\kern0pt}\ {\isacharasterisk}{\kern0pt}v\ {\isacharparenleft}{\kern0pt}r{\isacharunderscore}{\kern0pt}vec\ d\ {\isacharplus}{\kern0pt}\ R{\isacharunderscore}{\kern0pt}mat\ d\ {\isacharasterisk}{\kern0pt}v\ v{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}%
-\begin{isamarkuptext}%
-\isa{Q{\isacharunderscore}{\kern0pt}mat} can be expressed as an infinite sum of \isa{P{\isacharunderscore}{\kern0pt}lower}. 
-  It is therefore lower triangular.%
-\end{isamarkuptext}\isamarkuptrue%
-\isacommand{lemma}\isamarkupfalse%
-\ inv{\isacharunderscore}{\kern0pt}Q{\isacharunderscore}{\kern0pt}mat{\isacharunderscore}{\kern0pt}suminf{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}matrix{\isacharunderscore}{\kern0pt}inv\ {\isacharparenleft}{\kern0pt}Q{\isacharunderscore}{\kern0pt}mat\ d{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSum}k{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}matpow\ {\isacharparenleft}{\kern0pt}l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isacharparenleft}{\kern0pt}P{\isacharunderscore}{\kern0pt}lower\ d{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ k{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}matrix{\isacharunderscore}{\kern0pt}inv\ {\isacharparenleft}{\kern0pt}Q{\isacharunderscore}{\kern0pt}mat\ d{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ {\isacharparenleft}{\kern0pt}inv\isactrlsub L\ {\isacharparenleft}{\kern0pt}Q{\isacharunderscore}{\kern0pt}GS\ d{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}inverse{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}\ is{\isacharunderscore}{\kern0pt}splitting{\isacharunderscore}{\kern0pt}blin{\isacharunderscore}{\kern0pt}def{\isacharprime}{\kern0pt}\ splitting{\isacharunderscore}{\kern0pt}gauss\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isacharequal}{\kern0pt}\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ {\isacharparenleft}{\kern0pt}{\isasymSum}i{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isasymP}\isactrlsub L\ d{\isacharparenright}{\kern0pt}{\isacharcircum}{\kern0pt}{\isacharcircum}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ norm{\isacharunderscore}{\kern0pt}{\isasymP}\isactrlsub L{\isacharunderscore}{\kern0pt}less{\isacharunderscore}{\kern0pt}one\ \isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ Q{\isacharunderscore}{\kern0pt}GS{\isacharunderscore}{\kern0pt}def\ inv\isactrlsub L{\isacharunderscore}{\kern0pt}inf{\isacharunderscore}{\kern0pt}sum{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSum}k{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isasymP}\isactrlsub L\ d{\isacharparenright}{\kern0pt}{\isacharcircum}{\kern0pt}{\isacharcircum}{\kern0pt}k{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ summable{\isacharunderscore}{\kern0pt}inv{\isacharunderscore}{\kern0pt}Q{\isacharbrackleft}{\kern0pt}of\ {\isachardoublequoteopen}Q{\isacharunderscore}{\kern0pt}GS\ d{\isachardoublequoteclose}{\isacharbrackright}{\kern0pt}\ norm{\isacharunderscore}{\kern0pt}{\isasymP}\isactrlsub L{\isacharunderscore}{\kern0pt}less{\isacharunderscore}{\kern0pt}one\ bounded{\isacharunderscore}{\kern0pt}linear{\isacharunderscore}{\kern0pt}blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\isanewline
-\ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ Q{\isacharunderscore}{\kern0pt}GS{\isacharunderscore}{\kern0pt}def\ row{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}subst\ bounded{\isacharunderscore}{\kern0pt}linear{\isachardot}{\kern0pt}suminf{\isacharparenright}{\kern0pt}\ auto\isanewline
-\ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSum}k{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}matpow\ {\isacharparenleft}{\kern0pt}l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isacharparenleft}{\kern0pt}P{\isacharunderscore}{\kern0pt}lower\ d{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ k{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}scaleR\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}matpow\ {\isasymP}\isactrlsub L{\isacharunderscore}{\kern0pt}def\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}inv{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{finally}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isacommand{{\isachardot}{\kern0pt}}\isamarkupfalse%
-\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ lt{\isacharunderscore}{\kern0pt}Q{\isacharunderscore}{\kern0pt}inv{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}lower{\isacharunderscore}{\kern0pt}triangular{\isacharunderscore}{\kern0pt}mat\ {\isacharparenleft}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}inv\ {\isacharparenleft}{\kern0pt}Q{\isacharunderscore}{\kern0pt}mat\ d{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ inv{\isacharunderscore}{\kern0pt}Q{\isacharunderscore}{\kern0pt}mat{\isacharunderscore}{\kern0pt}suminf\isanewline
-\ \ \isacommand{using}\isamarkupfalse%
-\ summable{\isacharunderscore}{\kern0pt}inv{\isacharunderscore}{\kern0pt}Q{\isacharbrackleft}{\kern0pt}of\ {\isachardoublequoteopen}Q{\isacharunderscore}{\kern0pt}GS\ d{\isachardoublequoteclose}{\isacharbrackright}{\kern0pt}\ norm{\isacharunderscore}{\kern0pt}{\isasymP}\isactrlsub L{\isacharunderscore}{\kern0pt}less{\isacharunderscore}{\kern0pt}one\ summable{\isacharunderscore}{\kern0pt}blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix{\isacharbrackleft}{\kern0pt}of\ {\isachardoublequoteopen}{\isasymlambda}i{\isachardot}{\kern0pt}\ \ {\isacharparenleft}{\kern0pt}l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isasymP}\isactrlsub L\ d{\isacharparenright}{\kern0pt}{\isacharcircum}{\kern0pt}{\isacharcircum}{\kern0pt}i{\isachardoublequoteclose}{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}intro\ lower{\isacharunderscore}{\kern0pt}triangular{\isacharunderscore}{\kern0pt}suminf\ lower{\isacharunderscore}{\kern0pt}triangular{\isacharunderscore}{\kern0pt}pow{\isacharparenright}{\kern0pt}\ \isanewline
-\ \ \ \ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ lower{\isacharunderscore}{\kern0pt}triangular{\isacharunderscore}{\kern0pt}mat{\isacharunderscore}{\kern0pt}def\ P{\isacharunderscore}{\kern0pt}lower{\isacharunderscore}{\kern0pt}def\ Q{\isacharunderscore}{\kern0pt}GS{\isacharunderscore}{\kern0pt}def\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}scaleR\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}matpow\ {\isasymP}\isactrlsub L{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\begin{isamarkuptext}%
-Each row of the matrix \isa{Q{\isacharunderscore}{\kern0pt}mat\ d} only depends on \isa{d}'s actions in lower states.%
-\end{isamarkuptext}\isamarkuptrue%
-\isacommand{lemma}\isamarkupfalse%
-\ inv{\isacharunderscore}{\kern0pt}Q{\isacharunderscore}{\kern0pt}mat{\isacharunderscore}{\kern0pt}indep{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}{\isasymAnd}i{\isachardot}{\kern0pt}\ i\ {\isasymle}\ s\ {\isasymLongrightarrow}\ d\ i\ {\isacharequal}{\kern0pt}\ d{\isacharprime}{\kern0pt}\ i{\isachardoublequoteclose}\ {\isachardoublequoteopen}i\ {\isasymle}\ s{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ \ {\isachardoublequoteopen}row\ i\ {\isacharparenleft}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}inv\ {\isacharparenleft}{\kern0pt}Q{\isacharunderscore}{\kern0pt}mat\ d{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ row\ i\ {\isacharparenleft}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}inv\ {\isacharparenleft}{\kern0pt}Q{\isacharunderscore}{\kern0pt}mat\ d{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}row\ i\ {\isacharparenleft}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}inv\ {\isacharparenleft}{\kern0pt}Q{\isacharunderscore}{\kern0pt}mat\ d{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ row\ i\ {\isacharparenleft}{\kern0pt}blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ {\isacharparenleft}{\kern0pt}inv\isactrlsub L\ {\isacharparenleft}{\kern0pt}Q{\isacharunderscore}{\kern0pt}GS\ d{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}inverse{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}\ is{\isacharunderscore}{\kern0pt}splitting{\isacharunderscore}{\kern0pt}blin{\isacharunderscore}{\kern0pt}def{\isacharprime}{\kern0pt}\ splitting{\isacharunderscore}{\kern0pt}gauss\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isacharequal}{\kern0pt}\ row\ i\ {\isacharparenleft}{\kern0pt}blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ {\isacharparenleft}{\kern0pt}{\isasymSum}i{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isasymP}\isactrlsub L\ d{\isacharparenright}{\kern0pt}{\isacharcircum}{\kern0pt}{\isacharcircum}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ norm{\isacharunderscore}{\kern0pt}{\isasymP}\isactrlsub L{\isacharunderscore}{\kern0pt}less{\isacharunderscore}{\kern0pt}one\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ Q{\isacharunderscore}{\kern0pt}GS{\isacharunderscore}{\kern0pt}def\ inv\isactrlsub L{\isacharunderscore}{\kern0pt}inf{\isacharunderscore}{\kern0pt}sum{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSum}k{\isachardot}{\kern0pt}\ row\ i\ {\isacharparenleft}{\kern0pt}blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isasymP}\isactrlsub L\ d{\isacharparenright}{\kern0pt}{\isacharcircum}{\kern0pt}{\isacharcircum}{\kern0pt}k{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ summable{\isacharunderscore}{\kern0pt}inv{\isacharunderscore}{\kern0pt}Q{\isacharbrackleft}{\kern0pt}of\ {\isachardoublequoteopen}Q{\isacharunderscore}{\kern0pt}GS\ d{\isachardoublequoteclose}{\isacharbrackright}{\kern0pt}\ norm{\isacharunderscore}{\kern0pt}{\isasymP}\isactrlsub L{\isacharunderscore}{\kern0pt}less{\isacharunderscore}{\kern0pt}one\isanewline
-\ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ Q{\isacharunderscore}{\kern0pt}GS{\isacharunderscore}{\kern0pt}def\ row{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}subst\ bounded{\isacharunderscore}{\kern0pt}linear{\isachardot}{\kern0pt}suminf{\isacharbrackleft}{\kern0pt}OF\ bounded{\isacharunderscore}{\kern0pt}linear{\isacharunderscore}{\kern0pt}compose{\isacharbrackleft}{\kern0pt}OF\ {\isacharunderscore}{\kern0pt}\ bounded{\isacharunderscore}{\kern0pt}linear{\isacharunderscore}{\kern0pt}blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix{\isacharbrackright}{\kern0pt}{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\ auto\isanewline
-\ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSum}k{\isachardot}{\kern0pt}\ row\ i\ {\isacharparenleft}{\kern0pt}matpow\ {\isacharparenleft}{\kern0pt}l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isacharparenleft}{\kern0pt}P{\isacharunderscore}{\kern0pt}lower\ d{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ k{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}matpow\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}scaleR\ {\isasymP}\isactrlsub L{\isacharunderscore}{\kern0pt}def\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}inv{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSum}k{\isachardot}{\kern0pt}\ l{\isacharcircum}{\kern0pt}k\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ row\ i\ {\isacharparenleft}{\kern0pt}matpow\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}P{\isacharunderscore}{\kern0pt}lower\ d{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ k{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}subst\ matpow{\isacharunderscore}{\kern0pt}scaleR{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ row{\isacharunderscore}{\kern0pt}def\ scaleR{\isacharunderscore}{\kern0pt}vec{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\ \isanewline
-\ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSum}k{\isachardot}{\kern0pt}\ l{\isacharcircum}{\kern0pt}k\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ row\ i\ {\isacharparenleft}{\kern0pt}matpow\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}P{\isacharunderscore}{\kern0pt}lower\ d{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ k{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assms\ \isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}subst\ lower{\isacharunderscore}{\kern0pt}triangular{\isacharunderscore}{\kern0pt}pow{\isacharunderscore}{\kern0pt}eq{\isacharbrackleft}{\kern0pt}of\ {\isachardoublequoteopen}P{\isacharunderscore}{\kern0pt}lower\ d{\isacharprime}{\kern0pt}{\isachardoublequoteclose}{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ P{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}elem\ lt{\isacharunderscore}{\kern0pt}P{\isacharunderscore}{\kern0pt}lower\ row{\isacharunderscore}{\kern0pt}P{\isacharunderscore}{\kern0pt}lower{\isacharunderscore}{\kern0pt}indep{\isacharbrackleft}{\kern0pt}of\ d{\isacharprime}{\kern0pt}\ {\isacharunderscore}{\kern0pt}\ d{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSum}k{\isachardot}{\kern0pt}\ row\ i\ {\isacharparenleft}{\kern0pt}matpow\ {\isacharparenleft}{\kern0pt}l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isacharparenleft}{\kern0pt}P{\isacharunderscore}{\kern0pt}lower\ d{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ k{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}subst\ matpow{\isacharunderscore}{\kern0pt}scaleR{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ row{\isacharunderscore}{\kern0pt}def\ scaleR{\isacharunderscore}{\kern0pt}vec{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSum}k{\isachardot}{\kern0pt}\ row\ i\ {\isacharparenleft}{\kern0pt}blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isasymP}\isactrlsub L\ d{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharcircum}{\kern0pt}{\isacharcircum}{\kern0pt}k{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ {\isasymP}\isactrlsub L{\isacharunderscore}{\kern0pt}def\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}inv\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}matpow\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}scaleR{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isacharequal}{\kern0pt}\ row\ i\ {\isacharparenleft}{\kern0pt}blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ {\isacharparenleft}{\kern0pt}{\isasymSum}i{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isasymP}\isactrlsub L\ d{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharcircum}{\kern0pt}{\isacharcircum}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ summable{\isacharunderscore}{\kern0pt}inv{\isacharunderscore}{\kern0pt}Q{\isacharbrackleft}{\kern0pt}of\ {\isachardoublequoteopen}Q{\isacharunderscore}{\kern0pt}GS\ d{\isacharprime}{\kern0pt}{\isachardoublequoteclose}{\isacharbrackright}{\kern0pt}\ norm{\isacharunderscore}{\kern0pt}{\isasymP}\isactrlsub L{\isacharunderscore}{\kern0pt}less{\isacharunderscore}{\kern0pt}one\ \isanewline
-\ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ Q{\isacharunderscore}{\kern0pt}GS{\isacharunderscore}{\kern0pt}def\ row{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ bounded{\isacharunderscore}{\kern0pt}linear{\isachardot}{\kern0pt}suminf{\isacharbrackleft}{\kern0pt}symmetric{\isacharbrackright}{\kern0pt}\ \isanewline
-\ \ \ \ \ \ \ \ bounded{\isacharunderscore}{\kern0pt}linear{\isacharunderscore}{\kern0pt}compose{\isacharbrackleft}{\kern0pt}OF\ {\isacharunderscore}{\kern0pt}\ bounded{\isacharunderscore}{\kern0pt}linear{\isacharunderscore}{\kern0pt}blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isacharequal}{\kern0pt}\ row\ i\ {\isacharparenleft}{\kern0pt}blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ {\isacharparenleft}{\kern0pt}inv\isactrlsub L\ {\isacharparenleft}{\kern0pt}Q{\isacharunderscore}{\kern0pt}GS\ d{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ Q{\isacharunderscore}{\kern0pt}GS{\isacharunderscore}{\kern0pt}def\ inv\isactrlsub L{\isacharunderscore}{\kern0pt}inf{\isacharunderscore}{\kern0pt}sum\ norm{\isacharunderscore}{\kern0pt}{\isasymP}\isactrlsub L{\isacharunderscore}{\kern0pt}less{\isacharunderscore}{\kern0pt}one{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isacharequal}{\kern0pt}\ row\ i\ {\isacharparenleft}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}inv\ {\isacharparenleft}{\kern0pt}Q{\isacharunderscore}{\kern0pt}mat\ d{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}inverse{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}\ is{\isacharunderscore}{\kern0pt}splitting{\isacharunderscore}{\kern0pt}blin{\isacharunderscore}{\kern0pt}def{\isacharprime}{\kern0pt}\ splitting{\isacharunderscore}{\kern0pt}gauss{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{finally}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isacommand{{\isachardot}{\kern0pt}}\isamarkupfalse%
-\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\begin{isamarkuptext}%
-As a result, also \isa{GS{\isacharunderscore}{\kern0pt}inv} is independent of lower actions.%
-\end{isamarkuptext}\isamarkuptrue%
-\isacommand{lemma}\isamarkupfalse%
-\ GS{\isacharunderscore}{\kern0pt}indep{\isacharunderscore}{\kern0pt}high{\isacharunderscore}{\kern0pt}states{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}{\isasymAnd}s{\isacharprime}{\kern0pt}{\isachardot}{\kern0pt}\ s{\isacharprime}{\kern0pt}\ {\isasymle}\ s\ {\isasymLongrightarrow}\ d\ s{\isacharprime}{\kern0pt}\ {\isacharequal}{\kern0pt}\ d{\isacharprime}{\kern0pt}\ s{\isacharprime}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}GS{\isacharunderscore}{\kern0pt}inv\ d\ v\ {\isachardollar}{\kern0pt}\ s\ {\isacharequal}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}inv\ d{\isacharprime}{\kern0pt}\ v\ {\isachardollar}{\kern0pt}\ s{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}row\ i\ {\isacharparenleft}{\kern0pt}P{\isacharunderscore}{\kern0pt}upper\ d{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ row\ i\ {\isacharparenleft}{\kern0pt}P{\isacharunderscore}{\kern0pt}upper\ d{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{if}\ {\isachardoublequoteopen}i\ {\isasymle}\ s{\isachardoublequoteclose}\ \isakeyword{for}\ i\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assms\ that\ row{\isacharunderscore}{\kern0pt}P{\isacharunderscore}{\kern0pt}upper{\isacharunderscore}{\kern0pt}indep\ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ R{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}upto{\isacharunderscore}{\kern0pt}s{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}row\ i\ {\isacharparenleft}{\kern0pt}R{\isacharunderscore}{\kern0pt}mat\ d{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ row\ i\ {\isacharparenleft}{\kern0pt}R{\isacharunderscore}{\kern0pt}mat\ d{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{if}\ {\isachardoublequoteopen}i\ {\isasymle}\ s{\isachardoublequoteclose}\ \isakeyword{for}\ i\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ that\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ row{\isacharunderscore}{\kern0pt}def\ R{\isacharunderscore}{\kern0pt}mat{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ Qr{\isacharunderscore}{\kern0pt}eq{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}inv\ {\isacharparenleft}{\kern0pt}Q{\isacharunderscore}{\kern0pt}mat\ d{\isacharparenright}{\kern0pt}\ {\isacharasterisk}{\kern0pt}v\ r{\isacharunderscore}{\kern0pt}vec\ d{\isacharparenright}{\kern0pt}\ {\isachardollar}{\kern0pt}\ s\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}inv\ {\isacharparenleft}{\kern0pt}Q{\isacharunderscore}{\kern0pt}mat\ d{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharasterisk}{\kern0pt}v\ r{\isacharunderscore}{\kern0pt}vec\ d{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isachardollar}{\kern0pt}\ s{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}inv\ {\isacharparenleft}{\kern0pt}Q{\isacharunderscore}{\kern0pt}mat\ d{\isacharparenright}{\kern0pt}\ {\isacharasterisk}{\kern0pt}v\ r{\isacharunderscore}{\kern0pt}vec\ d{\isacharparenright}{\kern0pt}\ {\isachardollar}{\kern0pt}\ s\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSum}j{\isasymin}UNIV{\isachardot}{\kern0pt}\ matrix{\isacharunderscore}{\kern0pt}inv\ {\isacharparenleft}{\kern0pt}Q{\isacharunderscore}{\kern0pt}mat\ d{\isacharparenright}{\kern0pt}\ {\isachardollar}{\kern0pt}\ s\ {\isachardollar}{\kern0pt}\ j\ {\isacharasterisk}{\kern0pt}\ r{\isacharunderscore}{\kern0pt}vec\ d\ {\isachardollar}{\kern0pt}\ j{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \ \ \ \isanewline
-\ \ \ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ matrix{\isacharunderscore}{\kern0pt}vector{\isacharunderscore}{\kern0pt}mult{\isacharunderscore}{\kern0pt}def\ \isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSum}j{\isasymin}UNIV{\isachardot}{\kern0pt}\ if\ s\ {\isacharless}{\kern0pt}\ j\ then\ {\isadigit{0}}\ else\ matrix{\isacharunderscore}{\kern0pt}inv\ {\isacharparenleft}{\kern0pt}Q{\isacharunderscore}{\kern0pt}mat\ d{\isacharparenright}{\kern0pt}\ {\isachardollar}{\kern0pt}\ s\ {\isachardollar}{\kern0pt}\ j\ {\isacharasterisk}{\kern0pt}\ r{\isacharunderscore}{\kern0pt}vec\ d\ {\isachardollar}{\kern0pt}\ j{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ lt{\isacharunderscore}{\kern0pt}Q{\isacharunderscore}{\kern0pt}inv\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ sum{\isachardot}{\kern0pt}cong\ simp{\isacharcolon}{\kern0pt}\ lower{\isacharunderscore}{\kern0pt}triangular{\isacharunderscore}{\kern0pt}mat{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSum}j{\isasymin}UNIV{\isachardot}{\kern0pt}\ if\ s\ {\isacharless}{\kern0pt}\ j\ then\ {\isadigit{0}}\ else\ matrix{\isacharunderscore}{\kern0pt}inv\ {\isacharparenleft}{\kern0pt}Q{\isacharunderscore}{\kern0pt}mat\ d{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isachardollar}{\kern0pt}\ s\ {\isachardollar}{\kern0pt}\ j\ {\isacharasterisk}{\kern0pt}\ r{\isacharunderscore}{\kern0pt}vec\ d\ {\isachardollar}{\kern0pt}\ j{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ inv{\isacharunderscore}{\kern0pt}Q{\isacharunderscore}{\kern0pt}mat{\isacharunderscore}{\kern0pt}indep\ assms\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}fastforce\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ sum{\isachardot}{\kern0pt}cong\ simp{\isacharcolon}{\kern0pt}\ row{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}inv\ {\isacharparenleft}{\kern0pt}Q{\isacharunderscore}{\kern0pt}mat\ d{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharasterisk}{\kern0pt}v\ r{\isacharunderscore}{\kern0pt}vec\ d{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isachardollar}{\kern0pt}\ s{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ lt{\isacharunderscore}{\kern0pt}Q{\isacharunderscore}{\kern0pt}inv\ \isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ matrix{\isacharunderscore}{\kern0pt}vector{\isacharunderscore}{\kern0pt}mult{\isacharunderscore}{\kern0pt}def\ assms\ lower{\isacharunderscore}{\kern0pt}triangular{\isacharunderscore}{\kern0pt}mat{\isacharunderscore}{\kern0pt}def\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ sum{\isachardot}{\kern0pt}cong{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{finally}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isacommand{{\isachardot}{\kern0pt}}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ QR{\isacharunderscore}{\kern0pt}eq{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}row\ s\ {\isacharparenleft}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}inv\ {\isacharparenleft}{\kern0pt}Q{\isacharunderscore}{\kern0pt}mat\ d{\isacharparenright}{\kern0pt}\ {\isacharasterisk}{\kern0pt}{\isacharasterisk}{\kern0pt}\ R{\isacharunderscore}{\kern0pt}mat\ d{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ row\ s\ {\isacharparenleft}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}inv\ {\isacharparenleft}{\kern0pt}Q{\isacharunderscore}{\kern0pt}mat\ d{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharasterisk}{\kern0pt}{\isacharasterisk}{\kern0pt}\ R{\isacharunderscore}{\kern0pt}mat\ d{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\ \isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}matrix{\isacharunderscore}{\kern0pt}inv\ {\isacharparenleft}{\kern0pt}Q{\isacharunderscore}{\kern0pt}mat\ d{\isacharparenright}{\kern0pt}\ {\isachardollar}{\kern0pt}\ s\ {\isachardollar}{\kern0pt}\ k\ {\isacharasterisk}{\kern0pt}\ R{\isacharunderscore}{\kern0pt}mat\ d\ {\isachardollar}{\kern0pt}\ k\ {\isachardollar}{\kern0pt}\ j\ {\isacharequal}{\kern0pt}\ matrix{\isacharunderscore}{\kern0pt}inv\ {\isacharparenleft}{\kern0pt}Q{\isacharunderscore}{\kern0pt}mat\ d{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isachardollar}{\kern0pt}\ s\ {\isachardollar}{\kern0pt}\ k\ {\isacharasterisk}{\kern0pt}\ R{\isacharunderscore}{\kern0pt}mat\ d{\isacharprime}{\kern0pt}\ {\isachardollar}{\kern0pt}\ k\ {\isachardollar}{\kern0pt}\ j{\isachardoublequoteclose}\ \isakeyword{for}\ k\ j\isanewline
-\ \ \ \ \isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\ \isanewline
-\ \ \ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}matrix{\isacharunderscore}{\kern0pt}inv\ {\isacharparenleft}{\kern0pt}Q{\isacharunderscore}{\kern0pt}mat\ d{\isacharparenright}{\kern0pt}\ {\isachardollar}{\kern0pt}\ s\ {\isachardollar}{\kern0pt}\ k\ {\isacharasterisk}{\kern0pt}\ R{\isacharunderscore}{\kern0pt}mat\ d\ {\isachardollar}{\kern0pt}\ k\ {\isachardollar}{\kern0pt}\ j\ {\isacharequal}{\kern0pt}\ \isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}if\ s\ {\isacharless}{\kern0pt}\ k\ then\ {\isadigit{0}}\ else\ matrix{\isacharunderscore}{\kern0pt}inv\ {\isacharparenleft}{\kern0pt}Q{\isacharunderscore}{\kern0pt}mat\ d{\isacharparenright}{\kern0pt}\ {\isachardollar}{\kern0pt}\ s\ {\isachardollar}{\kern0pt}\ k\ {\isacharasterisk}{\kern0pt}\ R{\isacharunderscore}{\kern0pt}mat\ d\ {\isachardollar}{\kern0pt}\ k\ {\isachardollar}{\kern0pt}\ j{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ lower{\isacharunderscore}{\kern0pt}triangular{\isacharunderscore}{\kern0pt}mat{\isacharunderscore}{\kern0pt}def\ lt{\isacharunderscore}{\kern0pt}Q{\isacharunderscore}{\kern0pt}inv\ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}if\ s\ {\isacharless}{\kern0pt}\ k\ then\ {\isadigit{0}}\ else\ matrix{\isacharunderscore}{\kern0pt}inv\ {\isacharparenleft}{\kern0pt}Q{\isacharunderscore}{\kern0pt}mat\ d{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isachardollar}{\kern0pt}\ s\ {\isachardollar}{\kern0pt}\ k\ {\isacharasterisk}{\kern0pt}\ R{\isacharunderscore}{\kern0pt}mat\ d\ {\isachardollar}{\kern0pt}\ k\ {\isachardollar}{\kern0pt}\ j{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharcomma}{\kern0pt}\ lifting{\isacharparenright}{\kern0pt}\ Finite{\isacharunderscore}{\kern0pt}Cartesian{\isacharunderscore}{\kern0pt}Product{\isachardot}{\kern0pt}row{\isacharunderscore}{\kern0pt}def\ assms\ inv{\isacharunderscore}{\kern0pt}Q{\isacharunderscore}{\kern0pt}mat{\isacharunderscore}{\kern0pt}indep\ order{\isacharunderscore}{\kern0pt}refl\ vec{\isacharunderscore}{\kern0pt}lambda{\isacharunderscore}{\kern0pt}eta{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}if\ s\ {\isacharless}{\kern0pt}\ k\ {\isasymor}\ j\ {\isacharless}{\kern0pt}\ k\ then\ {\isadigit{0}}\ else\ {\isacharparenleft}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}inv\ {\isacharparenleft}{\kern0pt}Q{\isacharunderscore}{\kern0pt}mat\ d{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isachardollar}{\kern0pt}\ s\ {\isachardollar}{\kern0pt}\ k\ {\isacharasterisk}{\kern0pt}\ R{\isacharunderscore}{\kern0pt}mat\ d\ {\isachardollar}{\kern0pt}\ k\ {\isachardollar}{\kern0pt}\ j{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ triangular{\isacharunderscore}{\kern0pt}mat{\isacharunderscore}{\kern0pt}R\isanewline
-\ \ \ \ \ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ upper{\isacharunderscore}{\kern0pt}triangular{\isacharunderscore}{\kern0pt}mat{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ split{\isacharcolon}{\kern0pt}\ if{\isacharunderscore}{\kern0pt}splits{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}if\ s\ {\isacharless}{\kern0pt}\ k\ {\isasymor}\ j\ {\isacharless}{\kern0pt}\ k\ then\ {\isadigit{0}}\ else\ {\isacharparenleft}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}inv\ {\isacharparenleft}{\kern0pt}Q{\isacharunderscore}{\kern0pt}mat\ d{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isachardollar}{\kern0pt}\ s\ {\isachardollar}{\kern0pt}\ k\ {\isacharasterisk}{\kern0pt}\ R{\isacharunderscore}{\kern0pt}mat\ d{\isacharprime}{\kern0pt}\ {\isachardollar}{\kern0pt}\ k\ {\isachardollar}{\kern0pt}\ j{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ R{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}upto{\isacharunderscore}{\kern0pt}s\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ row{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isacharequal}{\kern0pt}\ matrix{\isacharunderscore}{\kern0pt}inv\ {\isacharparenleft}{\kern0pt}Q{\isacharunderscore}{\kern0pt}mat\ d{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isachardollar}{\kern0pt}\ s\ {\isachardollar}{\kern0pt}\ k\ {\isacharasterisk}{\kern0pt}\ R{\isacharunderscore}{\kern0pt}mat\ d{\isacharprime}{\kern0pt}\ {\isachardollar}{\kern0pt}\ k\ {\isachardollar}{\kern0pt}\ j{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ lower{\isacharunderscore}{\kern0pt}triangular{\isacharunderscore}{\kern0pt}mat{\isacharunderscore}{\kern0pt}def\ lt{\isacharunderscore}{\kern0pt}Q{\isacharunderscore}{\kern0pt}inv\ mult{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}zero\ triangular{\isacharunderscore}{\kern0pt}mat{\isacharunderscore}{\kern0pt}R\ upper{\isacharunderscore}{\kern0pt}triangular{\isacharunderscore}{\kern0pt}mat{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{finally}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isacommand{{\isachardot}{\kern0pt}}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ row{\isacharunderscore}{\kern0pt}def\ matrix{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}mult{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ QR{\isacharunderscore}{\kern0pt}eq\ Qr{\isacharunderscore}{\kern0pt}eq\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp\ add{\isacharcolon}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}inv{\isacharunderscore}{\kern0pt}def\ vec{\isachardot}{\kern0pt}add\ row{\isacharunderscore}{\kern0pt}def\ matrix{\isacharunderscore}{\kern0pt}vector{\isacharunderscore}{\kern0pt}mul{\isacharunderscore}{\kern0pt}assoc\ matrix{\isacharunderscore}{\kern0pt}vector{\isacharunderscore}{\kern0pt}mult{\isacharunderscore}{\kern0pt}code{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\begin{isamarkuptext}%
-This recursive definition mimics the computation of the GS iteration.%
-\end{isamarkuptext}\isamarkuptrue%
-\isacommand{lemma}\isamarkupfalse%
-\ GS{\isacharunderscore}{\kern0pt}inv{\isacharunderscore}{\kern0pt}rec{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}GS{\isacharunderscore}{\kern0pt}inv\ d\ v\ {\isacharequal}{\kern0pt}\ r{\isacharunderscore}{\kern0pt}vec\ d\ {\isacharplus}{\kern0pt}\ l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isacharparenleft}{\kern0pt}P{\isacharunderscore}{\kern0pt}upper\ d\ {\isacharasterisk}{\kern0pt}v\ v\ {\isacharplus}{\kern0pt}\ P{\isacharunderscore}{\kern0pt}lower\ d\ {\isacharasterisk}{\kern0pt}v\ {\isacharparenleft}{\kern0pt}GS{\isacharunderscore}{\kern0pt}inv\ d\ v{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}Q{\isacharunderscore}{\kern0pt}mat\ d\ {\isacharasterisk}{\kern0pt}v\ {\isacharparenleft}{\kern0pt}GS{\isacharunderscore}{\kern0pt}inv\ d\ v{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ r{\isacharunderscore}{\kern0pt}vec\ d\ {\isacharplus}{\kern0pt}\ R{\isacharunderscore}{\kern0pt}mat\ d\ {\isacharasterisk}{\kern0pt}v\ v{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ splitting{\isacharunderscore}{\kern0pt}gauss{\isacharbrackleft}{\kern0pt}of\ d{\isacharbrackright}{\kern0pt}\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}inverse{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ GS{\isacharunderscore}{\kern0pt}inv{\isacharunderscore}{\kern0pt}def\ matrix{\isacharunderscore}{\kern0pt}vector{\isacharunderscore}{\kern0pt}mul{\isacharunderscore}{\kern0pt}assoc\ is{\isacharunderscore}{\kern0pt}splitting{\isacharunderscore}{\kern0pt}blin{\isacharunderscore}{\kern0pt}def{\isacharprime}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}subst\ matrix{\isacharunderscore}{\kern0pt}inv{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ auto\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\ \isanewline
-\ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ Q{\isacharunderscore}{\kern0pt}mat{\isacharunderscore}{\kern0pt}def\ R{\isacharunderscore}{\kern0pt}mat{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ algebra{\isacharunderscore}{\kern0pt}simps\ scaleR{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}vector{\isacharunderscore}{\kern0pt}assoc{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ is{\isacharunderscore}{\kern0pt}am{\isacharunderscore}{\kern0pt}GS{\isacharunderscore}{\kern0pt}inv{\isacharunderscore}{\kern0pt}extend{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}{\isasymAnd}s{\isachardot}{\kern0pt}\ s\ {\isacharless}{\kern0pt}\ k\ {\isasymLongrightarrow}\ is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max\ {\isacharparenleft}{\kern0pt}{\isasymlambda}d{\isachardot}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}inv\ d\ v\ {\isachardollar}{\kern0pt}\ s{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymlambda}d{\isachardot}{\kern0pt}\ d\ {\isasymin}\ D\isactrlsub D{\isacharparenright}{\kern0pt}\ d{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isakeyword{and}\ {\isachardoublequoteopen}is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max\ {\isacharparenleft}{\kern0pt}{\isasymlambda}a{\isachardot}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}inv\ {\isacharparenleft}{\kern0pt}d\ {\isacharparenleft}{\kern0pt}k\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ a{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ v\ {\isachardollar}{\kern0pt}\ k{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymlambda}a{\isachardot}{\kern0pt}\ a\ {\isasymin}\ A\ k{\isacharparenright}{\kern0pt}\ a{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isakeyword{and}\ {\isachardoublequoteopen}s\ {\isasymle}\ k{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isakeyword{and}\ {\isachardoublequoteopen}d\ {\isasymin}\ D\isactrlsub D{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max\ {\isacharparenleft}{\kern0pt}{\isasymlambda}d{\isachardot}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}inv\ d\ v\ {\isachardollar}{\kern0pt}\ s{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymlambda}d{\isachardot}{\kern0pt}\ d\ {\isasymin}\ D\isactrlsub D{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}d\ {\isacharparenleft}{\kern0pt}k\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ a{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ am{\isacharunderscore}{\kern0pt}k{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max\ {\isacharparenleft}{\kern0pt}{\isasymlambda}d{\isachardot}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}inv\ d\ v\ {\isachardollar}{\kern0pt}\ k{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymlambda}d{\isachardot}{\kern0pt}\ d\ {\isasymin}\ D\isactrlsub D{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}d\ {\isacharparenleft}{\kern0pt}k\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ a{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}rule\ is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}linorderI{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{fix}\isamarkupfalse%
-\ y\isanewline
-\ \ \ \ \isacommand{assume}\isamarkupfalse%
-\ {\isachardoublequoteopen}y\ {\isasymin}\ D\isactrlsub D{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}GS{\isacharunderscore}{\kern0pt}inv\ y\ v\ {\isachardollar}{\kern0pt}\ k\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}r{\isacharunderscore}{\kern0pt}vec\ y\ {\isacharplus}{\kern0pt}\ l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isacharparenleft}{\kern0pt}P{\isacharunderscore}{\kern0pt}upper\ y\ {\isacharasterisk}{\kern0pt}v\ v\ {\isacharplus}{\kern0pt}\ P{\isacharunderscore}{\kern0pt}lower\ y\ {\isacharasterisk}{\kern0pt}v\ {\isacharparenleft}{\kern0pt}GS{\isacharunderscore}{\kern0pt}inv\ y\ v{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isachardollar}{\kern0pt}\ k{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ GS{\isacharunderscore}{\kern0pt}inv{\isacharunderscore}{\kern0pt}rec\ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isacharequal}{\kern0pt}\ r\ {\isacharparenleft}{\kern0pt}k{\isacharcomma}{\kern0pt}\ y\ k{\isacharparenright}{\kern0pt}\ {\isacharplus}{\kern0pt}\ l\ {\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}P{\isacharunderscore}{\kern0pt}upper\ y\ {\isacharasterisk}{\kern0pt}v\ v{\isacharparenright}{\kern0pt}\ {\isachardollar}{\kern0pt}\ k\ {\isacharplus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}P{\isacharunderscore}{\kern0pt}lower\ y\ {\isacharasterisk}{\kern0pt}v\ GS{\isacharunderscore}{\kern0pt}inv\ y\ v{\isacharparenright}{\kern0pt}\ {\isachardollar}{\kern0pt}\ k{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isasymle}\ r\ {\isacharparenleft}{\kern0pt}k{\isacharcomma}{\kern0pt}\ {\isacharparenleft}{\kern0pt}d{\isacharparenleft}{\kern0pt}k\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ y\ k{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ k{\isacharparenright}{\kern0pt}\ {\isacharplus}{\kern0pt}\ l\ {\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}P{\isacharunderscore}{\kern0pt}upper\ {\isacharparenleft}{\kern0pt}d{\isacharparenleft}{\kern0pt}k\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ y\ k{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharasterisk}{\kern0pt}v\ v{\isacharparenright}{\kern0pt}\ {\isachardollar}{\kern0pt}\ k\ {\isacharplus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}P{\isacharunderscore}{\kern0pt}lower\ {\isacharparenleft}{\kern0pt}d{\isacharparenleft}{\kern0pt}k\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ y\ k{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharasterisk}{\kern0pt}v\ GS{\isacharunderscore}{\kern0pt}inv\ {\isacharparenleft}{\kern0pt}d{\isacharparenleft}{\kern0pt}k\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ y\ k{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ v{\isacharparenright}{\kern0pt}\ {\isachardollar}{\kern0pt}\ k{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}rule\ add{\isacharunderscore}{\kern0pt}mono{\isacharcomma}{\kern0pt}\ goal{\isacharunderscore}{\kern0pt}cases{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{case}\isamarkupfalse%
-\ {\isadigit{2}}\isanewline
-\ \ \ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}case\isanewline
-\ \ \ \ \ \ \isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}intro\ mult{\isacharunderscore}{\kern0pt}left{\isacharunderscore}{\kern0pt}mono\ add{\isacharunderscore}{\kern0pt}mono{\isacharcomma}{\kern0pt}\ goal{\isacharunderscore}{\kern0pt}cases{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{case}\isamarkupfalse%
-\ {\isadigit{1}}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}case\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ matrix{\isacharunderscore}{\kern0pt}vector{\isacharunderscore}{\kern0pt}mult{\isacharunderscore}{\kern0pt}def\ P{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}elem\ fun{\isacharunderscore}{\kern0pt}upd{\isacharunderscore}{\kern0pt}same\ P{\isacharunderscore}{\kern0pt}upper{\isacharunderscore}{\kern0pt}def\ cong{\isacharcolon}{\kern0pt}\ if{\isacharunderscore}{\kern0pt}cong{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ \isacommand{case}\isamarkupfalse%
-\ {\isadigit{2}}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}case\ \isanewline
-\ \ \ \ \ \ \ \ \isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}P{\isacharunderscore}{\kern0pt}lower\ y\ {\isacharasterisk}{\kern0pt}v\ GS{\isacharunderscore}{\kern0pt}inv\ y\ v{\isacharparenright}{\kern0pt}\ {\isachardollar}{\kern0pt}\ k\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}P{\isacharunderscore}{\kern0pt}lower\ {\isacharparenleft}{\kern0pt}d{\isacharparenleft}{\kern0pt}k\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ y\ k{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharasterisk}{\kern0pt}v\ GS{\isacharunderscore}{\kern0pt}inv\ y\ v{\isacharparenright}{\kern0pt}\ {\isachardollar}{\kern0pt}\ k{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ matrix{\isacharunderscore}{\kern0pt}vector{\isacharunderscore}{\kern0pt}mult{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ P{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}elem\ fun{\isacharunderscore}{\kern0pt}upd{\isacharunderscore}{\kern0pt}same\ P{\isacharunderscore}{\kern0pt}lower{\isacharunderscore}{\kern0pt}def\ cong{\isacharcolon}{\kern0pt}\ if{\isacharunderscore}{\kern0pt}cong{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSum}j{\isasymin}UNIV{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}if\ j\ {\isacharless}{\kern0pt}\ k\ then\ pmf\ {\isacharparenleft}{\kern0pt}K\ {\isacharparenleft}{\kern0pt}k{\isacharcomma}{\kern0pt}\ y\ k{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ j\ \ {\isacharasterisk}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}inv\ y\ v\ {\isachardollar}{\kern0pt}\ j\ \ else\ {\isadigit{0}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ matrix{\isacharunderscore}{\kern0pt}vector{\isacharunderscore}{\kern0pt}mult{\isacharunderscore}{\kern0pt}def\ P{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}elem\ P{\isacharunderscore}{\kern0pt}lower{\isacharunderscore}{\kern0pt}def\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ sum{\isachardot}{\kern0pt}cong{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isasymle}\ {\isacharparenleft}{\kern0pt}{\isasymSum}j{\isasymin}UNIV{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}if\ j\ {\isacharless}{\kern0pt}\ k\ then\ pmf\ {\isacharparenleft}{\kern0pt}K\ {\isacharparenleft}{\kern0pt}k{\isacharcomma}{\kern0pt}\ y\ k{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ j\ \ {\isacharasterisk}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}inv\ d\ v\ {\isachardollar}{\kern0pt}\ j\ \ else\ {\isadigit{0}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assms\ {\isacartoucheopen}y{\isasymin}D\isactrlsub D{\isacartoucheclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}fastforce\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ sum{\isacharunderscore}{\kern0pt}mono\ mult{\isacharunderscore}{\kern0pt}left{\isacharunderscore}{\kern0pt}mono\ dest{\isacharcolon}{\kern0pt}\ is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}linorderD{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isacharequal}{\kern0pt}\ \ {\isacharparenleft}{\kern0pt}{\isasymSum}j{\isasymin}UNIV{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}if\ j\ {\isacharless}{\kern0pt}\ k\ then\ pmf\ {\isacharparenleft}{\kern0pt}K\ {\isacharparenleft}{\kern0pt}k{\isacharcomma}{\kern0pt}\ y\ k{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ j\ \ {\isacharasterisk}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}inv\ {\isacharparenleft}{\kern0pt}d{\isacharparenleft}{\kern0pt}k\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ y\ k{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ v\ {\isachardollar}{\kern0pt}\ j\ else\ {\isadigit{0}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \ \isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ GS{\isacharunderscore}{\kern0pt}indep{\isacharunderscore}{\kern0pt}high{\isacharunderscore}{\kern0pt}states{\isacharbrackleft}{\kern0pt}of\ {\isacharunderscore}{\kern0pt}\ {\isachardoublequoteopen}d{\isacharparenleft}{\kern0pt}k\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ y\ k{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ d{\isacharcomma}{\kern0pt}\ symmetric{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}fastforce\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ sum{\isachardot}{\kern0pt}cong\ dest{\isacharcolon}{\kern0pt}\ leD{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isacharequal}{\kern0pt}\ \ {\isacharparenleft}{\kern0pt}P{\isacharunderscore}{\kern0pt}lower\ {\isacharparenleft}{\kern0pt}d{\isacharparenleft}{\kern0pt}k\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ y\ k{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharasterisk}{\kern0pt}v\ GS{\isacharunderscore}{\kern0pt}inv\ {\isacharparenleft}{\kern0pt}d{\isacharparenleft}{\kern0pt}k\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ y\ k{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ v{\isacharparenright}{\kern0pt}\ {\isachardollar}{\kern0pt}\ k{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ matrix{\isacharunderscore}{\kern0pt}vector{\isacharunderscore}{\kern0pt}mult{\isacharunderscore}{\kern0pt}def\ P{\isacharunderscore}{\kern0pt}lower{\isacharunderscore}{\kern0pt}def\ P{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}elem\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}fastforce\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ sum{\isachardot}{\kern0pt}cong{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{finally}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isacommand{{\isachardot}{\kern0pt}}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \isacommand{qed}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \isacommand{qed}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}r{\isacharunderscore}{\kern0pt}vec\ {\isacharparenleft}{\kern0pt}d{\isacharparenleft}{\kern0pt}k\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ y\ k{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharplus}{\kern0pt}\ l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}P{\isacharunderscore}{\kern0pt}upper\ {\isacharparenleft}{\kern0pt}d{\isacharparenleft}{\kern0pt}k\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ y\ k{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharasterisk}{\kern0pt}v\ v{\isacharparenright}{\kern0pt}\ {\isacharplus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}P{\isacharunderscore}{\kern0pt}lower\ {\isacharparenleft}{\kern0pt}d{\isacharparenleft}{\kern0pt}k\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ y\ k{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharasterisk}{\kern0pt}v\ GS{\isacharunderscore}{\kern0pt}inv\ {\isacharparenleft}{\kern0pt}d{\isacharparenleft}{\kern0pt}k\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ y\ k{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ v{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isachardollar}{\kern0pt}\ k{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isacharequal}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}inv\ {\isacharparenleft}{\kern0pt}d{\isacharparenleft}{\kern0pt}k\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ y\ k{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ v\ {\isachardollar}{\kern0pt}\ k{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ GS{\isacharunderscore}{\kern0pt}inv{\isacharunderscore}{\kern0pt}rec\ \isacommand{by}\isamarkupfalse%
-\ presburger\isanewline
-\ \ \ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isasymle}\ GS{\isacharunderscore}{\kern0pt}inv\ {\isacharparenleft}{\kern0pt}d{\isacharparenleft}{\kern0pt}k\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ a{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ v\ {\isachardollar}{\kern0pt}\ k{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}linorderD{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isacharbrackleft}{\kern0pt}OF\ assms{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isacharbrackright}{\kern0pt}\ {\isacartoucheopen}y\ {\isasymin}\ D\isactrlsub D{\isacartoucheclose}\ is{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \ \ \isacommand{finally}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}GS{\isacharunderscore}{\kern0pt}inv\ y\ v\ {\isachardollar}{\kern0pt}\ k\ {\isasymle}\ GS{\isacharunderscore}{\kern0pt}inv\ {\isacharparenleft}{\kern0pt}d{\isacharparenleft}{\kern0pt}k\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ a{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ v\ {\isachardollar}{\kern0pt}\ k{\isachardoublequoteclose}\isacommand{{\isachardot}{\kern0pt}}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}d{\isacharparenleft}{\kern0pt}k\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isasymin}\ D\isactrlsub D{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assms\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ is{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det{\isacharunderscore}{\kern0pt}def\ is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}linorder{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}cases\ {\isachardoublequoteopen}s\ {\isacharless}{\kern0pt}\ k{\isachardoublequoteclose}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{case}\isamarkupfalse%
-\ True\isanewline
-\ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ am{\isacharunderscore}{\kern0pt}k\ assms{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}{\isacharbrackleft}{\kern0pt}OF\ True{\isacharbrackright}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}indep{\isacharunderscore}{\kern0pt}high{\isacharunderscore}{\kern0pt}states{\isacharbrackleft}{\kern0pt}of\ s\ {\isachardoublequoteopen}d\ {\isacharparenleft}{\kern0pt}k\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ a{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ d{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}fastforce\ dest{\isacharcolon}{\kern0pt}\ is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}linorderD\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}linorderI{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{case}\isamarkupfalse%
-\ False\isanewline
-\ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assms\ am{\isacharunderscore}{\kern0pt}k\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}GS{\isacharunderscore}{\kern0pt}le{\isacharcolon}{\kern0pt}\ \isanewline
-\ \ {\isachardoublequoteopen}{\isasymexists}d{\isachardot}{\kern0pt}\ {\isasymforall}s{\isasymle}k{\isachardot}{\kern0pt}\ is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max\ {\isacharparenleft}{\kern0pt}{\isasymlambda}d{\isachardot}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}inv\ d\ v\ {\isachardollar}{\kern0pt}\ s{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymlambda}d{\isachardot}{\kern0pt}\ d\ {\isasymin}\ D\isactrlsub D{\isacharparenright}{\kern0pt}\ d{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}induction\ k\ rule{\isacharcolon}{\kern0pt}\ less{\isacharunderscore}{\kern0pt}induct{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{case}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}less\ x{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}case\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}cases\ {\isachardoublequoteopen}{\isasymexists}y{\isachardot}{\kern0pt}\ y\ {\isacharless}{\kern0pt}\ x{\isachardoublequoteclose}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{case}\isamarkupfalse%
-\ True\isanewline
-\ \ \ \ \isacommand{define}\isamarkupfalse%
-\ y\ \isakeyword{where}\ {\isachardoublequoteopen}y\ {\isacharequal}{\kern0pt}\ Max\ {\isacharbraceleft}{\kern0pt}y{\isachardot}{\kern0pt}\ y\ {\isacharless}{\kern0pt}\ x{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\ \ \isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}y\ {\isacharless}{\kern0pt}\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ Max{\isacharunderscore}{\kern0pt}in\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ True\ y{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{obtain}\isamarkupfalse%
-\ d{\isacharunderscore}{\kern0pt}opt\ \isakeyword{where}\ d{\isacharunderscore}{\kern0pt}opt{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max\ {\isacharparenleft}{\kern0pt}{\isasymlambda}d{\isachardot}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}inv\ d\ v\ {\isachardollar}{\kern0pt}\ s{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymlambda}d{\isachardot}{\kern0pt}\ d\ {\isasymin}\ D\isactrlsub D{\isacharparenright}{\kern0pt}\ d{\isacharunderscore}{\kern0pt}opt{\isachardoublequoteclose}\ \isakeyword{if}\ {\isachardoublequoteopen}s\ {\isasymle}\ y{\isachardoublequoteclose}\ \isakeyword{for}\ s\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isacartoucheopen}y\ {\isacharless}{\kern0pt}\ x{\isacartoucheclose}\ less\ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\isanewline
-\ \ \ \ \isacommand{define}\isamarkupfalse%
-\ d{\isacharunderscore}{\kern0pt}act\ \isakeyword{where}\ d{\isacharunderscore}{\kern0pt}act{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}d{\isacharunderscore}{\kern0pt}act\ a\ {\isacharequal}{\kern0pt}\ d{\isacharunderscore}{\kern0pt}opt{\isacharparenleft}{\kern0pt}x\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ a{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{for}\ a\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ le{\isacharunderscore}{\kern0pt}y{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}a\ {\isacharless}{\kern0pt}\ x\ {\isasymLongrightarrow}\ a\ {\isasymle}\ y{\isachardoublequoteclose}\ \isakeyword{for}\ a\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ y{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}GS{\isacharunderscore}{\kern0pt}inv\ d\ v\ {\isacharequal}{\kern0pt}\ r{\isacharunderscore}{\kern0pt}vec\ d\ {\isacharplus}{\kern0pt}\ l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isacharparenleft}{\kern0pt}P{\isacharunderscore}{\kern0pt}upper\ d\ {\isacharasterisk}{\kern0pt}v\ v\ {\isacharplus}{\kern0pt}\ P{\isacharunderscore}{\kern0pt}lower\ d\ {\isacharasterisk}{\kern0pt}v\ {\isacharparenleft}{\kern0pt}GS{\isacharunderscore}{\kern0pt}inv\ d\ v{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{for}\ d\isanewline
-\ \ \ \ \isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}Q{\isacharunderscore}{\kern0pt}mat\ d\ {\isacharasterisk}{\kern0pt}v\ {\isacharparenleft}{\kern0pt}GS{\isacharunderscore}{\kern0pt}inv\ d\ v{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}R{\isacharunderscore}{\kern0pt}mat\ d\ {\isacharasterisk}{\kern0pt}v\ v\ {\isacharplus}{\kern0pt}\ r{\isacharunderscore}{\kern0pt}vec\ d{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ GS{\isacharunderscore}{\kern0pt}inv{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ splitting{\isacharunderscore}{\kern0pt}gauss{\isacharbrackleft}{\kern0pt}unfolded\ is{\isacharunderscore}{\kern0pt}splitting{\isacharunderscore}{\kern0pt}blin{\isacharunderscore}{\kern0pt}def{\isacharprime}{\kern0pt}{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ matrix{\isacharunderscore}{\kern0pt}vector{\isacharunderscore}{\kern0pt}mul{\isacharunderscore}{\kern0pt}assoc\ matrix{\isacharunderscore}{\kern0pt}inv{\isacharunderscore}{\kern0pt}right{\isacharbrackleft}{\kern0pt}OF\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}inverse{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ Q{\isacharunderscore}{\kern0pt}mat{\isacharunderscore}{\kern0pt}def\ R{\isacharunderscore}{\kern0pt}mat{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ scaleR{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}vector{\isacharunderscore}{\kern0pt}assoc\ algebra{\isacharunderscore}{\kern0pt}simps{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isasymSqunion}d\ {\isasymin}\ D\isactrlsub D{\isachardot}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}inv\ d\ v\ {\isachardollar}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSqunion}d\ {\isasymin}\ D\isactrlsub D{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}r{\isacharunderscore}{\kern0pt}vec\ d\ {\isacharplus}{\kern0pt}\ l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isacharparenleft}{\kern0pt}P{\isacharunderscore}{\kern0pt}upper\ d\ {\isacharasterisk}{\kern0pt}v\ v\ {\isacharplus}{\kern0pt}\ P{\isacharunderscore}{\kern0pt}lower\ d\ {\isacharasterisk}{\kern0pt}v\ {\isacharparenleft}{\kern0pt}GS{\isacharunderscore}{\kern0pt}inv\ d\ v{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isachardollar}{\kern0pt}\ x{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isadigit{1}}\ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSqunion}a\ {\isasymin}\ A\ x{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}r{\isacharunderscore}{\kern0pt}vec\ {\isacharparenleft}{\kern0pt}d{\isacharunderscore}{\kern0pt}act\ a{\isacharparenright}{\kern0pt}\ {\isacharplus}{\kern0pt}\ l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isacharparenleft}{\kern0pt}P{\isacharunderscore}{\kern0pt}upper\ {\isacharparenleft}{\kern0pt}d{\isacharunderscore}{\kern0pt}act\ a{\isacharparenright}{\kern0pt}\ {\isacharasterisk}{\kern0pt}v\ v\ {\isacharplus}{\kern0pt}\ P{\isacharunderscore}{\kern0pt}lower\ {\isacharparenleft}{\kern0pt}d{\isacharunderscore}{\kern0pt}act\ a{\isacharparenright}{\kern0pt}\ {\isacharasterisk}{\kern0pt}v\ {\isacharparenleft}{\kern0pt}GS{\isacharunderscore}{\kern0pt}inv\ {\isacharparenleft}{\kern0pt}d{\isacharunderscore}{\kern0pt}act\ a{\isacharparenright}{\kern0pt}\ v{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isachardollar}{\kern0pt}\ x{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}rule\ antisym{\isacharcomma}{\kern0pt}\ rule\ cSUP{\isacharunderscore}{\kern0pt}mono{\isacharcomma}{\kern0pt}\ goal{\isacharunderscore}{\kern0pt}cases{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{case}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}{\isadigit{3}}\ n{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}P{\isacharunderscore}{\kern0pt}upper\ n\ {\isacharasterisk}{\kern0pt}v\ v{\isacharparenright}{\kern0pt}\ {\isachardollar}{\kern0pt}\ x\ {\isasymle}\ {\isacharparenleft}{\kern0pt}P{\isacharunderscore}{\kern0pt}upper\ {\isacharparenleft}{\kern0pt}d{\isacharunderscore}{\kern0pt}opt{\isacharparenleft}{\kern0pt}x\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ n\ x{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharasterisk}{\kern0pt}v\ v{\isacharparenright}{\kern0pt}\ {\isachardollar}{\kern0pt}\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ P{\isacharunderscore}{\kern0pt}upper{\isacharunderscore}{\kern0pt}def\ matrix{\isacharunderscore}{\kern0pt}vector{\isacharunderscore}{\kern0pt}mult{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ P{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}elem\ cong{\isacharcolon}{\kern0pt}\ if{\isacharunderscore}{\kern0pt}cong{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isanewline
-\ \ \ \ \ \ \isacommand{{\isacharbraceleft}{\kern0pt}}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymAnd}j{\isachardot}{\kern0pt}\ j\ {\isacharless}{\kern0pt}\ x\ {\isasymLongrightarrow}\ GS{\isacharunderscore}{\kern0pt}inv\ n\ v\ {\isachardollar}{\kern0pt}\ j\ {\isasymle}\ GS{\isacharunderscore}{\kern0pt}inv\ {\isacharparenleft}{\kern0pt}d{\isacharunderscore}{\kern0pt}opt{\isacharparenleft}{\kern0pt}x\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ n\ x{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ v\ {\isachardollar}{\kern0pt}\ j{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ d{\isacharunderscore}{\kern0pt}opt{\isacharbrackleft}{\kern0pt}OF\ le{\isacharunderscore}{\kern0pt}y{\isacharbrackright}{\kern0pt}\ {\isadigit{3}}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}subst\ GS{\isacharunderscore}{\kern0pt}indep{\isacharunderscore}{\kern0pt}high{\isacharunderscore}{\kern0pt}states{\isacharbrackleft}{\kern0pt}of\ {\isacharunderscore}{\kern0pt}\ {\isachardoublequoteopen}d{\isacharunderscore}{\kern0pt}opt{\isacharparenleft}{\kern0pt}x\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ n\ x{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ d{\isacharunderscore}{\kern0pt}opt{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}linorder{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}P{\isacharunderscore}{\kern0pt}lower\ n\ {\isacharasterisk}{\kern0pt}v\ GS{\isacharunderscore}{\kern0pt}inv\ n\ v{\isacharparenright}{\kern0pt}\ {\isachardollar}{\kern0pt}\ x\ {\isasymle}\ {\isacharparenleft}{\kern0pt}P{\isacharunderscore}{\kern0pt}lower\ {\isacharparenleft}{\kern0pt}d{\isacharunderscore}{\kern0pt}opt{\isacharparenleft}{\kern0pt}x\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ n\ x{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharasterisk}{\kern0pt}v\ GS{\isacharunderscore}{\kern0pt}inv\ {\isacharparenleft}{\kern0pt}d{\isacharunderscore}{\kern0pt}opt{\isacharparenleft}{\kern0pt}x\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ n\ x{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ v{\isacharparenright}{\kern0pt}\ {\isachardollar}{\kern0pt}\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ matrix{\isacharunderscore}{\kern0pt}vector{\isacharunderscore}{\kern0pt}mult{\isacharunderscore}{\kern0pt}def\ P{\isacharunderscore}{\kern0pt}lower{\isacharunderscore}{\kern0pt}def\ P{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}elem\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}fastforce\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ mult{\isacharunderscore}{\kern0pt}left{\isacharunderscore}{\kern0pt}mono\ sum{\isacharunderscore}{\kern0pt}mono{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{{\isacharbraceright}{\kern0pt}}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}case\isanewline
-\ \ \ \ \ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ d{\isacharunderscore}{\kern0pt}act\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ bexI{\isacharbrackleft}{\kern0pt}of\ {\isacharunderscore}{\kern0pt}\ {\isachardoublequoteopen}n\ x{\isachardoublequoteclose}{\isacharbrackright}{\kern0pt}\ mult{\isacharunderscore}{\kern0pt}left{\isacharunderscore}{\kern0pt}mono\ add{\isacharunderscore}{\kern0pt}mono\ simp{\isacharcolon}{\kern0pt}\ is{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \isacommand{case}\isamarkupfalse%
-\ {\isadigit{4}}\isanewline
-\ \ \ \ \ \ \isacommand{then}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}case\isanewline
-\ \ \ \ \ \ \isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}rule\ cSUP{\isacharunderscore}{\kern0pt}mono{\isacharcomma}{\kern0pt}\ goal{\isacharunderscore}{\kern0pt}cases{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{case}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}{\isadigit{3}}\ n{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{then}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}case\ \isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ d{\isacharunderscore}{\kern0pt}opt\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}fastforce\ simp{\isacharcolon}{\kern0pt}\ d{\isacharunderscore}{\kern0pt}act\ is{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det{\isacharunderscore}{\kern0pt}def\ is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}linorder\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ bexI{\isacharbrackleft}{\kern0pt}of\ {\isacharunderscore}{\kern0pt}\ {\isachardoublequoteopen}d{\isacharunderscore}{\kern0pt}act\ n{\isachardoublequoteclose}{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{qed}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ A{\isacharunderscore}{\kern0pt}ne{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{qed}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSqunion}a\ {\isasymin}\ A\ x{\isachardot}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}inv\ {\isacharparenleft}{\kern0pt}d{\isacharunderscore}{\kern0pt}act\ a{\isacharparenright}{\kern0pt}\ v\ {\isachardollar}{\kern0pt}\ x{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isadigit{1}}\ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \isacommand{finally}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isacharasterisk}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isasymSqunion}d\ {\isasymin}\ D\isactrlsub D{\isachardot}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}inv\ d\ v\ {\isachardollar}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSqunion}a\ {\isasymin}\ A\ x{\isachardot}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}inv\ {\isacharparenleft}{\kern0pt}d{\isacharunderscore}{\kern0pt}act\ a{\isacharparenright}{\kern0pt}\ v\ {\isachardollar}{\kern0pt}\ x{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isacommand{{\isachardot}{\kern0pt}}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{then}\isamarkupfalse%
-\ \isacommand{obtain}\isamarkupfalse%
-\ a{\isacharunderscore}{\kern0pt}opt\ \isakeyword{where}\ a{\isacharunderscore}{\kern0pt}opt{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max\ {\isacharparenleft}{\kern0pt}{\isasymlambda}a{\isachardot}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}inv\ {\isacharparenleft}{\kern0pt}d{\isacharunderscore}{\kern0pt}act\ a{\isacharparenright}{\kern0pt}\ v\ {\isachardollar}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymlambda}a{\isachardot}{\kern0pt}\ a\ {\isasymin}\ A\ x{\isacharparenright}{\kern0pt}\ a{\isacharunderscore}{\kern0pt}opt{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ A{\isacharunderscore}{\kern0pt}ne\ finite\ finite{\isacharunderscore}{\kern0pt}is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isasymSqunion}d\ {\isasymin}\ D\isactrlsub D{\isachardot}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}inv\ d\ v\ {\isachardollar}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}inv\ {\isacharparenleft}{\kern0pt}d{\isacharunderscore}{\kern0pt}act\ a{\isacharunderscore}{\kern0pt}opt{\isacharparenright}{\kern0pt}\ v\ {\isachardollar}{\kern0pt}\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharasterisk}{\kern0pt}\ arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}SUP{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ am{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}opt{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max\ {\isacharparenleft}{\kern0pt}{\isasymlambda}d{\isachardot}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}inv\ d\ v\ {\isachardollar}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymlambda}d{\isachardot}{\kern0pt}\ d\ {\isasymin}\ D\isactrlsub D{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}d{\isacharunderscore}{\kern0pt}act\ a{\isacharunderscore}{\kern0pt}opt{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ a{\isacharunderscore}{\kern0pt}opt\ d{\isacharunderscore}{\kern0pt}opt\ d{\isacharunderscore}{\kern0pt}act\ \isacommand{unfolding}\isamarkupfalse%
-\ is{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}fastforce\ dest{\isacharcolon}{\kern0pt}\ is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}linorderD{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ SUP{\isacharunderscore}{\kern0pt}is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max\ {\isacharparenleft}{\kern0pt}{\isasymlambda}d{\isachardot}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}inv\ d\ v\ {\isachardollar}{\kern0pt}\ x{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymlambda}d{\isachardot}{\kern0pt}\ d\ {\isasymin}\ D\isactrlsub D{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}d{\isacharunderscore}{\kern0pt}act\ a{\isacharunderscore}{\kern0pt}opt{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{if}\ {\isachardoublequoteopen}x{\isacharprime}{\kern0pt}\ {\isacharless}{\kern0pt}\ x{\isachardoublequoteclose}\ \isakeyword{for}\ x{\isacharprime}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}s{\isacharprime}{\kern0pt}\ {\isasymle}\ x{\isacharprime}{\kern0pt}\ {\isasymLongrightarrow}\ d{\isacharunderscore}{\kern0pt}act\ a{\isacharunderscore}{\kern0pt}opt\ s{\isacharprime}{\kern0pt}\ {\isacharequal}{\kern0pt}\ d{\isacharunderscore}{\kern0pt}opt\ s{\isacharprime}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{for}\ s{\isacharprime}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ d{\isacharunderscore}{\kern0pt}act\ that\ is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}linorderD{\isacharbrackleft}{\kern0pt}OF\ d{\isacharunderscore}{\kern0pt}opt{\isacharbrackleft}{\kern0pt}OF\ le{\isacharunderscore}{\kern0pt}y{\isacharbrackleft}{\kern0pt}OF\ that{\isacharbrackright}{\kern0pt}{\isacharbrackright}{\kern0pt}{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ am{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}opt\ is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}linorderD{\isacharbrackleft}{\kern0pt}OF\ d{\isacharunderscore}{\kern0pt}opt{\isacharbrackleft}{\kern0pt}OF\ le{\isacharunderscore}{\kern0pt}y{\isacharbrackleft}{\kern0pt}OF\ that{\isacharbrackright}{\kern0pt}{\isacharbrackright}{\kern0pt}{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}indep{\isacharunderscore}{\kern0pt}high{\isacharunderscore}{\kern0pt}states{\isacharbrackleft}{\kern0pt}of\ {\isacharunderscore}{\kern0pt}\ {\isachardoublequoteopen}d{\isacharunderscore}{\kern0pt}act\ a{\isacharunderscore}{\kern0pt}opt{\isachardoublequoteclose}\ d{\isacharunderscore}{\kern0pt}opt{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ am{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}opt\ antisym{\isacharunderscore}{\kern0pt}conv{\isadigit{1}}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{case}\isamarkupfalse%
-\ False\isanewline
-\ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ finite{\isacharunderscore}{\kern0pt}is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharbrackleft}{\kern0pt}OF\ finite{\isacharunderscore}{\kern0pt}D\isactrlsub D{\isacharbrackright}{\kern0pt}\ \isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}fastforce\ simp{\isacharcolon}{\kern0pt}\ arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}def\ someI{\isacharunderscore}{\kern0pt}ex\ dest{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ le{\isacharunderscore}{\kern0pt}neq{\isacharunderscore}{\kern0pt}trans{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ ex{\isacharunderscore}{\kern0pt}is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}GS{\isacharcolon}{\kern0pt}\ \isanewline
-\ \ {\isachardoublequoteopen}{\isasymexists}d{\isachardot}{\kern0pt}\ {\isasymforall}s{\isachardot}{\kern0pt}\ is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max\ {\isacharparenleft}{\kern0pt}{\isasymlambda}d{\isachardot}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}inv\ d\ v\ {\isachardollar}{\kern0pt}\ s{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymlambda}d{\isachardot}{\kern0pt}\ d\ {\isasymin}\ D\isactrlsub D{\isacharparenright}{\kern0pt}\ d{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}GS{\isacharunderscore}{\kern0pt}le{\isacharbrackleft}{\kern0pt}of\ {\isachardoublequoteopen}Max\ UNIV{\isachardoublequoteclose}{\isacharbrackright}{\kern0pt}\ \ \isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{function}\isamarkupfalse%
-\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}fun\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}fun\ v\ s\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSqunion}a\ {\isasymin}\ A\ s{\isachardot}{\kern0pt}\ r\ {\isacharparenleft}{\kern0pt}s{\isacharcomma}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isacharplus}{\kern0pt}\ l\ {\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}\isanewline
-\ \ {\isacharparenleft}{\kern0pt}{\isasymSum}s{\isacharprime}{\kern0pt}\ {\isacharless}{\kern0pt}\ s{\isachardot}{\kern0pt}\ pmf\ {\isacharparenleft}{\kern0pt}K\ {\isacharparenleft}{\kern0pt}s{\isacharcomma}{\kern0pt}a{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ s{\isacharprime}{\kern0pt}\ {\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}fun\ v\ s{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharplus}{\kern0pt}\ \isanewline
-\ \ {\isacharparenleft}{\kern0pt}{\isasymSum}s{\isacharprime}{\kern0pt}\ {\isasymin}\ {\isacharbraceleft}{\kern0pt}s{\isacharprime}{\kern0pt}{\isachardot}{\kern0pt}\ s\ {\isasymle}\ s{\isacharprime}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ pmf\ {\isacharparenleft}{\kern0pt}K\ {\isacharparenleft}{\kern0pt}s{\isacharcomma}{\kern0pt}a{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ s{\isacharprime}{\kern0pt}\ {\isacharasterisk}{\kern0pt}\ v\ s{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isacommand{termination}\isamarkupfalse%
-\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}relation\ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}{\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}y{\isacharparenright}{\kern0pt}{\isachardot}{\kern0pt}\ snd\ x\ {\isacharless}{\kern0pt}\ snd\ y{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}{\isacharcomma}{\kern0pt}\ rule\ wfI{\isacharunderscore}{\kern0pt}min{\isacharcomma}{\kern0pt}\ goal{\isacharunderscore}{\kern0pt}cases{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{case}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}\ x\ Q{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\ {\isachardoublequoteopen}\ x\ {\isasymin}\ Q{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isacharasterisk}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}u{\isachardot}{\kern0pt}\ {\isasymexists}a{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}a{\isacharcomma}{\kern0pt}\ u{\isacharparenright}{\kern0pt}\ {\isasymin}\ Q{\isacharbraceright}{\kern0pt}\ {\isasymnoteq}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}mono{\isacharunderscore}{\kern0pt}tags{\isacharcomma}{\kern0pt}\ lifting{\isacharparenright}{\kern0pt}\ {\isacartoucheopen}x{\isasymin}Q{\isacartoucheclose}\ prod{\isachardot}{\kern0pt}collapse\ Collect{\isacharunderscore}{\kern0pt}empty{\isacharunderscore}{\kern0pt}eq{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymexists}a\ x{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}a{\isacharcomma}{\kern0pt}x{\isacharparenright}{\kern0pt}{\isasymin}Q\ {\isasymand}\ x\ {\isacharequal}{\kern0pt}\ Min\ {\isacharparenleft}{\kern0pt}snd\ {\isacharbackquote}{\kern0pt}\ Q{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ image{\isacharunderscore}{\kern0pt}iff{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}mono{\isacharunderscore}{\kern0pt}tags{\isacharcomma}{\kern0pt}\ lifting{\isacharparenright}{\kern0pt}\ equals{\isadigit{0}}D\ Min{\isacharunderscore}{\kern0pt}in{\isacharbrackleft}{\kern0pt}OF\ finite{\isacharbrackright}{\kern0pt}\ prod{\isachardot}{\kern0pt}collapse\ image{\isacharunderscore}{\kern0pt}iff{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{then}\isamarkupfalse%
-\ \isacommand{obtain}\isamarkupfalse%
-\ x\ \isakeyword{where}\ {\isachardoublequoteopen}x\ {\isasymin}\ Q{\isachardoublequoteclose}\ {\isachardoublequoteopen}snd\ x\ {\isacharequal}{\kern0pt}\ Min\ {\isacharbraceleft}{\kern0pt}snd\ x{\isacharbar}{\kern0pt}\ x{\isachardot}{\kern0pt}\ x\ {\isasymin}\ Q{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ Setcompr{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}image\ snd{\isacharunderscore}{\kern0pt}conv{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}case\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isacharasterisk}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}intro\ bexI{\isacharbrackleft}{\kern0pt}of\ {\isacharunderscore}{\kern0pt}\ x{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\ auto\isanewline
-\isacommand{qed}\isamarkupfalse%
-\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{declare}\isamarkupfalse%
-\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}fun{\isachardot}{\kern0pt}simps{\isacharbrackleft}{\kern0pt}simp\ del{\isacharbrackright}{\kern0pt}\ \isanewline
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ {\isachardoublequoteopen}GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}elem\ v\ s\ a\ {\isacharequal}{\kern0pt}\ r\ {\isacharparenleft}{\kern0pt}s{\isacharcomma}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isacharplus}{\kern0pt}\ l\ {\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}\isanewline
-\ \ {\isacharparenleft}{\kern0pt}{\isasymSum}s{\isacharprime}{\kern0pt}\ {\isacharless}{\kern0pt}\ s{\isachardot}{\kern0pt}\ pmf\ {\isacharparenleft}{\kern0pt}K\ {\isacharparenleft}{\kern0pt}s{\isacharcomma}{\kern0pt}a{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ s{\isacharprime}{\kern0pt}\ {\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}fun\ v\ s{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharplus}{\kern0pt}\ \isanewline
-\ \ {\isacharparenleft}{\kern0pt}{\isasymSum}s{\isacharprime}{\kern0pt}\ {\isasymin}\ {\isacharbraceleft}{\kern0pt}s{\isacharprime}{\kern0pt}{\isachardot}{\kern0pt}\ s\ {\isasymle}\ s{\isacharprime}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ pmf\ {\isacharparenleft}{\kern0pt}K\ {\isacharparenleft}{\kern0pt}s{\isacharcomma}{\kern0pt}a{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ s{\isacharprime}{\kern0pt}\ {\isacharasterisk}{\kern0pt}\ v\ s{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}fun{\isacharunderscore}{\kern0pt}elem{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}fun\ v\ s\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSqunion}a\ {\isasymin}\ A\ s{\isachardot}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}elem\ v\ s\ a{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}elem{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \isacommand{using}\isamarkupfalse%
-\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}fun{\isachardot}{\kern0pt}simps\ \isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ blast%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ {\isachardoublequoteopen}GS{\isacharunderscore}{\kern0pt}rec\ v\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymchi}\ s{\isachardot}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}fun\ {\isacharparenleft}{\kern0pt}vec{\isacharunderscore}{\kern0pt}nth\ v{\isacharparenright}{\kern0pt}\ s{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}def{\isacharprime}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}GS{\isacharunderscore}{\kern0pt}rec\ v\ {\isachardollar}{\kern0pt}\ s\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSqunion}a\ {\isasymin}\ A\ s{\isachardot}{\kern0pt}\ r\ {\isacharparenleft}{\kern0pt}s{\isacharcomma}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isacharplus}{\kern0pt}\ l\ {\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}\isanewline
-\ \ {\isacharparenleft}{\kern0pt}{\isasymSum}s{\isacharprime}{\kern0pt}\ {\isacharless}{\kern0pt}\ s{\isachardot}{\kern0pt}\ pmf\ {\isacharparenleft}{\kern0pt}K\ {\isacharparenleft}{\kern0pt}s{\isacharcomma}{\kern0pt}a{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ s{\isacharprime}{\kern0pt}\ {\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}GS{\isacharunderscore}{\kern0pt}rec\ v\ {\isachardollar}{\kern0pt}\ s{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharplus}{\kern0pt}\ \isanewline
-\ \ {\isacharparenleft}{\kern0pt}{\isasymSum}s{\isacharprime}{\kern0pt}\ {\isasymin}\ {\isacharbraceleft}{\kern0pt}s{\isacharprime}{\kern0pt}{\isachardot}{\kern0pt}\ s\ {\isasymle}\ s{\isacharprime}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ pmf\ {\isacharparenleft}{\kern0pt}K\ {\isacharparenleft}{\kern0pt}s{\isacharcomma}{\kern0pt}a{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ s{\isacharprime}{\kern0pt}\ {\isacharasterisk}{\kern0pt}\ v\ {\isachardollar}{\kern0pt}\ s{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}fun{\isachardot}{\kern0pt}simps{\isacharbrackleft}{\kern0pt}of\ {\isacharunderscore}{\kern0pt}\ s{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}eq{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}GS{\isacharunderscore}{\kern0pt}rec\ v\ {\isachardollar}{\kern0pt}\ s\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSqunion}a\ {\isasymin}\ A\ s{\isachardot}{\kern0pt}\ r\ {\isacharparenleft}{\kern0pt}s{\isacharcomma}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isacharplus}{\kern0pt}\ l\ {\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}\isanewline
-\ \ {\isacharparenleft}{\kern0pt}P{\isacharunderscore}{\kern0pt}lower\ {\isacharparenleft}{\kern0pt}d{\isacharparenleft}{\kern0pt}s\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ a{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharasterisk}{\kern0pt}v\ {\isacharparenleft}{\kern0pt}GS{\isacharunderscore}{\kern0pt}rec\ v{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isachardollar}{\kern0pt}\ s\ {\isacharplus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}P{\isacharunderscore}{\kern0pt}upper\ {\isacharparenleft}{\kern0pt}d{\isacharparenleft}{\kern0pt}s\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ a{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharasterisk}{\kern0pt}v\ v{\isacharparenright}{\kern0pt}\ {\isachardollar}{\kern0pt}\ s{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}def{\isacharprime}{\kern0pt}{\isacharbrackleft}{\kern0pt}of\ v\ s{\isacharbrackright}{\kern0pt}\ P{\isacharunderscore}{\kern0pt}lower{\isacharunderscore}{\kern0pt}def\ P{\isacharunderscore}{\kern0pt}upper{\isacharunderscore}{\kern0pt}def\ P{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}elem\ matrix{\isacharunderscore}{\kern0pt}vector{\isacharunderscore}{\kern0pt}mult{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ if{\isacharunderscore}{\kern0pt}distrib{\isacharbrackleft}{\kern0pt}\isakeyword{where}\ f\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}{\isasymlambda}x{\isachardot}{\kern0pt}\ x\ {\isacharasterisk}{\kern0pt}\ {\isacharunderscore}{\kern0pt}\ {\isachardollar}{\kern0pt}\ {\isacharunderscore}{\kern0pt}{\isachardoublequoteclose}{\isacharbrackright}{\kern0pt}\ sum{\isachardot}{\kern0pt}If{\isacharunderscore}{\kern0pt}cases\ lessThan{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isacommand{definition}\isamarkupfalse%
-\ {\isachardoublequoteopen}GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}step\ d\ v\ {\isasymequiv}\ r{\isacharunderscore}{\kern0pt}vec\ d\ {\isacharplus}{\kern0pt}\ l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isacharparenleft}{\kern0pt}P{\isacharunderscore}{\kern0pt}lower\ d\ {\isacharasterisk}{\kern0pt}v\ GS{\isacharunderscore}{\kern0pt}rec\ v\ {\isacharplus}{\kern0pt}\ P{\isacharunderscore}{\kern0pt}upper\ d\ {\isacharasterisk}{\kern0pt}v\ v{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}eq{\isacharprime}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}GS{\isacharunderscore}{\kern0pt}rec\ v\ {\isachardollar}{\kern0pt}\ s\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSqunion}a\ {\isasymin}\ A\ s{\isachardot}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}step\ {\isacharparenleft}{\kern0pt}d{\isacharparenleft}{\kern0pt}s{\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ a{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ v\ {\isachardollar}{\kern0pt}\ s{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}eq\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}step{\isacharunderscore}{\kern0pt}def\ \isacommand{by}\isamarkupfalse%
-\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}vec{\isacharcolon}{\kern0pt}\isanewline
-\ \ {\isachardoublequoteopen}GS{\isacharunderscore}{\kern0pt}rec\ v\ {\isachardollar}{\kern0pt}\ s\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSqunion}d{\isasymin}D\isactrlsub D{\isachardot}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}step\ d\ v\ {\isachardollar}{\kern0pt}\ s{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{obtain}\isamarkupfalse%
-\ d\ \isakeyword{where}\ d{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max\ {\isacharparenleft}{\kern0pt}{\isasymlambda}d{\isachardot}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}step\ d\ v\ {\isachardollar}{\kern0pt}\ s{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymlambda}d{\isachardot}{\kern0pt}\ d\ {\isasymin}D\isactrlsub D{\isacharparenright}{\kern0pt}\ d{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ finite{\isacharunderscore}{\kern0pt}is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharbrackleft}{\kern0pt}OF\ finite{\isacharcomma}{\kern0pt}\ of\ {\isachardoublequoteopen}D\isactrlsub D{\isachardoublequoteclose}\ {\isacharbrackright}{\kern0pt}\ ex{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}GS{\isacharunderscore}{\kern0pt}rec\ v\ {\isachardollar}{\kern0pt}\ s\ {\isacharequal}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}step\ d\ v\ {\isachardollar}{\kern0pt}\ s{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}eq{\isacharprime}{\kern0pt}{\isacharbrackleft}{\kern0pt}of\ {\isacharunderscore}{\kern0pt}\ {\isacharunderscore}{\kern0pt}\ d{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}intro\ antisym\ cSUP{\isacharunderscore}{\kern0pt}least{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymAnd}x{\isachardot}{\kern0pt}\ x\ {\isasymin}\ A\ s\ {\isasymLongrightarrow}\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}step\ {\isacharparenleft}{\kern0pt}d{\isacharparenleft}{\kern0pt}s\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ x{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ v\ {\isachardollar}{\kern0pt}\ s\ {\isasymle}\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}step\ d\ v\ {\isachardollar}{\kern0pt}\ s{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ A{\isacharunderscore}{\kern0pt}ne\ d\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}intro\ is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}linorderD{\isacharbrackleft}{\kern0pt}OF\ d{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ is{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det{\isacharunderscore}{\kern0pt}def\ is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}linorder{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}step\ d\ v\ {\isachardollar}{\kern0pt}\ s\ {\isasymle}\ {\isacharparenleft}{\kern0pt}{\isasymSqunion}a{\isasymin}A\ s{\isachardot}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}step\ {\isacharparenleft}{\kern0pt}d{\isacharparenleft}{\kern0pt}s\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ a{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ v\ {\isachardollar}{\kern0pt}\ s{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ d\ \isacommand{unfolding}\isamarkupfalse%
-\ is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}linorder\ is{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det{\isacharunderscore}{\kern0pt}def\ fun{\isacharunderscore}{\kern0pt}upd{\isacharunderscore}{\kern0pt}triv\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ cSUP{\isacharunderscore}{\kern0pt}upper{\isadigit{2}}{\isacharbrackleft}{\kern0pt}of\ {\isacharunderscore}{\kern0pt}\ {\isacharunderscore}{\kern0pt}\ {\isachardoublequoteopen}d\ s{\isachardoublequoteclose}{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ A{\isacharunderscore}{\kern0pt}ne{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ d\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}subst\ arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}SUP{\isacharbrackleft}{\kern0pt}symmetric{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\ auto\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{lift{\isacharunderscore}{\kern0pt}definition}\isamarkupfalse%
-\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}fun\isactrlsub b\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}s\ {\isasymRightarrow}\isactrlsub b\ real{\isacharparenright}{\kern0pt}\ {\isasymRightarrow}\ {\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}s\ {\isasymRightarrow}\isactrlsub b\ real{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{is}\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}fun\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ {\isachardoublequoteopen}GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}fun{\isacharunderscore}{\kern0pt}inner\ {\isacharparenleft}{\kern0pt}v\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isacharprime}{\kern0pt}s\ {\isasymRightarrow}\isactrlsub b\ real{\isacharparenright}{\kern0pt}\ s\ a\ {\isasymequiv}\ r\ {\isacharparenleft}{\kern0pt}s{\isacharcomma}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isacharplus}{\kern0pt}\ l\ {\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}\isanewline
-\ \ {\isacharparenleft}{\kern0pt}{\isasymSum}s{\isacharprime}{\kern0pt}\ {\isacharless}{\kern0pt}\ s{\isachardot}{\kern0pt}\ pmf\ {\isacharparenleft}{\kern0pt}K\ {\isacharparenleft}{\kern0pt}s{\isacharcomma}{\kern0pt}a{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ s{\isacharprime}{\kern0pt}\ {\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}fun\isactrlsub b\ v\ s{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharplus}{\kern0pt}\ \isanewline
-\ \ {\isacharparenleft}{\kern0pt}{\isasymSum}s{\isacharprime}{\kern0pt}\ {\isasymin}\ {\isacharbraceleft}{\kern0pt}s{\isacharprime}{\kern0pt}{\isachardot}{\kern0pt}\ s\ {\isasymle}\ s{\isacharprime}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ pmf\ {\isacharparenleft}{\kern0pt}K\ {\isacharparenleft}{\kern0pt}s{\isacharcomma}{\kern0pt}a{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ s{\isacharprime}{\kern0pt}\ {\isacharasterisk}{\kern0pt}\ v\ s{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}iter\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}iter\ v\ s\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSqunion}a\ {\isasymin}\ A\ s{\isachardot}{\kern0pt}\ r\ {\isacharparenleft}{\kern0pt}s{\isacharcomma}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isacharplus}{\kern0pt}\ l\ {\isacharasterisk}{\kern0pt}\isanewline
-\ \ {\isacharparenleft}{\kern0pt}{\isasymSum}s{\isacharprime}{\kern0pt}\ {\isasymin}\ UNIV{\isachardot}{\kern0pt}\ pmf\ {\isacharparenleft}{\kern0pt}K\ {\isacharparenleft}{\kern0pt}s{\isacharcomma}{\kern0pt}a{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ s{\isacharprime}{\kern0pt}\ {\isacharasterisk}{\kern0pt}\ v\ s{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}fun{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}GS{\isacharunderscore}{\kern0pt}iter{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}{\isasymforall}s{\isacharprime}{\kern0pt}\ {\isacharless}{\kern0pt}\ s{\isachardot}{\kern0pt}\ v{\isacharunderscore}{\kern0pt}next\ s{\isacharprime}{\kern0pt}\ {\isacharequal}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}fun\ v\ s{\isacharprime}{\kern0pt}{\isachardoublequoteclose}\ {\isachardoublequoteopen}{\isasymforall}s{\isacharprime}{\kern0pt}\ {\isasymin}\ {\isacharbraceleft}{\kern0pt}s{\isacharprime}{\kern0pt}{\isachardot}{\kern0pt}\ s\ {\isasymle}\ s{\isacharprime}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ v{\isacharunderscore}{\kern0pt}next\ s{\isacharprime}{\kern0pt}\ {\isacharequal}{\kern0pt}\ v\ s{\isacharprime}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}fun\ v\ s\ {\isacharequal}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}iter\ v{\isacharunderscore}{\kern0pt}next\ s{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}s{\isacharprime}{\kern0pt}{\isachardot}{\kern0pt}\ s{\isacharprime}{\kern0pt}\ {\isacharless}{\kern0pt}\ s{\isacharbraceright}{\kern0pt}\ {\isasymunion}\ {\isacharbraceleft}{\kern0pt}s{\isacharprime}{\kern0pt}{\isachardot}{\kern0pt}\ s\ {\isasymle}\ s{\isacharprime}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ UNIV{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isacharasterisk}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isasymSum}s{\isacharprime}{\kern0pt}{\isacharless}{\kern0pt}s{\isachardot}{\kern0pt}\ f\ s{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharplus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSum}s{\isacharprime}{\kern0pt}{\isasymin}Collect\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}{\isasymle}{\isacharparenright}{\kern0pt}\ s{\isacharparenright}{\kern0pt}{\isachardot}{\kern0pt}\ f\ s{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSum}s{\isacharprime}{\kern0pt}\ {\isasymin}\ UNIV{\isachardot}{\kern0pt}\ f\ s{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{for}\ f\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}subst\ sum{\isachardot}{\kern0pt}union{\isacharunderscore}{\kern0pt}disjoint{\isacharbrackleft}{\kern0pt}symmetric{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}auto\ simp\ add{\isacharcolon}{\kern0pt}\ lessThan{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}fun\ v\ s\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSqunion}a{\isasymin}A\ s{\isachardot}{\kern0pt}\ r\ {\isacharparenleft}{\kern0pt}s{\isacharcomma}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isacharplus}{\kern0pt}\ l\ {\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}{\isasymSum}s{\isacharprime}{\kern0pt}{\isacharless}{\kern0pt}s{\isachardot}{\kern0pt}\ pmf\ {\isacharparenleft}{\kern0pt}K\ {\isacharparenleft}{\kern0pt}s{\isacharcomma}{\kern0pt}\ a{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ s{\isacharprime}{\kern0pt}\ {\isacharasterisk}{\kern0pt}\ v{\isacharunderscore}{\kern0pt}next\ s{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharplus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSum}s{\isacharprime}{\kern0pt}{\isasymin}Collect\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}{\isasymle}{\isacharparenright}{\kern0pt}\ s{\isacharparenright}{\kern0pt}{\isachardot}{\kern0pt}\ pmf\ {\isacharparenleft}{\kern0pt}K\ {\isacharparenleft}{\kern0pt}s{\isacharcomma}{\kern0pt}\ a{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ s{\isacharprime}{\kern0pt}\ {\isacharasterisk}{\kern0pt}\ v\ s{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assms\ \isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}subst\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}fun{\isachardot}{\kern0pt}simps{\isacharparenright}{\kern0pt}\ auto\isanewline
-\ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSqunion}a{\isasymin}A\ s{\isachardot}{\kern0pt}\ r\ {\isacharparenleft}{\kern0pt}s{\isacharcomma}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isacharplus}{\kern0pt}\ l\ {\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}{\isasymSum}s{\isacharprime}{\kern0pt}{\isacharless}{\kern0pt}s{\isachardot}{\kern0pt}\ pmf\ {\isacharparenleft}{\kern0pt}K\ {\isacharparenleft}{\kern0pt}s{\isacharcomma}{\kern0pt}\ a{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ s{\isacharprime}{\kern0pt}\ {\isacharasterisk}{\kern0pt}\ v{\isacharunderscore}{\kern0pt}next\ s{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharplus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSum}s{\isacharprime}{\kern0pt}{\isasymin}Collect\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}{\isasymle}{\isacharparenright}{\kern0pt}\ s{\isacharparenright}{\kern0pt}{\isachardot}{\kern0pt}\ pmf\ {\isacharparenleft}{\kern0pt}K\ {\isacharparenleft}{\kern0pt}s{\isacharcomma}{\kern0pt}\ a{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ s{\isacharprime}{\kern0pt}\ {\isacharasterisk}{\kern0pt}\ v{\isacharunderscore}{\kern0pt}next\ s{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assms\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isacharequal}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}iter\ v{\isacharunderscore}{\kern0pt}next\ s{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ {\isacharasterisk}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}iter{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{finally}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\ \isacommand{{\isachardot}{\kern0pt}}\isamarkupfalse%
-\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ foldl{\isacharunderscore}{\kern0pt}upd{\isacharunderscore}{\kern0pt}notin{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymnotin}\ set\ X\ {\isasymLongrightarrow}\ foldl\ {\isacharparenleft}{\kern0pt}{\isasymlambda}f\ y{\isachardot}{\kern0pt}\ f{\isacharparenleft}{\kern0pt}y\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ g\ f\ y{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ c\ X\ x\ {\isacharequal}{\kern0pt}\ c\ x{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}induction\ X\ arbitrary{\isacharcolon}{\kern0pt}\ c{\isacharparenright}{\kern0pt}\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ foldl{\isacharunderscore}{\kern0pt}upd{\isacharunderscore}{\kern0pt}notin{\isacharprime}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymnotin}\ set\ Y\ {\isasymLongrightarrow}\ foldl\ {\isacharparenleft}{\kern0pt}{\isasymlambda}f\ y{\isachardot}{\kern0pt}\ f{\isacharparenleft}{\kern0pt}y\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ g\ f\ y{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ c\ {\isacharparenleft}{\kern0pt}X{\isacharat}{\kern0pt}Y{\isacharparenright}{\kern0pt}\ x\ {\isacharequal}{\kern0pt}\ foldl\ {\isacharparenleft}{\kern0pt}{\isasymlambda}f\ y{\isachardot}{\kern0pt}\ f{\isacharparenleft}{\kern0pt}y\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ g\ f\ y{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ c\ X\ x{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}induction\ X\ arbitrary{\isacharcolon}{\kern0pt}\ x\ c\ Y{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}auto\ simp\ add{\isacharcolon}{\kern0pt}\ foldl{\isacharunderscore}{\kern0pt}upd{\isacharunderscore}{\kern0pt}notin{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ sorted{\isacharunderscore}{\kern0pt}list{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}set{\isacharunderscore}{\kern0pt}split{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}finite\ X{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}sorted{\isacharunderscore}{\kern0pt}list{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}set\ X\ {\isacharequal}{\kern0pt}\ sorted{\isacharunderscore}{\kern0pt}list{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}set\ {\isacharbraceleft}{\kern0pt}x\ {\isasymin}\ X{\isachardot}{\kern0pt}\ x\ {\isacharless}{\kern0pt}\ y{\isacharbraceright}{\kern0pt}\ {\isacharat}{\kern0pt}\ sorted{\isacharunderscore}{\kern0pt}list{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}set\ {\isacharbraceleft}{\kern0pt}x\ {\isasymin}\ X{\isachardot}{\kern0pt}\ y\ {\isasymle}\ x{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ assms\isanewline
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}induction\ {\isachardoublequoteopen}card\ X{\isachardoublequoteclose}\ arbitrary{\isacharcolon}{\kern0pt}\ X{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{case}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}Suc\ n\ X{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}sorted{\isacharunderscore}{\kern0pt}list{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}set\ X\ {\isacharequal}{\kern0pt}\ Min\ X\ {\isacharhash}{\kern0pt}\ sorted{\isacharunderscore}{\kern0pt}list{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}set\ {\isacharparenleft}{\kern0pt}X\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}Min\ X{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ Suc\ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ intro{\isacharcolon}{\kern0pt}\ sorted{\isacharunderscore}{\kern0pt}list{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}set{\isacharunderscore}{\kern0pt}nonempty{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isacharequal}{\kern0pt}\ Min\ X\ {\isacharhash}{\kern0pt}\ sorted{\isacharunderscore}{\kern0pt}list{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}set\ {\isacharbraceleft}{\kern0pt}x\ {\isasymin}\ {\isacharparenleft}{\kern0pt}X\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}Min\ X{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardot}{\kern0pt}\ x\ {\isacharless}{\kern0pt}\ y{\isacharbraceright}{\kern0pt}\ {\isacharat}{\kern0pt}\ sorted{\isacharunderscore}{\kern0pt}list{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}set\ {\isacharbraceleft}{\kern0pt}x\ {\isasymin}\ {\isacharparenleft}{\kern0pt}X\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}Min\ X{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardot}{\kern0pt}\ y\ {\isasymle}\ x{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ Suc\ card{\isachardot}{\kern0pt}remove{\isacharbrackleft}{\kern0pt}OF\ Suc{\isacharparenleft}{\kern0pt}{\isadigit{3}}{\isacharparenright}{\kern0pt}\ Min{\isacharunderscore}{\kern0pt}in{\isacharbrackright}{\kern0pt}\ card{\isachardot}{\kern0pt}empty\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}fastforce\ simp{\isacharcolon}{\kern0pt}\ Suc{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharplus}{\kern0pt}\isanewline
-\ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isacharequal}{\kern0pt}\ sorted{\isacharunderscore}{\kern0pt}list{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}set\ {\isacharparenleft}{\kern0pt}{\isacharbraceleft}{\kern0pt}x\ {\isasymin}\ X{\isachardot}{\kern0pt}\ x\ {\isacharless}{\kern0pt}\ y{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharat}{\kern0pt}\ sorted{\isacharunderscore}{\kern0pt}list{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}set\ {\isacharbraceleft}{\kern0pt}x\ {\isasymin}\ X{\isachardot}{\kern0pt}\ y\ {\isasymle}\ x{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}cases\ {\isachardoublequoteopen}Min\ X\ {\isacharless}{\kern0pt}\ y{\isachardoublequoteclose}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{case}\isamarkupfalse%
-\ True\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ Min{\isacharunderscore}{\kern0pt}eq{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}Min\ X\ {\isacharequal}{\kern0pt}\ Min\ {\isacharbraceleft}{\kern0pt}x\ {\isasymin}\ X{\isachardot}{\kern0pt}\ x\ {\isacharless}{\kern0pt}\ y{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ True\ Suc\ Min{\isacharunderscore}{\kern0pt}in\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}subst\ eq{\isacharunderscore}{\kern0pt}Min{\isacharunderscore}{\kern0pt}iff{\isacharparenright}{\kern0pt}\ fastforce{\isacharplus}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}x\ {\isasymin}\ {\isacharparenleft}{\kern0pt}X\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}Min\ X{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardot}{\kern0pt}\ x\ {\isacharless}{\kern0pt}\ y{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}x\ {\isasymin}\ X{\isachardot}{\kern0pt}\ x\ {\isacharless}{\kern0pt}\ y{\isacharbraceright}{\kern0pt}\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}Min\ {\isacharbraceleft}{\kern0pt}x\ {\isasymin}\ X{\isachardot}{\kern0pt}\ x\ {\isacharless}{\kern0pt}\ y{\isacharbraceright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ Min{\isacharunderscore}{\kern0pt}eq\ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}Min\ X\ {\isacharhash}{\kern0pt}\ sorted{\isacharunderscore}{\kern0pt}list{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}set\ {\isacharbraceleft}{\kern0pt}x\ {\isasymin}\ {\isacharparenleft}{\kern0pt}X\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}Min\ X{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardot}{\kern0pt}\ x\ {\isacharless}{\kern0pt}\ y{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ \isanewline
-\ \ \ \ \ \ Min\ {\isacharbraceleft}{\kern0pt}x\ {\isasymin}\ X{\isachardot}{\kern0pt}\ x\ {\isacharless}{\kern0pt}\ y{\isacharbraceright}{\kern0pt}\ {\isacharhash}{\kern0pt}\ sorted{\isacharunderscore}{\kern0pt}list{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}set\ {\isacharparenleft}{\kern0pt}{\isacharbraceleft}{\kern0pt}x\ {\isasymin}\ X{\isachardot}{\kern0pt}\ x\ {\isacharless}{\kern0pt}\ y{\isacharbraceright}{\kern0pt}\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}Min\ {\isacharbraceleft}{\kern0pt}x\ {\isasymin}\ X{\isachardot}{\kern0pt}\ x\ {\isacharless}{\kern0pt}\ y{\isacharbraceright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ Min{\isacharunderscore}{\kern0pt}eq\ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isacharequal}{\kern0pt}\ sorted{\isacharunderscore}{\kern0pt}list{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}set\ {\isacharparenleft}{\kern0pt}{\isacharbraceleft}{\kern0pt}x\ {\isasymin}\ X{\isachardot}{\kern0pt}\ x\ {\isacharless}{\kern0pt}\ y{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ Suc\ True\ Min{\isacharunderscore}{\kern0pt}in\ Min{\isacharunderscore}{\kern0pt}eq\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}subst\ sorted{\isacharunderscore}{\kern0pt}list{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}set{\isacharunderscore}{\kern0pt}nonempty{\isacharbrackleft}{\kern0pt}symmetric{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\ fastforce{\isacharplus}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{finally}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}Min\ X\ {\isacharhash}{\kern0pt}\ sorted{\isacharunderscore}{\kern0pt}list{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}set\ {\isacharbraceleft}{\kern0pt}x\ {\isasymin}\ {\isacharparenleft}{\kern0pt}X\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}Min\ X{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardot}{\kern0pt}\ x\ {\isacharless}{\kern0pt}\ y{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ sorted{\isacharunderscore}{\kern0pt}list{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}set\ {\isacharparenleft}{\kern0pt}{\isacharbraceleft}{\kern0pt}x\ {\isasymin}\ X{\isachardot}{\kern0pt}\ x\ {\isacharless}{\kern0pt}\ y{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isacommand{{\isachardot}{\kern0pt}}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}Min\ X\ {\isacharhash}{\kern0pt}\ sorted{\isacharunderscore}{\kern0pt}list{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}set\ {\isacharbraceleft}{\kern0pt}x\ {\isasymin}\ {\isacharparenleft}{\kern0pt}X\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}Min\ X{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardot}{\kern0pt}\ x\ {\isacharless}{\kern0pt}\ y{\isacharbraceright}{\kern0pt}\ {\isacharat}{\kern0pt}\ sorted{\isacharunderscore}{\kern0pt}list{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}set\ {\isacharbraceleft}{\kern0pt}x\ {\isasymin}\ {\isacharparenleft}{\kern0pt}X\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}Min\ X{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardot}{\kern0pt}\ y\ {\isasymle}\ x{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ \isanewline
-\ \ \ \ \ \ sorted{\isacharunderscore}{\kern0pt}list{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}set\ {\isacharparenleft}{\kern0pt}{\isacharbraceleft}{\kern0pt}x\ {\isasymin}\ X{\isachardot}{\kern0pt}\ x\ {\isacharless}{\kern0pt}\ y{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharat}{\kern0pt}\ sorted{\isacharunderscore}{\kern0pt}list{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}set\ {\isacharbraceleft}{\kern0pt}x\ {\isasymin}\ {\isacharparenleft}{\kern0pt}X\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}Min\ X{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardot}{\kern0pt}\ y\ {\isasymle}\ x{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \isacommand{then}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ True\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ append{\isacharunderscore}{\kern0pt}Cons{\isacharbrackleft}{\kern0pt}symmetric{\isacharbrackright}{\kern0pt}\ simp\ del{\isacharcolon}{\kern0pt}\ append{\isacharunderscore}{\kern0pt}Cons\ dest{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ leD\ intro{\isacharcolon}{\kern0pt}\ arg{\isacharunderscore}{\kern0pt}cong{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{case}\isamarkupfalse%
-\ False\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ Min{\isacharunderscore}{\kern0pt}eq{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}Min\ X\ {\isacharequal}{\kern0pt}\ Min\ {\isacharbraceleft}{\kern0pt}x\ {\isasymin}\ X{\isachardot}{\kern0pt}\ y\ {\isasymle}\ x{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ False\ Suc\ Min{\isacharunderscore}{\kern0pt}in\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}subst\ eq{\isacharunderscore}{\kern0pt}Min{\isacharunderscore}{\kern0pt}iff{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}fastforce\ simp{\isacharcolon}{\kern0pt}\ linorder{\isacharunderscore}{\kern0pt}class{\isachardot}{\kern0pt}not{\isacharunderscore}{\kern0pt}less{\isacharparenright}{\kern0pt}{\isacharplus}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{2}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}x\ {\isasymin}\ {\isacharparenleft}{\kern0pt}X\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}Min\ X{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardot}{\kern0pt}\ y\ {\isasymle}\ x{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}x\ {\isasymin}\ X{\isachardot}{\kern0pt}\ y\ {\isasymle}\ x{\isacharbraceright}{\kern0pt}\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}Min\ {\isacharbraceleft}{\kern0pt}x\ {\isasymin}\ X{\isachardot}{\kern0pt}\ y\ {\isasymle}\ x{\isacharbraceright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ Min{\isacharunderscore}{\kern0pt}eq\ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}x\ {\isasymin}\ X\ {\isasymLongrightarrow}\ {\isasymnot}\ x\ {\isacharless}{\kern0pt}\ y{\isachardoublequoteclose}\ \isakeyword{for}\ x\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ False\ Min{\isacharunderscore}{\kern0pt}less{\isacharunderscore}{\kern0pt}iff\ Suc{\isacharparenleft}{\kern0pt}{\isadigit{3}}{\isacharparenright}{\kern0pt}\ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}x\ {\isasymin}\ X{\isachardot}{\kern0pt}\ x\ {\isacharless}{\kern0pt}\ y{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\ \isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}Min\ X\ {\isacharhash}{\kern0pt}\ sorted{\isacharunderscore}{\kern0pt}list{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}set\ {\isacharbraceleft}{\kern0pt}x\ {\isasymin}\ X\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}Min\ X{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ x\ {\isacharless}{\kern0pt}\ y{\isacharbraceright}{\kern0pt}\ {\isacharat}{\kern0pt}\ sorted{\isacharunderscore}{\kern0pt}list{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}set\ {\isacharbraceleft}{\kern0pt}x\ {\isasymin}\ X\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}Min\ X{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ y\ {\isasymle}\ x{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ Min\ X\ {\isacharhash}{\kern0pt}\ sorted{\isacharunderscore}{\kern0pt}list{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}set\ {\isacharbraceleft}{\kern0pt}x\ {\isasymin}\ X\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}Min\ X{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ y\ {\isasymle}\ x{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ Suc\ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isacharequal}{\kern0pt}\ Min\ {\isacharbraceleft}{\kern0pt}x\ {\isasymin}\ X{\isachardot}{\kern0pt}\ y\ {\isasymle}\ x{\isacharbraceright}{\kern0pt}\ {\isacharhash}{\kern0pt}\ sorted{\isacharunderscore}{\kern0pt}list{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}set\ {\isacharparenleft}{\kern0pt}{\isacharbraceleft}{\kern0pt}x\ {\isasymin}\ X{\isachardot}{\kern0pt}\ y\ {\isasymle}\ x{\isacharbraceright}{\kern0pt}\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}Min\ {\isacharbraceleft}{\kern0pt}x\ {\isasymin}\ X{\isachardot}{\kern0pt}\ y\ {\isasymle}\ x{\isacharbraceright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ Min{\isacharunderscore}{\kern0pt}eq\ {\isadigit{2}}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isacharequal}{\kern0pt}\ sorted{\isacharunderscore}{\kern0pt}list{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}set\ {\isacharparenleft}{\kern0pt}{\isacharbraceleft}{\kern0pt}x\ {\isasymin}\ X{\isachardot}{\kern0pt}\ y\ {\isasymle}\ x{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \ \ \ \ \ \isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ Suc\ False\ Min{\isacharunderscore}{\kern0pt}in\ Min{\isacharunderscore}{\kern0pt}eq\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}subst\ sorted{\isacharunderscore}{\kern0pt}list{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}set{\isacharunderscore}{\kern0pt}nonempty{\isacharbrackleft}{\kern0pt}symmetric{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\ fastforce{\isacharplus}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isacharequal}{\kern0pt}\ sorted{\isacharunderscore}{\kern0pt}list{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}set\ {\isacharparenleft}{\kern0pt}{\isacharbraceleft}{\kern0pt}x\ {\isasymin}\ X{\isachardot}{\kern0pt}\ x\ {\isacharless}{\kern0pt}\ y{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharat}{\kern0pt}\ sorted{\isacharunderscore}{\kern0pt}list{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}set\ {\isacharparenleft}{\kern0pt}{\isacharbraceleft}{\kern0pt}x\ {\isasymin}\ X{\isachardot}{\kern0pt}\ y\ {\isasymle}\ x{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ {\isacartoucheopen}{\isacharbraceleft}{\kern0pt}x\ {\isasymin}\ X{\isachardot}{\kern0pt}\ x\ {\isacharless}{\kern0pt}\ y{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacartoucheclose}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{finally}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isacommand{{\isachardot}{\kern0pt}}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{finally}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}case\isacommand{{\isachardot}{\kern0pt}}\isamarkupfalse%
-\isanewline
-\isacommand{qed}\isamarkupfalse%
-\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ sorted{\isacharunderscore}{\kern0pt}list{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}set{\isacharunderscore}{\kern0pt}split{\isacharprime}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}finite\ X{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}sorted{\isacharunderscore}{\kern0pt}list{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}set\ X\ {\isacharequal}{\kern0pt}\ sorted{\isacharunderscore}{\kern0pt}list{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}set\ {\isacharbraceleft}{\kern0pt}x\ {\isasymin}\ X{\isachardot}{\kern0pt}\ x\ {\isasymle}\ y{\isacharbraceright}{\kern0pt}\ {\isacharat}{\kern0pt}\ sorted{\isacharunderscore}{\kern0pt}list{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}set\ {\isacharbraceleft}{\kern0pt}x\ {\isasymin}\ X{\isachardot}{\kern0pt}\ y\ {\isacharless}{\kern0pt}\ x{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ sorted{\isacharunderscore}{\kern0pt}list{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}set{\isacharunderscore}{\kern0pt}split{\isacharbrackleft}{\kern0pt}of\ X{\isacharbrackright}{\kern0pt}\isanewline
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}cases\ {\isachardoublequoteopen}{\isasymexists}x\ {\isasymin}\ X{\isachardot}{\kern0pt}\ y\ {\isacharless}{\kern0pt}\ x{\isachardoublequoteclose}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{case}\isamarkupfalse%
-\ True\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}x\ {\isasymin}\ X{\isachardot}{\kern0pt}\ x\ {\isasymle}\ y{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}x\ {\isasymin}\ X{\isachardot}{\kern0pt}\ x\ {\isacharless}{\kern0pt}\ Min\ {\isacharbraceleft}{\kern0pt}x\ {\isasymin}\ X{\isachardot}{\kern0pt}\ y\ {\isacharless}{\kern0pt}\ x{\isacharbraceright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assms\ True\ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}subst\ Min{\isacharunderscore}{\kern0pt}gr{\isacharunderscore}{\kern0pt}iff{\isacharparenright}{\kern0pt}\ auto\isanewline
-\ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}x\ {\isasymin}\ X{\isachardot}{\kern0pt}\ y\ {\isacharless}{\kern0pt}\ x{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}x\ {\isasymin}\ X{\isachardot}{\kern0pt}\ Min\ {\isacharbraceleft}{\kern0pt}x\ {\isasymin}\ X{\isachardot}{\kern0pt}\ y\ {\isacharless}{\kern0pt}\ x{\isacharbraceright}{\kern0pt}\ {\isasymle}\ \ x{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assms\ True\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}subst\ Min{\isacharunderscore}{\kern0pt}le{\isacharunderscore}{\kern0pt}iff{\isacharparenright}{\kern0pt}\ auto\isanewline
-\ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ sorted{\isacharunderscore}{\kern0pt}list{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}set{\isacharunderscore}{\kern0pt}split{\isacharbrackleft}{\kern0pt}OF\ assms{\isacharcomma}{\kern0pt}\ of\ {\isachardoublequoteopen}Min\ {\isacharbraceleft}{\kern0pt}x\ {\isasymin}\ X{\isachardot}{\kern0pt}\ y\ {\isacharless}{\kern0pt}\ x{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{case}\isamarkupfalse%
-\ False\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isacharasterisk}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}x\ {\isasymin}\ X{\isachardot}{\kern0pt}\ y\ {\isacharless}{\kern0pt}\ x{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}x\ {\isasymin}\ X{\isachardot}{\kern0pt}\ x\ {\isasymle}\ y{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ X{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp\ add{\isacharcolon}{\kern0pt}linorder{\isacharunderscore}{\kern0pt}class{\isachardot}{\kern0pt}not{\isacharunderscore}{\kern0pt}less{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ False\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ {\isacharasterisk}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}fun{\isacharunderscore}{\kern0pt}code{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}fun\ v\ s\ {\isacharequal}{\kern0pt}\ foldl\ {\isacharparenleft}{\kern0pt}{\isasymlambda}v\ s{\isachardot}{\kern0pt}\ v{\isacharparenleft}{\kern0pt}s\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}iter\ v\ s{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ v\ {\isacharparenleft}{\kern0pt}sorted{\isacharunderscore}{\kern0pt}list{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}set\ {\isacharbraceleft}{\kern0pt}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}s{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ s{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}induction\ s\ rule{\isacharcolon}{\kern0pt}\ less{\isacharunderscore}{\kern0pt}induct{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{case}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}less\ s{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}foldl\ {\isacharparenleft}{\kern0pt}{\isasymlambda}v\ s{\isachardot}{\kern0pt}\ v{\isacharparenleft}{\kern0pt}s\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}iter\ v\ s{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ v\ {\isacharparenleft}{\kern0pt}sorted{\isacharunderscore}{\kern0pt}list{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}set\ {\isacharbraceleft}{\kern0pt}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}s{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ s\isanewline
-\ \ \ \ {\isacharequal}{\kern0pt}\ foldl\ {\isacharparenleft}{\kern0pt}{\isasymlambda}v\ s{\isachardot}{\kern0pt}\ v{\isacharparenleft}{\kern0pt}s\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}iter\ v\ s{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ v\ {\isacharparenleft}{\kern0pt}sorted{\isacharunderscore}{\kern0pt}list{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}set\ {\isacharbraceleft}{\kern0pt}x\ {\isasymin}\ {\isacharbraceleft}{\kern0pt}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}s{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ x\ {\isacharless}{\kern0pt}\ s{\isacharbraceright}{\kern0pt}\ {\isacharat}{\kern0pt}\ sorted{\isacharunderscore}{\kern0pt}list{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}set\ {\isacharbraceleft}{\kern0pt}x\ {\isasymin}\ {\isacharbraceleft}{\kern0pt}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}s{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ s\ {\isasymle}\ x{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ s{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}subst\ sorted{\isacharunderscore}{\kern0pt}list{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}set{\isacharunderscore}{\kern0pt}split{\isacharbrackleft}{\kern0pt}of\ {\isacharunderscore}{\kern0pt}\ s{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\ auto\isanewline
-\ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isacharequal}{\kern0pt}\ foldl\ {\isacharparenleft}{\kern0pt}{\isasymlambda}v\ s{\isachardot}{\kern0pt}\ v{\isacharparenleft}{\kern0pt}s\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}iter\ v\ s{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ v\ {\isacharparenleft}{\kern0pt}sorted{\isacharunderscore}{\kern0pt}list{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}set\ {\isacharbraceleft}{\kern0pt}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharless}{\kern0pt}s{\isacharbraceright}{\kern0pt}\ {\isacharat}{\kern0pt}\ sorted{\isacharunderscore}{\kern0pt}list{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}set\ {\isacharbraceleft}{\kern0pt}s{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ s{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}x\ {\isasymin}\ {\isacharbraceleft}{\kern0pt}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}s{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ x\ {\isacharless}{\kern0pt}s{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharless}{\kern0pt}s{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}x\ {\isasymin}\ {\isacharbraceleft}{\kern0pt}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}s{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ s\ {\isasymle}\ x{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}s{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isacharequal}{\kern0pt}\ \ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}iter\ {\isacharparenleft}{\kern0pt}foldl\ {\isacharparenleft}{\kern0pt}{\isasymlambda}v\ s{\isachardot}{\kern0pt}\ v{\isacharparenleft}{\kern0pt}s\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}iter\ v\ s{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ v\ {\isacharparenleft}{\kern0pt}sorted{\isacharunderscore}{\kern0pt}list{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}set\ {\isacharbraceleft}{\kern0pt}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharless}{\kern0pt}s{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ s{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isacharequal}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}fun\ v\ s{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}intro\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}fun{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}GS{\isacharunderscore}{\kern0pt}iter{\isacharbrackleft}{\kern0pt}symmetric{\isacharbrackright}{\kern0pt}{\isacharcomma}{\kern0pt}\ safe{\isacharcomma}{\kern0pt}\ goal{\isacharunderscore}{\kern0pt}cases{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{case}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}\ s{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{assume}\isamarkupfalse%
-\ {\isachardoublequoteopen}s{\isacharprime}{\kern0pt}\ {\isacharless}{\kern0pt}\ s{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ {\isacharasterisk}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}Collect\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}{\isacharless}{\kern0pt}{\isacharparenright}{\kern0pt}\ s{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymnoteq}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\ \isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}x\ {\isasymin}\ {\isacharbraceleft}{\kern0pt}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharless}{\kern0pt}s{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ x\ {\isacharless}{\kern0pt}\ Min\ {\isacharparenleft}{\kern0pt}Collect\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}{\isacharless}{\kern0pt}{\isacharparenright}{\kern0pt}\ s{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}s{\isacharprime}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ leI\ {\isadigit{1}}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp\ add{\isacharcolon}{\kern0pt}\ Min{\isacharunderscore}{\kern0pt}gr{\isacharunderscore}{\kern0pt}iff{\isacharbrackleft}{\kern0pt}OF\ finite{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}x\ {\isasymin}\ {\isacharbraceleft}{\kern0pt}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharless}{\kern0pt}s{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ Min\ {\isacharparenleft}{\kern0pt}Collect\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}{\isacharless}{\kern0pt}{\isacharparenright}{\kern0pt}\ s{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymle}\ x{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}s{\isacharprime}{\kern0pt}{\isacharless}{\kern0pt}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharless}{\kern0pt}s{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isacharasterisk}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp\ add{\isacharcolon}{\kern0pt}\ Min{\isacharunderscore}{\kern0pt}le{\isacharunderscore}{\kern0pt}iff{\isacharbrackleft}{\kern0pt}OF\ finite{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}foldl\ {\isacharparenleft}{\kern0pt}{\isasymlambda}v\ s{\isachardot}{\kern0pt}\ v{\isacharparenleft}{\kern0pt}s\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}iter\ v\ s{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ v\ {\isacharparenleft}{\kern0pt}sorted{\isacharunderscore}{\kern0pt}list{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}set\ {\isacharbraceleft}{\kern0pt}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharless}{\kern0pt}s{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ s{\isacharprime}{\kern0pt}\ \isanewline
-\ \ {\isacharequal}{\kern0pt}\ foldl\ {\isacharparenleft}{\kern0pt}{\isasymlambda}v\ s{\isachardot}{\kern0pt}\ v{\isacharparenleft}{\kern0pt}s\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}iter\ v\ s{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ v\ {\isacharparenleft}{\kern0pt}sorted{\isacharunderscore}{\kern0pt}list{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}set\ {\isacharbraceleft}{\kern0pt}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}s{\isacharprime}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharat}{\kern0pt}\ sorted{\isacharunderscore}{\kern0pt}list{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}set\ {\isacharbraceleft}{\kern0pt}s{\isacharprime}{\kern0pt}{\isacharless}{\kern0pt}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharless}{\kern0pt}s{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ s{\isacharprime}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}subst\ sorted{\isacharunderscore}{\kern0pt}list{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}set{\isacharunderscore}{\kern0pt}split{\isacharbrackleft}{\kern0pt}of\ {\isacharunderscore}{\kern0pt}\ {\isachardoublequoteopen}Min{\isacharbraceleft}{\kern0pt}s{\isachardot}{\kern0pt}\ s{\isacharprime}{\kern0pt}\ {\isacharless}{\kern0pt}\ s{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\ auto\isanewline
-\ \ \ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isacharequal}{\kern0pt}\ \ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}fun\ v\ s{\isacharprime}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isadigit{1}}{\isachardoublequoteclose}\ less{\isachardot}{\kern0pt}IH\ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}subst\ foldl{\isacharunderscore}{\kern0pt}upd{\isacharunderscore}{\kern0pt}notin{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}\ fastforce{\isacharplus}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{finally}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}case\isacommand{{\isachardot}{\kern0pt}}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ intro{\isacharcolon}{\kern0pt}\ foldl{\isacharunderscore}{\kern0pt}upd{\isacharunderscore}{\kern0pt}notin{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{finally}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}case\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}fun{\isacharunderscore}{\kern0pt}code{\isacharprime}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}fun\ v\ s\ {\isacharequal}{\kern0pt}\ foldl\ {\isacharparenleft}{\kern0pt}{\isasymlambda}v\ s{\isachardot}{\kern0pt}\ v{\isacharparenleft}{\kern0pt}s\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}iter\ v\ s{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ v\ {\isacharparenleft}{\kern0pt}sorted{\isacharunderscore}{\kern0pt}list{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}set\ UNIV{\isacharparenright}{\kern0pt}\ s{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}cases\ {\isachardoublequoteopen}s\ {\isacharequal}{\kern0pt}\ Max\ UNIV{\isachardoublequoteclose}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{case}\isamarkupfalse%
-\ True\isanewline
-\ \ \isacommand{then}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\ \isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}fun{\isacharunderscore}{\kern0pt}code\ atMost{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{case}\isamarkupfalse%
-\ False\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isacharasterisk}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}Collect\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}{\isacharless}{\kern0pt}{\isacharparenright}{\kern0pt}\ s{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymnoteq}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ not{\isacharunderscore}{\kern0pt}le\ eq{\isacharunderscore}{\kern0pt}Max{\isacharunderscore}{\kern0pt}iff{\isacharbrackleft}{\kern0pt}OF\ finite{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}x{\isachardot}{\kern0pt}\ x\ {\isacharless}{\kern0pt}\ Min\ {\isacharparenleft}{\kern0pt}Collect\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}{\isacharless}{\kern0pt}{\isacharparenright}{\kern0pt}\ s{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}s{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ Min{\isacharunderscore}{\kern0pt}less{\isacharunderscore}{\kern0pt}iff{\isacharbrackleft}{\kern0pt}OF\ finite\ {\isacharasterisk}{\kern0pt}{\isacharbrackright}{\kern0pt}\ intro{\isacharcolon}{\kern0pt}\ leI{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{then}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ sorted{\isacharunderscore}{\kern0pt}list{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}set{\isacharunderscore}{\kern0pt}split{\isacharbrackleft}{\kern0pt}of\ UNIV\ {\isachardoublequoteopen}Min{\isacharbraceleft}{\kern0pt}s{\isacharprime}{\kern0pt}{\isachardot}{\kern0pt}\ s\ {\isacharless}{\kern0pt}\ s{\isacharprime}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}{\isacharcomma}{\kern0pt}\ OF\ finite{\isacharbrackright}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}fun{\isacharunderscore}{\kern0pt}code\ \isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}subst\ foldl{\isacharunderscore}{\kern0pt}upd{\isacharunderscore}{\kern0pt}notin{\isacharprime}{\kern0pt}{\isacharbrackleft}{\kern0pt}of\ s{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\ auto\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}fun{\isacharunderscore}{\kern0pt}code{\isacharprime}{\kern0pt}{\isacharprime}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}fun\ v\ {\isacharequal}{\kern0pt}\ foldl\ {\isacharparenleft}{\kern0pt}{\isasymlambda}v\ s{\isachardot}{\kern0pt}\ v{\isacharparenleft}{\kern0pt}s\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}iter\ v\ s{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ v\ {\isacharparenleft}{\kern0pt}sorted{\isacharunderscore}{\kern0pt}list{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}set\ UNIV{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}fun{\isacharunderscore}{\kern0pt}code{\isacharprime}{\kern0pt}\ \isacommand{by}\isamarkupfalse%
-\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}elem{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}GS{\isacharunderscore}{\kern0pt}rec\ v\ {\isachardollar}{\kern0pt}\ s\ {\isacharequal}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}fun\ {\isacharparenleft}{\kern0pt}vec{\isacharunderscore}{\kern0pt}nth\ v{\isacharparenright}{\kern0pt}\ s{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}step{\isacharunderscore}{\kern0pt}elem{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}step\ d\ v\ {\isachardollar}{\kern0pt}\ s\ {\isacharequal}{\kern0pt}\ r\ {\isacharparenleft}{\kern0pt}s{\isacharcomma}{\kern0pt}\ d\ s{\isacharparenright}{\kern0pt}\ {\isacharplus}{\kern0pt}\ l\ {\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}{\isasymSum}s{\isacharprime}{\kern0pt}\ {\isacharless}{\kern0pt}\ s{\isachardot}{\kern0pt}\ pmf\ {\isacharparenleft}{\kern0pt}K\ {\isacharparenleft}{\kern0pt}s{\isacharcomma}{\kern0pt}\ d\ s{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ s{\isacharprime}{\kern0pt}\ {\isacharasterisk}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}rec\ v\ {\isachardollar}{\kern0pt}\ s{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharplus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSum}s{\isacharprime}{\kern0pt}\ {\isasymin}\ {\isacharbraceleft}{\kern0pt}s{\isacharprime}{\kern0pt}{\isachardot}{\kern0pt}\ s\ {\isasymle}\ s{\isacharprime}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ pmf\ {\isacharparenleft}{\kern0pt}K\ {\isacharparenleft}{\kern0pt}s{\isacharcomma}{\kern0pt}\ d\ s{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ s{\isacharprime}{\kern0pt}\ {\isacharasterisk}{\kern0pt}\ v\ {\isachardollar}{\kern0pt}\ s{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}step{\isacharunderscore}{\kern0pt}def\ P{\isacharunderscore}{\kern0pt}upper{\isacharunderscore}{\kern0pt}def\ P{\isacharunderscore}{\kern0pt}lower{\isacharunderscore}{\kern0pt}def\ lessThan{\isacharunderscore}{\kern0pt}def\ P{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}elem\ matrix{\isacharunderscore}{\kern0pt}vector{\isacharunderscore}{\kern0pt}mult{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ sum{\isachardot}{\kern0pt}If{\isacharunderscore}{\kern0pt}cases\ algebra{\isacharunderscore}{\kern0pt}simps\ if{\isacharunderscore}{\kern0pt}distrib{\isacharbrackleft}{\kern0pt}of\ {\isachardoublequoteopen}{\isasymlambda}x{\isachardot}{\kern0pt}\ {\isacharunderscore}{\kern0pt}\ {\isachardollar}{\kern0pt}\ {\isacharunderscore}{\kern0pt}\ {\isacharasterisk}{\kern0pt}\ x{\isachardoublequoteclose}{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}step{\isacharunderscore}{\kern0pt}act{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}d\ {\isasymin}D\isactrlsub D{\isachardoublequoteclose}\ {\isachardoublequoteopen}is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max\ {\isacharparenleft}{\kern0pt}{\isasymlambda}a{\isachardot}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}step\ {\isacharparenleft}{\kern0pt}d{\isacharprime}{\kern0pt}{\isacharparenleft}{\kern0pt}s\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ a{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ v\ {\isachardollar}{\kern0pt}\ s{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymlambda}a{\isachardot}{\kern0pt}\ a\ {\isasymin}A\ s{\isacharparenright}{\kern0pt}\ a{\isachardoublequoteclose}\ \isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max\ {\isacharparenleft}{\kern0pt}{\isasymlambda}d{\isachardot}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}step\ d\ v\ {\isachardollar}{\kern0pt}\ s{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymlambda}d{\isachardot}{\kern0pt}\ d\ {\isasymin}D\isactrlsub D{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}d{\isacharparenleft}{\kern0pt}s\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ a{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ assms\isanewline
-\ \ \isacommand{unfolding}\isamarkupfalse%
-\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}step{\isacharunderscore}{\kern0pt}elem\ is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}linorder\ is{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}step{\isacharunderscore}{\kern0pt}act{\isacharprime}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}d\ {\isasymin}D\isactrlsub D{\isachardoublequoteclose}\ {\isachardoublequoteopen}is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max\ {\isacharparenleft}{\kern0pt}{\isasymlambda}a{\isachardot}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}step\ {\isacharparenleft}{\kern0pt}d{\isacharprime}{\kern0pt}{\isacharparenleft}{\kern0pt}s\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ a{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ v\ {\isachardollar}{\kern0pt}\ s{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymlambda}a{\isachardot}{\kern0pt}\ a\ {\isasymin}A\ s{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}d\ s{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max\ {\isacharparenleft}{\kern0pt}{\isasymlambda}d{\isachardot}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}step\ d\ v\ {\isachardollar}{\kern0pt}\ s{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymlambda}d{\isachardot}{\kern0pt}\ d\ {\isasymin}D\isactrlsub D{\isacharparenright}{\kern0pt}\ d{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}step{\isacharunderscore}{\kern0pt}act{\isacharbrackleft}{\kern0pt}OF\ assms{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ fastforce%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\isanewline
-\ \ is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}GS{\isacharunderscore}{\kern0pt}rec{\isacharcolon}{\kern0pt}\ \isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}{\isasymAnd}s{\isachardot}{\kern0pt}\ is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max\ {\isacharparenleft}{\kern0pt}{\isasymlambda}d{\isachardot}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}step\ d\ v\ {\isachardollar}{\kern0pt}\ s{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymlambda}d{\isachardot}{\kern0pt}\ d\ {\isasymin}\ D\isactrlsub D{\isacharparenright}{\kern0pt}\ d{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}GS{\isacharunderscore}{\kern0pt}rec\ v\ {\isacharequal}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}step\ d\ v{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}SUP{\isacharbrackleft}{\kern0pt}OF\ assms{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ vec{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}iff\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}vec\ {\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\isanewline
-\ \ is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}GS{\isacharunderscore}{\kern0pt}rec{\isacharprime}{\kern0pt}{\isacharcolon}{\kern0pt}\ \isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max\ {\isacharparenleft}{\kern0pt}{\isasymlambda}d{\isachardot}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}step\ d\ v\ {\isachardollar}{\kern0pt}\ s{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymlambda}d{\isachardot}{\kern0pt}\ d\ {\isasymin}\ D\isactrlsub D{\isacharparenright}{\kern0pt}\ d{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}GS{\isacharunderscore}{\kern0pt}rec\ v\ {\isachardollar}{\kern0pt}\ s\ {\isacharequal}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}step\ d\ v\ {\isachardollar}{\kern0pt}\ s{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ assms\ \isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}vec\ arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}SUP{\isacharbrackleft}{\kern0pt}symmetric{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\isanewline
-\ \ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}GS{\isacharunderscore}{\kern0pt}inv{\isacharcolon}{\kern0pt}\ \isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}{\isasymAnd}s{\isachardot}{\kern0pt}\ is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max\ {\isacharparenleft}{\kern0pt}{\isasymlambda}d{\isachardot}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}step\ d\ v\ {\isachardollar}{\kern0pt}\ s{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymlambda}d{\isachardot}{\kern0pt}\ d\ {\isasymin}\ D\isactrlsub D{\isacharparenright}{\kern0pt}\ d{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}GS{\isacharunderscore}{\kern0pt}rec\ v\ {\isacharequal}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}inv\ d\ v{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}GS{\isacharunderscore}{\kern0pt}rec\ v\ {\isacharequal}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}step\ d\ v{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}GS{\isacharunderscore}{\kern0pt}rec{\isacharbrackleft}{\kern0pt}OF\ assms{\isacharbrackright}{\kern0pt}\ \isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}GS{\isacharunderscore}{\kern0pt}rec\ v\ {\isacharequal}{\kern0pt}\ r{\isacharunderscore}{\kern0pt}vec\ d\ {\isacharplus}{\kern0pt}\ R{\isacharunderscore}{\kern0pt}mat\ d\ {\isacharasterisk}{\kern0pt}v\ v\ {\isacharplus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ P{\isacharunderscore}{\kern0pt}lower\ d{\isacharparenright}{\kern0pt}\ {\isacharasterisk}{\kern0pt}v\ GS{\isacharunderscore}{\kern0pt}rec\ v{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ R{\isacharunderscore}{\kern0pt}mat{\isacharunderscore}{\kern0pt}def\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}step{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ scaleR{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}vector{\isacharunderscore}{\kern0pt}assoc\ algebra{\isacharunderscore}{\kern0pt}simps{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}Q{\isacharunderscore}{\kern0pt}mat\ d\ {\isacharasterisk}{\kern0pt}v\ GS{\isacharunderscore}{\kern0pt}rec\ v\ {\isacharequal}{\kern0pt}\ r{\isacharunderscore}{\kern0pt}vec\ d\ {\isacharplus}{\kern0pt}\ R{\isacharunderscore}{\kern0pt}mat\ d\ {\isacharasterisk}{\kern0pt}v\ v{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ Q{\isacharunderscore}{\kern0pt}mat{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharcomma}{\kern0pt}\ lifting{\isacharparenright}{\kern0pt}\ add{\isacharunderscore}{\kern0pt}diff{\isacharunderscore}{\kern0pt}cancel\ matrix{\isacharunderscore}{\kern0pt}vector{\isacharunderscore}{\kern0pt}mult{\isacharunderscore}{\kern0pt}diff{\isacharunderscore}{\kern0pt}rdistrib\ matrix{\isacharunderscore}{\kern0pt}vector{\isacharunderscore}{\kern0pt}mul{\isacharunderscore}{\kern0pt}lid{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}inv\ {\isacharparenleft}{\kern0pt}Q{\isacharunderscore}{\kern0pt}mat\ d{\isacharparenright}{\kern0pt}\ {\isacharasterisk}{\kern0pt}{\isacharasterisk}{\kern0pt}\ Q{\isacharunderscore}{\kern0pt}mat\ d{\isacharparenright}{\kern0pt}\ {\isacharasterisk}{\kern0pt}v\ GS{\isacharunderscore}{\kern0pt}rec\ v\ {\isacharequal}{\kern0pt}\ matrix{\isacharunderscore}{\kern0pt}inv\ {\isacharparenleft}{\kern0pt}Q{\isacharunderscore}{\kern0pt}mat\ d{\isacharparenright}{\kern0pt}\ {\isacharasterisk}{\kern0pt}v\ {\isacharparenleft}{\kern0pt}r{\isacharunderscore}{\kern0pt}vec\ d\ {\isacharplus}{\kern0pt}\ R{\isacharunderscore}{\kern0pt}mat\ d\ {\isacharasterisk}{\kern0pt}v\ v{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ matrix{\isacharunderscore}{\kern0pt}vector{\isacharunderscore}{\kern0pt}mul{\isacharunderscore}{\kern0pt}assoc{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}GS{\isacharunderscore}{\kern0pt}rec\ v\ {\isacharequal}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}inv\ d\ v{\isachardoublequoteclose}\ \isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ splitting{\isacharunderscore}{\kern0pt}gauss\isanewline
-\ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ GS{\isacharunderscore}{\kern0pt}inv{\isacharunderscore}{\kern0pt}def\ is{\isacharunderscore}{\kern0pt}splitting{\isacharunderscore}{\kern0pt}blin{\isacharunderscore}{\kern0pt}def{\isacharprime}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}subst\ {\isacharparenleft}{\kern0pt}asm{\isacharparenright}{\kern0pt}\ matrix{\isacharunderscore}{\kern0pt}inv{\isacharunderscore}{\kern0pt}left{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}fastforce\ intro{\isacharcolon}{\kern0pt}\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}inverse{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharplus}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\isanewline
-\ \ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}step{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}GS{\isacharunderscore}{\kern0pt}inv{\isacharcolon}{\kern0pt}\ \isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}{\isasymAnd}s{\isachardot}{\kern0pt}\ is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max\ {\isacharparenleft}{\kern0pt}{\isasymlambda}d{\isachardot}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}step\ d\ v\ {\isachardollar}{\kern0pt}\ s{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymlambda}d{\isachardot}{\kern0pt}\ d\ {\isasymin}\ D\isactrlsub D{\isacharparenright}{\kern0pt}\ d{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}step\ d\ v\ {\isacharequal}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}inv\ d\ v{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}GS{\isacharunderscore}{\kern0pt}inv{\isacharbrackleft}{\kern0pt}OF\ assms{\isacharbrackright}{\kern0pt}\ is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}GS{\isacharunderscore}{\kern0pt}rec{\isacharbrackleft}{\kern0pt}OF\ assms{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ strict{\isacharunderscore}{\kern0pt}lower{\isacharunderscore}{\kern0pt}triangular{\isacharunderscore}{\kern0pt}mat{\isacharunderscore}{\kern0pt}mult{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}strict{\isacharunderscore}{\kern0pt}lower{\isacharunderscore}{\kern0pt}triangular{\isacharunderscore}{\kern0pt}mat\ M{\isachardoublequoteclose}\ {\isachardoublequoteopen}{\isasymAnd}i{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ j\ {\isasymLongrightarrow}\ v\ {\isachardollar}{\kern0pt}\ i\ {\isacharequal}{\kern0pt}\ v{\isacharprime}{\kern0pt}\ {\isachardollar}{\kern0pt}\ i{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}M\ {\isacharasterisk}{\kern0pt}v\ v{\isacharparenright}{\kern0pt}\ {\isachardollar}{\kern0pt}\ j\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}M\ {\isacharasterisk}{\kern0pt}v\ v{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isachardollar}{\kern0pt}\ j{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}M\ {\isacharasterisk}{\kern0pt}v\ v{\isacharparenright}{\kern0pt}\ {\isachardollar}{\kern0pt}\ j\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSum}i{\isasymin}UNIV{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}if\ j\ {\isasymle}\ i\ then\ {\isadigit{0}}\ else\ \ M\ {\isachardollar}{\kern0pt}\ j\ {\isachardollar}{\kern0pt}\ i\ {\isacharasterisk}{\kern0pt}\ v\ {\isachardollar}{\kern0pt}\ i{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assms\ \isacommand{unfolding}\isamarkupfalse%
-\ strict{\isacharunderscore}{\kern0pt}lower{\isacharunderscore}{\kern0pt}triangular{\isacharunderscore}{\kern0pt}mat{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ matrix{\isacharunderscore}{\kern0pt}vector{\isacharunderscore}{\kern0pt}mult{\isacharunderscore}{\kern0pt}def\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ sum{\isachardot}{\kern0pt}cong{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSum}i{\isasymin}UNIV{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}if\ j\ {\isasymle}\ i\ then\ {\isadigit{0}}\ else\ \ M\ {\isachardollar}{\kern0pt}\ j\ {\isachardollar}{\kern0pt}\ i\ {\isacharasterisk}{\kern0pt}\ v{\isacharprime}{\kern0pt}\ {\isachardollar}{\kern0pt}\ i{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assms\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ sum{\isachardot}{\kern0pt}cong{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}M\ {\isacharasterisk}{\kern0pt}v\ v{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isachardollar}{\kern0pt}\ j{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assms\ \isacommand{unfolding}\isamarkupfalse%
-\ strict{\isacharunderscore}{\kern0pt}lower{\isacharunderscore}{\kern0pt}triangular{\isacharunderscore}{\kern0pt}mat{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ matrix{\isacharunderscore}{\kern0pt}vector{\isacharunderscore}{\kern0pt}mult{\isacharunderscore}{\kern0pt}def\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ sum{\isachardot}{\kern0pt}cong{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{finally}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isacommand{{\isachardot}{\kern0pt}}\isamarkupfalse%
-\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ Q{\isacharunderscore}{\kern0pt}mat{\isacharunderscore}{\kern0pt}invertible{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}invertible\ {\isacharparenleft}{\kern0pt}Q{\isacharunderscore}{\kern0pt}mat\ d{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}meson\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}inverse{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}\ is{\isacharunderscore}{\kern0pt}splitting{\isacharunderscore}{\kern0pt}blin{\isacharunderscore}{\kern0pt}def{\isacharprime}{\kern0pt}\ splitting{\isacharunderscore}{\kern0pt}gauss{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ GS{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}GS{\isacharunderscore}{\kern0pt}inv{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}{\isasymAnd}s{\isachardot}{\kern0pt}\ s\ {\isasymle}\ k\ {\isasymLongrightarrow}\ is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max\ {\isacharparenleft}{\kern0pt}{\isasymlambda}d{\isachardot}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}step\ d\ v\ {\isachardollar}{\kern0pt}\ s{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymlambda}d{\isachardot}{\kern0pt}\ d\ {\isasymin}\ D\isactrlsub D{\isacharparenright}{\kern0pt}\ d{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}s\ {\isasymle}\ k{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}step\ d\ v\ {\isachardollar}{\kern0pt}\ s\ {\isacharequal}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}inv\ d\ v\ {\isachardollar}{\kern0pt}\ s{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isacharasterisk}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}GS{\isacharunderscore}{\kern0pt}rec\ v\ {\isachardollar}{\kern0pt}\ s\ {\isacharequal}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}step\ d\ v\ {\isachardollar}{\kern0pt}\ s{\isachardoublequoteclose}\ \isakeyword{if}\ {\isachardoublequoteopen}s\ {\isasymle}\ k{\isachardoublequoteclose}\ \isakeyword{for}\ s\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assms\ is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}GS{\isacharunderscore}{\kern0pt}rec{\isacharprime}{\kern0pt}\ that\ \isacommand{by}\isamarkupfalse%
-\ presburger\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}GS{\isacharunderscore}{\kern0pt}rec\ v\ {\isachardollar}{\kern0pt}\ s\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}r{\isacharunderscore}{\kern0pt}vec\ d\ {\isacharplus}{\kern0pt}\ R{\isacharunderscore}{\kern0pt}mat\ d\ {\isacharasterisk}{\kern0pt}v\ v\ {\isacharplus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ P{\isacharunderscore}{\kern0pt}lower\ d{\isacharparenright}{\kern0pt}\ {\isacharasterisk}{\kern0pt}v\ GS{\isacharunderscore}{\kern0pt}rec\ v{\isacharparenright}{\kern0pt}\ {\isachardollar}{\kern0pt}\ s{\isachardoublequoteclose}\ \isakeyword{if}\ {\isachardoublequoteopen}s\ {\isasymle}\ k{\isachardoublequoteclose}\ \isakeyword{for}\ s\isanewline
-\ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ R{\isacharunderscore}{\kern0pt}mat{\isacharunderscore}{\kern0pt}def\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}step{\isacharunderscore}{\kern0pt}def\ \isacommand{using}\isamarkupfalse%
-\ that\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ scaleR{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}vector{\isacharunderscore}{\kern0pt}assoc\ pth{\isacharunderscore}{\kern0pt}{\isadigit{6}}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}Q{\isacharunderscore}{\kern0pt}mat\ d\ {\isacharasterisk}{\kern0pt}v\ GS{\isacharunderscore}{\kern0pt}rec\ v{\isacharparenright}{\kern0pt}\ {\isachardollar}{\kern0pt}\ s\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}r{\isacharunderscore}{\kern0pt}vec\ d\ {\isacharplus}{\kern0pt}\ R{\isacharunderscore}{\kern0pt}mat\ d\ {\isacharasterisk}{\kern0pt}v\ v{\isacharparenright}{\kern0pt}\ {\isachardollar}{\kern0pt}\ s{\isachardoublequoteclose}\ \isakeyword{if}\ {\isachardoublequoteopen}s\ {\isasymle}\ k{\isachardoublequoteclose}\ \isakeyword{for}\ s\isanewline
-\ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ Q{\isacharunderscore}{\kern0pt}mat{\isacharunderscore}{\kern0pt}def\ \isacommand{using}\isamarkupfalse%
-\ that\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ matrix{\isacharunderscore}{\kern0pt}vector{\isacharunderscore}{\kern0pt}mult{\isacharunderscore}{\kern0pt}diff{\isacharunderscore}{\kern0pt}rdistrib{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}inv\ {\isacharparenleft}{\kern0pt}Q{\isacharunderscore}{\kern0pt}mat\ d{\isacharparenright}{\kern0pt}\ {\isacharasterisk}{\kern0pt}v\ {\isacharparenleft}{\kern0pt}Q{\isacharunderscore}{\kern0pt}mat\ d\ {\isacharasterisk}{\kern0pt}v\ GS{\isacharunderscore}{\kern0pt}rec\ v{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isachardollar}{\kern0pt}\ s\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}inv\ {\isacharparenleft}{\kern0pt}Q{\isacharunderscore}{\kern0pt}mat\ d{\isacharparenright}{\kern0pt}\ {\isacharasterisk}{\kern0pt}v\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}r{\isacharunderscore}{\kern0pt}vec\ d\ {\isacharplus}{\kern0pt}\ R{\isacharunderscore}{\kern0pt}mat\ d\ {\isacharasterisk}{\kern0pt}v\ v{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isachardollar}{\kern0pt}\ s{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assms\ lt{\isacharunderscore}{\kern0pt}Q{\isacharunderscore}{\kern0pt}inv\ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ intro{\isacharcolon}{\kern0pt}\ lower{\isacharunderscore}{\kern0pt}triangular{\isacharunderscore}{\kern0pt}mat{\isacharunderscore}{\kern0pt}mult{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}step\ d\ v\ {\isachardollar}{\kern0pt}\ s\ {\isacharequal}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}inv\ d\ v\ {\isachardollar}{\kern0pt}\ s{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ GS{\isacharunderscore}{\kern0pt}inv{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ matrix{\isacharunderscore}{\kern0pt}inv{\isacharunderscore}{\kern0pt}left{\isacharbrackleft}{\kern0pt}OF\ Q{\isacharunderscore}{\kern0pt}mat{\isacharunderscore}{\kern0pt}invertible{\isacharbrackright}{\kern0pt}\ assms\ \ {\isacharasterisk}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ matrix{\isacharunderscore}{\kern0pt}vector{\isacharunderscore}{\kern0pt}mul{\isacharunderscore}{\kern0pt}assoc{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}GS{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}splitting{\isacharprime}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}{\isasymAnd}s{\isachardot}{\kern0pt}\ s\ {\isasymle}\ k\ {\isasymLongrightarrow}\ is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max\ {\isacharparenleft}{\kern0pt}{\isasymlambda}d{\isachardot}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}step\ d\ v\ {\isachardollar}{\kern0pt}\ s{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymlambda}d{\isachardot}{\kern0pt}\ d\ {\isasymin}\ D\isactrlsub D{\isacharparenright}{\kern0pt}\ d{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}s\ {\isasymle}\ k{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max\ {\isacharparenleft}{\kern0pt}{\isasymlambda}d{\isachardot}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}inv\ d\ v\ {\isachardollar}{\kern0pt}\ s{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymlambda}d{\isachardot}{\kern0pt}\ d\ {\isasymin}\ D\isactrlsub D{\isacharparenright}{\kern0pt}\ d{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ assms\isanewline
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}induction\ k\ arbitrary{\isacharcolon}{\kern0pt}\ s\ rule{\isacharcolon}{\kern0pt}\ less{\isacharunderscore}{\kern0pt}induct{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{case}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}less\ x{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ d{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}d\ {\isasymin}\ D\isactrlsub D{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assms{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}\ is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}linorderD\ \isacommand{by}\isamarkupfalse%
-\ fast\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max\ {\isacharparenleft}{\kern0pt}{\isasymlambda}a{\isachardot}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}inv\ {\isacharparenleft}{\kern0pt}d{\isacharparenleft}{\kern0pt}s\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ a{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ v\ {\isachardollar}{\kern0pt}\ s{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymlambda}a{\isachardot}{\kern0pt}\ a\ {\isasymin}\ A\ s{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}d\ s{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{if}\ {\isachardoublequoteopen}s\ {\isasymle}\ x{\isachardoublequoteclose}\ \isakeyword{for}\ s\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max\ {\isacharparenleft}{\kern0pt}{\isasymlambda}a{\isachardot}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}step\ {\isacharparenleft}{\kern0pt}d{\isacharparenleft}{\kern0pt}s\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ a{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ v\ {\isachardollar}{\kern0pt}\ s{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymlambda}a{\isachardot}{\kern0pt}\ a\ {\isasymin}\ A\ s{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}d\ s{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ less{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isacharbrackleft}{\kern0pt}OF\ that{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ is{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det{\isacharunderscore}{\kern0pt}def\ is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}linorder\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ {\isacharasterisk}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max\ {\isacharparenleft}{\kern0pt}{\isasymlambda}a{\isachardot}{\kern0pt}\ r\ {\isacharparenleft}{\kern0pt}s{\isacharcomma}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isacharplus}{\kern0pt}\ l\ {\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}P{\isacharunderscore}{\kern0pt}lower\ {\isacharparenleft}{\kern0pt}d{\isacharparenleft}{\kern0pt}s\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ a{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharasterisk}{\kern0pt}v\ GS{\isacharunderscore}{\kern0pt}rec\ v{\isacharparenright}{\kern0pt}\ {\isachardollar}{\kern0pt}\ s\ {\isacharplus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}P{\isacharunderscore}{\kern0pt}upper\ {\isacharparenleft}{\kern0pt}d{\isacharparenleft}{\kern0pt}s\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ a{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharasterisk}{\kern0pt}v\ v{\isacharparenright}{\kern0pt}\ {\isachardollar}{\kern0pt}\ s{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymlambda}a{\isachardot}{\kern0pt}\ a\ {\isasymin}\ A\ s{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}d\ s{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}step{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max\ {\isacharparenleft}{\kern0pt}{\isasymlambda}a{\isachardot}{\kern0pt}\ r\ {\isacharparenleft}{\kern0pt}s{\isacharcomma}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isacharplus}{\kern0pt}\ l\ {\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}P{\isacharunderscore}{\kern0pt}lower\ {\isacharparenleft}{\kern0pt}d{\isacharparenleft}{\kern0pt}s\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ a{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharasterisk}{\kern0pt}v\ GS{\isacharunderscore}{\kern0pt}inv\ {\isacharparenleft}{\kern0pt}d{\isacharparenleft}{\kern0pt}s\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ a{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ v{\isacharparenright}{\kern0pt}\ {\isachardollar}{\kern0pt}\ s\ {\isacharplus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}P{\isacharunderscore}{\kern0pt}upper\ {\isacharparenleft}{\kern0pt}d{\isacharparenleft}{\kern0pt}s\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ a{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharasterisk}{\kern0pt}v\ v{\isacharparenright}{\kern0pt}\ {\isachardollar}{\kern0pt}\ s{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymlambda}a{\isachardot}{\kern0pt}\ a\ {\isasymin}\ A\ s{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}d\ s{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}P{\isacharunderscore}{\kern0pt}lower\ {\isacharparenleft}{\kern0pt}d{\isacharparenleft}{\kern0pt}s\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ a{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharasterisk}{\kern0pt}v\ GS{\isacharunderscore}{\kern0pt}rec\ v{\isacharparenright}{\kern0pt}\ {\isachardollar}{\kern0pt}\ s\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}P{\isacharunderscore}{\kern0pt}lower\ {\isacharparenleft}{\kern0pt}d{\isacharparenleft}{\kern0pt}s\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ a{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharasterisk}{\kern0pt}v\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}step\ d\ v{\isacharparenright}{\kern0pt}\ {\isachardollar}{\kern0pt}\ s{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{for}\ a\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}GS{\isacharunderscore}{\kern0pt}rec{\isacharprime}{\kern0pt}\ less{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}\ that\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ lower{\isacharunderscore}{\kern0pt}triangular{\isacharunderscore}{\kern0pt}mat{\isacharunderscore}{\kern0pt}mult{\isacharbrackleft}{\kern0pt}OF\ lt{\isacharunderscore}{\kern0pt}P{\isacharunderscore}{\kern0pt}lower{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}P{\isacharunderscore}{\kern0pt}lower\ {\isacharparenleft}{\kern0pt}d{\isacharparenleft}{\kern0pt}s\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ a{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharasterisk}{\kern0pt}v\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}step\ d\ v{\isacharparenright}{\kern0pt}\ {\isachardollar}{\kern0pt}\ s{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}P{\isacharunderscore}{\kern0pt}lower\ {\isacharparenleft}{\kern0pt}d{\isacharparenleft}{\kern0pt}s\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ a{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharasterisk}{\kern0pt}v\ GS{\isacharunderscore}{\kern0pt}inv\ d\ v{\isacharparenright}{\kern0pt}\ {\isachardollar}{\kern0pt}\ s{\isachardoublequoteclose}\ \isakeyword{for}\ a\ \ \ \ \ \ \isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ less{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}\ that\ GS{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}GS{\isacharunderscore}{\kern0pt}inv\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}fastforce\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ lower{\isacharunderscore}{\kern0pt}triangular{\isacharunderscore}{\kern0pt}mat{\isacharunderscore}{\kern0pt}mult{\isacharbrackleft}{\kern0pt}OF\ lt{\isacharunderscore}{\kern0pt}P{\isacharunderscore}{\kern0pt}lower{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}P{\isacharunderscore}{\kern0pt}lower\ {\isacharparenleft}{\kern0pt}d{\isacharparenleft}{\kern0pt}s\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ a{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharasterisk}{\kern0pt}v\ GS{\isacharunderscore}{\kern0pt}inv\ d\ v{\isacharparenright}{\kern0pt}\ {\isachardollar}{\kern0pt}\ s\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}P{\isacharunderscore}{\kern0pt}lower\ {\isacharparenleft}{\kern0pt}d{\isacharparenleft}{\kern0pt}s\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ a{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharasterisk}{\kern0pt}v\ GS{\isacharunderscore}{\kern0pt}inv\ {\isacharparenleft}{\kern0pt}d{\isacharparenleft}{\kern0pt}s\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ a{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ v{\isacharparenright}{\kern0pt}\ {\isachardollar}{\kern0pt}\ s{\isachardoublequoteclose}\ \isakeyword{for}\ a\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ GS{\isacharunderscore}{\kern0pt}indep{\isacharunderscore}{\kern0pt}high{\isacharunderscore}{\kern0pt}states{\isacharbrackleft}{\kern0pt}of\ {\isacharunderscore}{\kern0pt}\ d\ {\isachardoublequoteopen}d{\isacharparenleft}{\kern0pt}s\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ a{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}{\isacharbrackright}{\kern0pt}\ \isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}fastforce\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ strict{\isacharunderscore}{\kern0pt}lower{\isacharunderscore}{\kern0pt}triangular{\isacharunderscore}{\kern0pt}mat{\isacharunderscore}{\kern0pt}mult{\isacharbrackleft}{\kern0pt}OF\ slt{\isacharunderscore}{\kern0pt}P{\isacharunderscore}{\kern0pt}lower{\isacharbrackright}{\kern0pt}\ dest{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ leD{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isacharasterisk}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max\ {\isacharparenleft}{\kern0pt}{\isasymlambda}a{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}r{\isacharunderscore}{\kern0pt}vec\ {\isacharparenleft}{\kern0pt}d{\isacharparenleft}{\kern0pt}s\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ a{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharplus}{\kern0pt}\ l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}P{\isacharunderscore}{\kern0pt}lower\ {\isacharparenleft}{\kern0pt}d{\isacharparenleft}{\kern0pt}s\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ a{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharasterisk}{\kern0pt}v\ GS{\isacharunderscore}{\kern0pt}inv\ {\isacharparenleft}{\kern0pt}d{\isacharparenleft}{\kern0pt}s\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ a{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ v{\isacharparenright}{\kern0pt}\ {\isacharplus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}P{\isacharunderscore}{\kern0pt}upper\ {\isacharparenleft}{\kern0pt}d{\isacharparenleft}{\kern0pt}s\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ a{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharasterisk}{\kern0pt}v\ v{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isachardollar}{\kern0pt}\ s{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymlambda}a{\isachardot}{\kern0pt}\ a\ {\isasymin}\ A\ s{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}d\ s{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ {\isacharasterisk}{\kern0pt}{\isacharasterisk}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max\ {\isacharparenleft}{\kern0pt}{\isasymlambda}a{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}r{\isacharunderscore}{\kern0pt}vec\ {\isacharparenleft}{\kern0pt}d{\isacharparenleft}{\kern0pt}s\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ a{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharplus}{\kern0pt}\ R{\isacharunderscore}{\kern0pt}mat\ {\isacharparenleft}{\kern0pt}d{\isacharparenleft}{\kern0pt}s\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ a{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharasterisk}{\kern0pt}v\ v{\isacharparenright}{\kern0pt}\ {\isacharplus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ P{\isacharunderscore}{\kern0pt}lower\ {\isacharparenleft}{\kern0pt}d{\isacharparenleft}{\kern0pt}s\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ a{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharasterisk}{\kern0pt}v\ GS{\isacharunderscore}{\kern0pt}inv\ {\isacharparenleft}{\kern0pt}d{\isacharparenleft}{\kern0pt}s\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ a{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ v{\isacharparenright}{\kern0pt}\ {\isacharparenright}{\kern0pt}\ {\isachardollar}{\kern0pt}\ s{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymlambda}a{\isachardot}{\kern0pt}\ a\ {\isasymin}\ A\ s{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}d\ s{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ R{\isacharunderscore}{\kern0pt}mat{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ algebra{\isacharunderscore}{\kern0pt}simps\ \ scaleR{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}vector{\isacharunderscore}{\kern0pt}assoc{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{proof}\isamarkupfalse%
-{\isacharminus}{\kern0pt}\ \isanewline
-\ \ \ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}r{\isacharunderscore}{\kern0pt}vec\ d\ {\isacharplus}{\kern0pt}\ R{\isacharunderscore}{\kern0pt}mat\ d\ {\isacharasterisk}{\kern0pt}v\ v{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ Q{\isacharunderscore}{\kern0pt}mat\ d\ {\isacharasterisk}{\kern0pt}v\ {\isacharparenleft}{\kern0pt}GS{\isacharunderscore}{\kern0pt}inv\ d\ v{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{for}\ d\ v\isanewline
-\ \ \ \ \ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ GS{\isacharunderscore}{\kern0pt}inv{\isacharunderscore}{\kern0pt}def\ matrix{\isacharunderscore}{\kern0pt}vector{\isacharunderscore}{\kern0pt}mul{\isacharunderscore}{\kern0pt}assoc\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharcomma}{\kern0pt}\ lifting{\isacharparenright}{\kern0pt}\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}inverse{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}\ is{\isacharunderscore}{\kern0pt}splitting{\isacharunderscore}{\kern0pt}blin{\isacharunderscore}{\kern0pt}def{\isacharprime}{\kern0pt}\ matrix{\isacharunderscore}{\kern0pt}inv{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}\ matrix{\isacharunderscore}{\kern0pt}vector{\isacharunderscore}{\kern0pt}mul{\isacharunderscore}{\kern0pt}lid\ splitting{\isacharunderscore}{\kern0pt}gauss{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}r{\isacharunderscore}{\kern0pt}vec\ d\ {\isacharplus}{\kern0pt}\ R{\isacharunderscore}{\kern0pt}mat\ d\ {\isacharasterisk}{\kern0pt}v\ v{\isacharparenright}{\kern0pt}\ {\isacharplus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ P{\isacharunderscore}{\kern0pt}lower\ d{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharasterisk}{\kern0pt}v\ GS{\isacharunderscore}{\kern0pt}inv\ d\ v{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}inv\ d\ v{\isachardoublequoteclose}\ \isakeyword{for}\ d\isanewline
-\ \ \ \ \ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ Q{\isacharunderscore}{\kern0pt}mat{\isacharunderscore}{\kern0pt}def\ \isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ matrix{\isacharunderscore}{\kern0pt}vector{\isacharunderscore}{\kern0pt}mult{\isacharunderscore}{\kern0pt}diff{\isacharunderscore}{\kern0pt}rdistrib{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\ \isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isacharasterisk}{\kern0pt}{\isacharasterisk}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ presburger\isanewline
-\ \ \ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}case\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ less\ d\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}fastforce\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ is{\isacharunderscore}{\kern0pt}am{\isacharunderscore}{\kern0pt}GS{\isacharunderscore}{\kern0pt}inv{\isacharunderscore}{\kern0pt}extend{\isacharbrackleft}{\kern0pt}of\ x\ v\ d\ {\isachardoublequoteopen}d\ x{\isachardoublequoteclose}\ s{\isacharcomma}{\kern0pt}\ unfolded\ fun{\isacharunderscore}{\kern0pt}upd{\isacharunderscore}{\kern0pt}triv{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ is{\isacharunderscore}{\kern0pt}am{\isacharunderscore}{\kern0pt}GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}step{\isacharunderscore}{\kern0pt}indep{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}d\ s\ {\isacharequal}{\kern0pt}\ d{\isacharprime}{\kern0pt}\ s{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max\ {\isacharparenleft}{\kern0pt}{\isasymlambda}d{\isachardot}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}step\ d\ v\ {\isachardollar}{\kern0pt}\ s{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymlambda}d{\isachardot}{\kern0pt}\ d\ {\isasymin}\ D\isactrlsub D{\isacharparenright}{\kern0pt}\ d{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}GS{\isacharunderscore}{\kern0pt}rec\ v\ {\isachardollar}{\kern0pt}\ s\ {\isacharequal}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}step\ d{\isacharprime}{\kern0pt}\ v\ {\isachardollar}{\kern0pt}\ s{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}GS{\isacharunderscore}{\kern0pt}rec\ v\ {\isachardollar}{\kern0pt}\ s\ {\isacharequal}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}step\ d\ v\ {\isachardollar}{\kern0pt}\ s{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}GS{\isacharunderscore}{\kern0pt}rec{\isacharprime}{\kern0pt}\ assms{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}\ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}step\ d\ v\ {\isachardollar}{\kern0pt}\ s\ {\isacharequal}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}step\ d{\isacharprime}{\kern0pt}\ v\ {\isachardollar}{\kern0pt}\ s{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}step{\isacharunderscore}{\kern0pt}elem\ assms{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}\ \isacommand{by}\isamarkupfalse%
-\ fastforce\isanewline
-\ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ is{\isacharunderscore}{\kern0pt}am{\isacharunderscore}{\kern0pt}GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}step{\isacharunderscore}{\kern0pt}indep{\isacharprime}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}d\ s\ {\isacharequal}{\kern0pt}\ d{\isacharprime}{\kern0pt}\ s{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max\ {\isacharparenleft}{\kern0pt}{\isasymlambda}d{\isachardot}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}step\ d\ v\ {\isachardollar}{\kern0pt}\ s{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymlambda}d{\isachardot}{\kern0pt}\ d\ {\isasymin}\ D\isactrlsub D{\isacharparenright}{\kern0pt}\ d{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}GS{\isacharunderscore}{\kern0pt}rec\ v\ {\isachardollar}{\kern0pt}\ s\ {\isacharequal}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}step\ d{\isacharprime}{\kern0pt}\ v\ {\isachardollar}{\kern0pt}\ s{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}GS{\isacharunderscore}{\kern0pt}rec\ v\ {\isachardollar}{\kern0pt}\ s\ {\isacharequal}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}step\ d\ v\ {\isachardollar}{\kern0pt}\ s{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}GS{\isacharunderscore}{\kern0pt}rec{\isacharprime}{\kern0pt}\ assms{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}\ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}step\ d\ v\ {\isachardollar}{\kern0pt}\ s\ {\isacharequal}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}step\ d{\isacharprime}{\kern0pt}\ v\ {\isachardollar}{\kern0pt}\ s{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}step{\isacharunderscore}{\kern0pt}elem\ assms{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}\ \isacommand{by}\isamarkupfalse%
-\ fastforce\isanewline
-\ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}GS{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}splitting{\isacharprime}{\kern0pt}{\isacharprime}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}{\isasymAnd}s{\isachardot}{\kern0pt}\ s\ {\isasymle}\ k\ {\isasymLongrightarrow}\ is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max\ {\isacharparenleft}{\kern0pt}{\isasymlambda}d{\isachardot}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}inv\ d\ v\ {\isachardollar}{\kern0pt}\ s{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymlambda}d{\isachardot}{\kern0pt}\ d\ {\isasymin}\ D\isactrlsub D{\isacharparenright}{\kern0pt}\ d{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}s\ {\isasymle}\ k{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max\ {\isacharparenleft}{\kern0pt}{\isasymlambda}d{\isachardot}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}step\ d\ v\ {\isachardollar}{\kern0pt}\ s{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymlambda}d{\isachardot}{\kern0pt}\ d\ {\isasymin}\ D\isactrlsub D{\isacharparenright}{\kern0pt}\ d\ {\isasymand}\ GS{\isacharunderscore}{\kern0pt}inv\ d\ v\ {\isachardollar}{\kern0pt}\ s\ {\isacharequal}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}rec\ v\ {\isachardollar}{\kern0pt}\ s{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ assms\isanewline
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}induction\ k\ arbitrary{\isacharcolon}{\kern0pt}\ s\ rule{\isacharcolon}{\kern0pt}\ less{\isacharunderscore}{\kern0pt}induct{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{case}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}less\ x{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ d{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}d\ {\isasymin}\ D\isactrlsub D{\isachardoublequoteclose}\ \isacommand{using}\isamarkupfalse%
-\ assms\ \isacommand{unfolding}\isamarkupfalse%
-\ is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}linorder\ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max\ {\isacharparenleft}{\kern0pt}{\isasymlambda}a{\isachardot}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}step\ {\isacharparenleft}{\kern0pt}d{\isacharparenleft}{\kern0pt}s\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ a{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ v\ {\isachardollar}{\kern0pt}\ s{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymlambda}a{\isachardot}{\kern0pt}\ a\ {\isasymin}\ A\ s{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}d\ s{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{if}\ {\isachardoublequoteopen}s\ {\isasymle}\ x{\isachardoublequoteclose}\ \isakeyword{for}\ s\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max\ {\isacharparenleft}{\kern0pt}{\isasymlambda}a{\isachardot}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}inv\ {\isacharparenleft}{\kern0pt}d{\isacharparenleft}{\kern0pt}s\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ a{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ v\ {\isachardollar}{\kern0pt}\ s{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymlambda}a{\isachardot}{\kern0pt}\ a\ {\isasymin}\ A\ s{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}d\ s{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ less{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isacharbrackleft}{\kern0pt}OF\ that{\isacharbrackright}{\kern0pt}\ \isanewline
-\ \ \ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ is{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det{\isacharunderscore}{\kern0pt}def\ is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}linorder\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ {\isacharasterisk}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max\ {\isacharparenleft}{\kern0pt}{\isasymlambda}a{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}r{\isacharunderscore}{\kern0pt}vec\ {\isacharparenleft}{\kern0pt}d{\isacharparenleft}{\kern0pt}s\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ a{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharplus}{\kern0pt}\ l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isacharparenleft}{\kern0pt}P{\isacharunderscore}{\kern0pt}lower\ {\isacharparenleft}{\kern0pt}d{\isacharparenleft}{\kern0pt}s\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ a{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharasterisk}{\kern0pt}v\ {\isacharparenleft}{\kern0pt}GS{\isacharunderscore}{\kern0pt}inv\ {\isacharparenleft}{\kern0pt}d{\isacharparenleft}{\kern0pt}s\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ a{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ v{\isacharparenright}{\kern0pt}\ {\isacharplus}{\kern0pt}\ P{\isacharunderscore}{\kern0pt}upper\ {\isacharparenleft}{\kern0pt}d{\isacharparenleft}{\kern0pt}s\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ a{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharasterisk}{\kern0pt}v\ v{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isachardollar}{\kern0pt}\ s{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymlambda}a{\isachardot}{\kern0pt}\ a\ {\isasymin}\ A\ s{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}d\ s{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}subst\ {\isacharparenleft}{\kern0pt}asm{\isacharparenright}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}inv{\isacharunderscore}{\kern0pt}rec{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ add{\isachardot}{\kern0pt}commute{\isacharparenright}{\kern0pt}\isanewline
-\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ {\isacharasterisk}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max\ {\isacharparenleft}{\kern0pt}{\isasymlambda}a{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}r{\isacharunderscore}{\kern0pt}vec\ {\isacharparenleft}{\kern0pt}d{\isacharparenleft}{\kern0pt}s\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ a{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharplus}{\kern0pt}\ l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isacharparenleft}{\kern0pt}P{\isacharunderscore}{\kern0pt}lower\ {\isacharparenleft}{\kern0pt}d{\isacharparenleft}{\kern0pt}s\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ a{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharasterisk}{\kern0pt}v\ {\isacharparenleft}{\kern0pt}GS{\isacharunderscore}{\kern0pt}inv\ d\ v{\isacharparenright}{\kern0pt}\ {\isacharplus}{\kern0pt}\ P{\isacharunderscore}{\kern0pt}upper\ {\isacharparenleft}{\kern0pt}d{\isacharparenleft}{\kern0pt}s\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ a{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharasterisk}{\kern0pt}v\ v{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isachardollar}{\kern0pt}\ s{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymlambda}a{\isachardot}{\kern0pt}\ a\ {\isasymin}\ A\ s{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}d\ s{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}P{\isacharunderscore}{\kern0pt}lower\ {\isacharparenleft}{\kern0pt}d{\isacharparenleft}{\kern0pt}s\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ a{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharasterisk}{\kern0pt}v\ {\isacharparenleft}{\kern0pt}GS{\isacharunderscore}{\kern0pt}inv\ {\isacharparenleft}{\kern0pt}d{\isacharparenleft}{\kern0pt}s\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ a{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ v{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isachardollar}{\kern0pt}\ s\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}P{\isacharunderscore}{\kern0pt}lower\ {\isacharparenleft}{\kern0pt}d{\isacharparenleft}{\kern0pt}s\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ a{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharasterisk}{\kern0pt}v\ {\isacharparenleft}{\kern0pt}GS{\isacharunderscore}{\kern0pt}inv\ d\ v{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isachardollar}{\kern0pt}\ s{\isachardoublequoteclose}\ \isakeyword{for}\ a\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ GS{\isacharunderscore}{\kern0pt}indep{\isacharunderscore}{\kern0pt}high{\isacharunderscore}{\kern0pt}states{\isacharbrackleft}{\kern0pt}of\ {\isacharunderscore}{\kern0pt}\ {\isachardoublequoteopen}d{\isacharparenleft}{\kern0pt}s\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ a{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ d\ v{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}rule\ strict{\isacharunderscore}{\kern0pt}lower{\isacharunderscore}{\kern0pt}triangular{\isacharunderscore}{\kern0pt}mat{\isacharunderscore}{\kern0pt}mult{\isacharbrackleft}{\kern0pt}OF\ slt{\isacharunderscore}{\kern0pt}P{\isacharunderscore}{\kern0pt}lower{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}metis\ array{\isacharunderscore}{\kern0pt}rules{\isacharparenleft}{\kern0pt}{\isadigit{4}}{\isacharparenright}{\kern0pt}\ leD{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\ \isacommand{using}\isamarkupfalse%
-\ {\isacharasterisk}{\kern0pt}\ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max\ {\isacharparenleft}{\kern0pt}{\isasymlambda}a{\isachardot}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}step\ {\isacharparenleft}{\kern0pt}d{\isacharparenleft}{\kern0pt}s\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ a{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ v\ {\isachardollar}{\kern0pt}\ s{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymlambda}a{\isachardot}{\kern0pt}\ a\ {\isasymin}\ A\ s{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}d\ s{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}P{\isacharunderscore}{\kern0pt}lower\ {\isacharparenleft}{\kern0pt}d{\isacharparenleft}{\kern0pt}s\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ a{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharasterisk}{\kern0pt}v\ {\isacharparenleft}{\kern0pt}GS{\isacharunderscore}{\kern0pt}inv\ d\ v{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isachardollar}{\kern0pt}\ s\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}P{\isacharunderscore}{\kern0pt}lower\ {\isacharparenleft}{\kern0pt}d{\isacharparenleft}{\kern0pt}s\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ a{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharasterisk}{\kern0pt}v\ {\isacharparenleft}{\kern0pt}GS{\isacharunderscore}{\kern0pt}rec\ v{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isachardollar}{\kern0pt}\ s{\isachardoublequoteclose}\ \isakeyword{for}\ a\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ less{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}\ less{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}that\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}intro\ strict{\isacharunderscore}{\kern0pt}lower{\isacharunderscore}{\kern0pt}triangular{\isacharunderscore}{\kern0pt}mat{\isacharunderscore}{\kern0pt}mult{\isacharbrackleft}{\kern0pt}OF\ slt{\isacharunderscore}{\kern0pt}P{\isacharunderscore}{\kern0pt}lower{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\ force\isanewline
-\ \ \ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\ \isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isacharasterisk}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}step{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isacharasterisk}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymAnd}s{\isachardot}{\kern0pt}\ s\ {\isasymle}\ x\ {\isasymLongrightarrow}\ is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max\ {\isacharparenleft}{\kern0pt}{\isasymlambda}d{\isachardot}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}step\ d\ v\ {\isachardollar}{\kern0pt}\ s{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymlambda}d{\isachardot}{\kern0pt}\ d\ {\isasymin}\ D\isactrlsub D{\isacharparenright}{\kern0pt}\ d{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ d\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}intro\ is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}step{\isacharunderscore}{\kern0pt}act{\isacharprime}{\kern0pt}{\isacharbrackleft}{\kern0pt}of\ d\ d{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\ auto\isanewline
-\ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}GS{\isacharunderscore}{\kern0pt}inv\ d\ v\ {\isachardollar}{\kern0pt}\ s\ {\isacharequal}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}rec\ v\ {\isachardollar}{\kern0pt}\ s{\isachardoublequoteclose}\ \isakeyword{if}\ {\isachardoublequoteopen}s\ {\isasymle}\ x{\isachardoublequoteclose}\ \isakeyword{for}\ s\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}GS{\isacharunderscore}{\kern0pt}rec\ v\ {\isachardollar}{\kern0pt}\ s\ {\isacharequal}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}step\ d\ v\ {\isachardollar}{\kern0pt}\ s{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isacharasterisk}{\kern0pt}{\isacharbrackleft}{\kern0pt}OF\ that{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}GS{\isacharunderscore}{\kern0pt}rec{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isacharasterisk}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}GS{\isacharunderscore}{\kern0pt}inv\ that\ \isacommand{by}\isamarkupfalse%
-\ presburger\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}case\ \isacommand{using}\isamarkupfalse%
-\ less\ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}GS{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}splitting{\isacharprime}{\kern0pt}{\isacharprime}{\kern0pt}{\isacharprime}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}{\isasymAnd}s{\isachardot}{\kern0pt}\ s\ {\isasymle}\ k\ {\isasymLongrightarrow}\ is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max\ {\isacharparenleft}{\kern0pt}{\isasymlambda}d{\isachardot}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}inv\ d\ v\ {\isachardollar}{\kern0pt}\ s{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymlambda}d{\isachardot}{\kern0pt}\ d\ {\isasymin}\ D\isactrlsub D{\isacharparenright}{\kern0pt}\ d{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}s\ {\isasymle}\ k{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max\ {\isacharparenleft}{\kern0pt}{\isasymlambda}d{\isachardot}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}step\ d\ v\ {\isachardollar}{\kern0pt}\ s{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymlambda}d{\isachardot}{\kern0pt}\ d\ {\isasymin}\ D\isactrlsub D{\isacharparenright}{\kern0pt}\ d{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ assms\ \ is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}GS{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}splitting{\isacharprime}{\kern0pt}{\isacharprime}{\kern0pt}\ \isacommand{by}\isamarkupfalse%
-\ blast%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}GS{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}splitting{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}{\isasymAnd}s{\isachardot}{\kern0pt}\ is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max\ {\isacharparenleft}{\kern0pt}{\isasymlambda}d{\isachardot}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}step\ d\ v\ {\isachardollar}{\kern0pt}\ s{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymlambda}d{\isachardot}{\kern0pt}\ d\ {\isasymin}\ D\isactrlsub D{\isacharparenright}{\kern0pt}\ d{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max\ {\isacharparenleft}{\kern0pt}{\isasymlambda}d{\isachardot}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}inv\ d\ v\ {\isachardollar}{\kern0pt}\ k{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymlambda}d{\isachardot}{\kern0pt}\ d\ {\isasymin}\ D\isactrlsub D{\isacharparenright}{\kern0pt}\ d{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ assms\ is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}GS{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}splitting{\isacharprime}{\kern0pt}{\isacharbrackleft}{\kern0pt}of\ {\isachardoublequoteopen}Max\ UNIV{\isachardoublequoteclose}{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}linorder{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}gs{\isacharunderscore}{\kern0pt}iff{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}d\ {\isasymin}\ D\isactrlsub D{\isachardoublequoteclose}\ \isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}\ {\isacharparenleft}{\kern0pt}{\isasymforall}s\ {\isasymle}\ k{\isachardot}{\kern0pt}\ is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max\ {\isacharparenleft}{\kern0pt}{\isasymlambda}d{\isachardot}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}inv\ d\ v\ {\isachardollar}{\kern0pt}\ s{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymlambda}d{\isachardot}{\kern0pt}\ d\ {\isasymin}\ D\isactrlsub D{\isacharparenright}{\kern0pt}\ d{\isacharparenright}{\kern0pt}\ {\isasymlongleftrightarrow}\isanewline
-\ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}s\ {\isasymle}\ k{\isachardot}{\kern0pt}\ is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max\ {\isacharparenleft}{\kern0pt}{\isasymlambda}d{\isachardot}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}step\ d\ v\ {\isachardollar}{\kern0pt}\ s{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymlambda}d{\isachardot}{\kern0pt}\ d\ {\isasymin}\ D\isactrlsub D{\isacharparenright}{\kern0pt}\ d{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}GS{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}splitting{\isacharprime}{\kern0pt}\ is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}GS{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}splitting{\isacharprime}{\kern0pt}{\isacharprime}{\kern0pt}\ \ \isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ meson%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ GS{\isacharunderscore}{\kern0pt}opt{\isacharunderscore}{\kern0pt}indep{\isacharunderscore}{\kern0pt}high{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isasymAnd}s{\isacharprime}{\kern0pt}{\isachardot}{\kern0pt}\ s{\isacharprime}{\kern0pt}\ {\isacharless}{\kern0pt}\ s\ {\isasymLongrightarrow}\ is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max\ {\isacharparenleft}{\kern0pt}{\isasymlambda}d{\isachardot}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}step\ d\ v\ {\isachardollar}{\kern0pt}\ s{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}\ is{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\ d{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ {\isachardoublequoteopen}s{\isacharprime}{\kern0pt}\ {\isacharless}{\kern0pt}\ s{\isachardoublequoteclose}\ {\isachardoublequoteopen}a\ {\isasymin}\ A\ s{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max\ {\isacharparenleft}{\kern0pt}{\isasymlambda}d{\isachardot}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}step\ d\ v\ {\isachardollar}{\kern0pt}\ s{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}\ is{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\ {\isacharparenleft}{\kern0pt}d{\isacharparenleft}{\kern0pt}s\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ a{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}rule\ is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}linorderI{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\ y\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\ {\isachardoublequoteopen}is{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\ y{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}step\ y\ v\ {\isachardollar}{\kern0pt}\ s{\isacharprime}{\kern0pt}\ {\isasymle}\ r\ {\isacharparenleft}{\kern0pt}s{\isacharprime}{\kern0pt}{\isacharcomma}{\kern0pt}\ d\ s{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharplus}{\kern0pt}\ l\ {\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}P{\isacharunderscore}{\kern0pt}lower\ d\ {\isacharasterisk}{\kern0pt}v\ GS{\isacharunderscore}{\kern0pt}rec\ v{\isacharparenright}{\kern0pt}\ {\isachardollar}{\kern0pt}\ s{\isacharprime}{\kern0pt}\ {\isacharplus}{\kern0pt}\ l\ {\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}P{\isacharunderscore}{\kern0pt}upper\ d\ {\isacharasterisk}{\kern0pt}v\ v{\isacharparenright}{\kern0pt}\ {\isachardollar}{\kern0pt}\ s{\isacharprime}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}linorderD{\isacharbrackleft}{\kern0pt}OF\ assms{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}step{\isacharunderscore}{\kern0pt}def\ algebra{\isacharunderscore}{\kern0pt}simps\ assms{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isacharequal}{\kern0pt}\ r\ {\isacharparenleft}{\kern0pt}s{\isacharprime}{\kern0pt}{\isacharcomma}{\kern0pt}\ {\isacharparenleft}{\kern0pt}d{\isacharparenleft}{\kern0pt}s\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ a{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ s{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharplus}{\kern0pt}\ l\ {\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}P{\isacharunderscore}{\kern0pt}lower\ {\isacharparenleft}{\kern0pt}d{\isacharparenleft}{\kern0pt}s\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ a{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharasterisk}{\kern0pt}v\ GS{\isacharunderscore}{\kern0pt}rec\ v{\isacharparenright}{\kern0pt}\ {\isachardollar}{\kern0pt}\ s{\isacharprime}{\kern0pt}\ {\isacharplus}{\kern0pt}\ l\ {\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}P{\isacharunderscore}{\kern0pt}upper\ {\isacharparenleft}{\kern0pt}d{\isacharparenleft}{\kern0pt}s\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ a{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharasterisk}{\kern0pt}v\ v{\isacharparenright}{\kern0pt}\ {\isachardollar}{\kern0pt}\ s{\isacharprime}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}P{\isacharunderscore}{\kern0pt}lower\ d\ {\isacharasterisk}{\kern0pt}v\ GS{\isacharunderscore}{\kern0pt}rec\ v{\isacharparenright}{\kern0pt}\ {\isachardollar}{\kern0pt}\ s{\isacharprime}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}P{\isacharunderscore}{\kern0pt}lower\ {\isacharparenleft}{\kern0pt}d{\isacharparenleft}{\kern0pt}s\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ a{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharasterisk}{\kern0pt}v\ GS{\isacharunderscore}{\kern0pt}rec\ v{\isacharparenright}{\kern0pt}\ {\isachardollar}{\kern0pt}\ s{\isacharprime}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assms\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}fastforce\ simp{\isacharcolon}{\kern0pt}\ matrix{\isacharunderscore}{\kern0pt}vector{\isacharunderscore}{\kern0pt}mult{\isacharunderscore}{\kern0pt}def\ P{\isacharunderscore}{\kern0pt}lower{\isacharunderscore}{\kern0pt}def\ P{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}elem\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ sum{\isachardot}{\kern0pt}cong{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}P{\isacharunderscore}{\kern0pt}upper\ d\ {\isacharasterisk}{\kern0pt}v\ v{\isacharparenright}{\kern0pt}\ {\isachardollar}{\kern0pt}\ s{\isacharprime}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}P{\isacharunderscore}{\kern0pt}upper\ {\isacharparenleft}{\kern0pt}d{\isacharparenleft}{\kern0pt}s\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ a{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharasterisk}{\kern0pt}v\ v{\isacharparenright}{\kern0pt}\ {\isachardollar}{\kern0pt}\ s{\isacharprime}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assms\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}fastforce\ simp{\isacharcolon}{\kern0pt}\ matrix{\isacharunderscore}{\kern0pt}vector{\isacharunderscore}{\kern0pt}mult{\isacharunderscore}{\kern0pt}def\ P{\isacharunderscore}{\kern0pt}upper{\isacharunderscore}{\kern0pt}def\ P{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}elem\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ sum{\isachardot}{\kern0pt}cong{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assms{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}\ \isacommand{by}\isamarkupfalse%
-\ force\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isacharequal}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}step\ {\isacharparenleft}{\kern0pt}d{\isacharparenleft}{\kern0pt}s\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ a{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ v\ {\isachardollar}{\kern0pt}\ s{\isacharprime}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}step{\isacharunderscore}{\kern0pt}def\ algebra{\isacharunderscore}{\kern0pt}simps{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{finally}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}step\ y\ v\ {\isachardollar}{\kern0pt}\ s{\isacharprime}{\kern0pt}\ {\isasymle}\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}step\ {\isacharparenleft}{\kern0pt}d{\isacharparenleft}{\kern0pt}s\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ a{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ v\ {\isachardollar}{\kern0pt}\ s{\isacharprime}{\kern0pt}{\isachardoublequoteclose}\isacommand{{\isachardot}{\kern0pt}}\isamarkupfalse%
-\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}is{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\ {\isacharparenleft}{\kern0pt}d{\isacharparenleft}{\kern0pt}s\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ a{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}linorderD{\isacharbrackleft}{\kern0pt}OF\ assms{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}{\isacharbrackleft}{\kern0pt}OF\ assms{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isacharbrackright}{\kern0pt}{\isacharbrackright}{\kern0pt}\ assms{\isacharparenleft}{\kern0pt}{\isadigit{3}}{\isacharparenright}{\kern0pt}\ is{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det{\isacharunderscore}{\kern0pt}def\ \isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ fastforce\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ mult{\isacharunderscore}{\kern0pt}mat{\isacharunderscore}{\kern0pt}vec{\isacharunderscore}{\kern0pt}nth{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}X\ {\isacharasterisk}{\kern0pt}v\ x{\isacharparenright}{\kern0pt}\ {\isachardollar}{\kern0pt}\ i\ {\isacharequal}{\kern0pt}\ scalar{\isacharunderscore}{\kern0pt}product\ {\isacharparenleft}{\kern0pt}row\ i\ X{\isacharparenright}{\kern0pt}\ x{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ matrix{\isacharunderscore}{\kern0pt}vector{\isacharunderscore}{\kern0pt}mult{\isacharunderscore}{\kern0pt}def\ row{\isacharunderscore}{\kern0pt}def\ scalar{\isacharunderscore}{\kern0pt}product{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ ext{\isacharunderscore}{\kern0pt}GS{\isacharunderscore}{\kern0pt}opt{\isacharunderscore}{\kern0pt}le{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isasymAnd}s{\isacharprime}{\kern0pt}{\isachardot}{\kern0pt}\ s{\isacharprime}{\kern0pt}\ {\isacharless}{\kern0pt}\ s\ {\isasymLongrightarrow}\ is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max\ {\isacharparenleft}{\kern0pt}{\isasymlambda}d{\isachardot}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}step\ d\ v\ {\isachardollar}{\kern0pt}\ s{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymlambda}d{\isachardot}{\kern0pt}\ d\ {\isasymin}\ D\isactrlsub D{\isacharparenright}{\kern0pt}\ d{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isakeyword{and}\ {\isachardoublequoteopen}is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max\ {\isacharparenleft}{\kern0pt}{\isasymlambda}a{\isachardot}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}step\ {\isacharparenleft}{\kern0pt}d{\isacharparenleft}{\kern0pt}s\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ a{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ v\ {\isachardollar}{\kern0pt}\ s{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymlambda}a{\isachardot}{\kern0pt}\ a\ {\isasymin}\ A\ s{\isacharparenright}{\kern0pt}\ a{\isachardoublequoteclose}\ {\isachardoublequoteopen}s{\isacharprime}{\kern0pt}\ {\isasymle}\ s{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isakeyword{and}\ {\isachardoublequoteopen}d\ {\isasymin}\ D\isactrlsub D{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max\ {\isacharparenleft}{\kern0pt}{\isasymlambda}d{\isachardot}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}step\ d\ v\ {\isachardollar}{\kern0pt}\ s{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymlambda}d{\isachardot}{\kern0pt}\ d\ {\isasymin}\ D\isactrlsub D{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}d{\isacharparenleft}{\kern0pt}s\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ a{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ assms\ is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}step{\isacharunderscore}{\kern0pt}act\ is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}linorderD{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}cases\ {\isachardoublequoteopen}s\ {\isacharequal}{\kern0pt}\ s{\isacharprime}{\kern0pt}{\isachardoublequoteclose}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}auto\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}opt{\isacharunderscore}{\kern0pt}indep{\isacharunderscore}{\kern0pt}high{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ ex{\isacharunderscore}{\kern0pt}GS{\isacharunderscore}{\kern0pt}opt{\isacharunderscore}{\kern0pt}le{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}{\isasymexists}d{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymforall}s{\isacharprime}{\kern0pt}\ {\isasymle}\ s{\isachardot}{\kern0pt}\ is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max\ {\isacharparenleft}{\kern0pt}{\isasymlambda}d{\isachardot}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}step\ d\ v\ {\isachardollar}{\kern0pt}\ s{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymlambda}d{\isachardot}{\kern0pt}\ d\ {\isasymin}\ D\isactrlsub D{\isacharparenright}{\kern0pt}\ d{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}induction\ s\ rule{\isacharcolon}{\kern0pt}\ less{\isacharunderscore}{\kern0pt}induct{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{case}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}less\ x{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}case\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}cases\ {\isachardoublequoteopen}{\isasymexists}y{\isachardot}{\kern0pt}\ y\ {\isacharless}{\kern0pt}\ x{\isachardoublequoteclose}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{case}\isamarkupfalse%
-\ True\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}y{\isachardot}{\kern0pt}\ y\ {\isacharless}{\kern0pt}\ x{\isacharbraceright}{\kern0pt}\ {\isasymnoteq}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\ \isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymAnd}y{\isachardot}{\kern0pt}\ y\ {\isasymle}\ Max\ {\isacharbraceleft}{\kern0pt}y{\isachardot}{\kern0pt}\ y\ {\isacharless}{\kern0pt}\ x{\isacharbraceright}{\kern0pt}\ {\isasymlongleftrightarrow}\ y\ {\isacharless}{\kern0pt}\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ True\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ Max{\isacharunderscore}{\kern0pt}ge{\isacharunderscore}{\kern0pt}iff{\isacharbrackleft}{\kern0pt}OF\ finite{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{obtain}\isamarkupfalse%
-\ d\ \isakeyword{where}\ d{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max\ {\isacharparenleft}{\kern0pt}{\isasymlambda}d{\isachardot}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}step\ d\ v\ {\isachardollar}{\kern0pt}\ s{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymlambda}d{\isachardot}{\kern0pt}\ d\ {\isasymin}\ D\isactrlsub D{\isacharparenright}{\kern0pt}\ d{\isachardoublequoteclose}\ \isakeyword{if}\ {\isachardoublequoteopen}s{\isacharprime}{\kern0pt}{\isacharless}{\kern0pt}\ x{\isachardoublequoteclose}\ \isakeyword{for}\ s{\isacharprime}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ less{\isacharbrackleft}{\kern0pt}of\ {\isachardoublequoteopen}Max\ {\isacharbraceleft}{\kern0pt}y{\isachardot}{\kern0pt}\ y\ {\isacharless}{\kern0pt}\ x{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}{\isacharbrackright}{\kern0pt}\ {\isadigit{1}}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \isacommand{obtain}\isamarkupfalse%
-\ a\ \isakeyword{where}\ a{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max\ {\isacharparenleft}{\kern0pt}{\isasymlambda}a{\isachardot}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}step\ {\isacharparenleft}{\kern0pt}d{\isacharparenleft}{\kern0pt}x\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ a{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ v\ {\isachardollar}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymlambda}a{\isachardot}{\kern0pt}\ a\ {\isasymin}\ A\ x{\isacharparenright}{\kern0pt}\ a{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ finite{\isacharunderscore}{\kern0pt}is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharbrackleft}{\kern0pt}OF\ finite\ A{\isacharunderscore}{\kern0pt}ne{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ d{\isacharprime}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max\ {\isacharparenleft}{\kern0pt}{\isasymlambda}d{\isachardot}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}step\ d\ v\ {\isachardollar}{\kern0pt}\ s{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymlambda}d{\isachardot}{\kern0pt}\ d\ {\isasymin}\ D\isactrlsub D{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}d{\isacharparenleft}{\kern0pt}x\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ a{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{if}\ {\isachardoublequoteopen}s{\isacharprime}{\kern0pt}\ {\isacharless}{\kern0pt}\ x{\isachardoublequoteclose}\ \isakeyword{for}\ s{\isacharprime}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ d\ GS{\isacharunderscore}{\kern0pt}opt{\isacharunderscore}{\kern0pt}indep{\isacharunderscore}{\kern0pt}high\ that\ is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}linorderD{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}{\isacharbrackleft}{\kern0pt}OF\ a{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ d{\isacharprime}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max\ {\isacharparenleft}{\kern0pt}{\isasymlambda}d{\isachardot}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}step\ d\ v\ {\isachardollar}{\kern0pt}\ s{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymlambda}d{\isachardot}{\kern0pt}\ d\ {\isasymin}\ D\isactrlsub D{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}d{\isacharparenleft}{\kern0pt}x\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ a{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{if}\ {\isachardoublequoteopen}s{\isacharprime}{\kern0pt}\ {\isasymle}\ x{\isachardoublequoteclose}\ \isakeyword{for}\ s{\isacharprime}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ \ that\ a\ is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}linorderD{\isacharbrackleft}{\kern0pt}OF\ d{\isacharbrackright}{\kern0pt}\ True\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}fastforce\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ ext{\isacharunderscore}{\kern0pt}GS{\isacharunderscore}{\kern0pt}opt{\isacharunderscore}{\kern0pt}le{\isacharbrackleft}{\kern0pt}OF\ d{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{case}\isamarkupfalse%
-\ False\isanewline
-\ \ \ \ \isacommand{define}\isamarkupfalse%
-\ d\ \isakeyword{where}\ {\isachardoublequoteopen}d\ y\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}SOME\ a{\isachardot}{\kern0pt}\ a\ {\isasymin}\ A\ y{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{for}\ y\isanewline
-\ \ \ \ \isacommand{obtain}\isamarkupfalse%
-\ a\ \isakeyword{where}\ a{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max\ {\isacharparenleft}{\kern0pt}{\isasymlambda}a{\isachardot}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}step\ {\isacharparenleft}{\kern0pt}d{\isacharparenleft}{\kern0pt}x\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ a{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ v\ {\isachardollar}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymlambda}a{\isachardot}{\kern0pt}\ a\ {\isasymin}\ A\ x{\isacharparenright}{\kern0pt}\ a{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ finite{\isacharunderscore}{\kern0pt}is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharbrackleft}{\kern0pt}OF\ finite\ A{\isacharunderscore}{\kern0pt}ne{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}y\ {\isasymle}\ x\ {\isasymLongrightarrow}\ y\ {\isacharequal}{\kern0pt}\ x{\isachardoublequoteclose}\ \isakeyword{for}\ y\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ False\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}meson\ le{\isacharunderscore}{\kern0pt}neq{\isacharunderscore}{\kern0pt}trans{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max\ {\isacharparenleft}{\kern0pt}{\isasymlambda}d{\isachardot}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}step\ d\ v\ {\isachardollar}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymlambda}d{\isachardot}{\kern0pt}\ d\ {\isasymin}\ D\isactrlsub D{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}d{\isacharparenleft}{\kern0pt}x\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ a{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ False\ a\ SOME{\isacharunderscore}{\kern0pt}is{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\ \isacommand{unfolding}\isamarkupfalse%
-\ d{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}fastforce\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}step{\isacharunderscore}{\kern0pt}act{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{then}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isadigit{1}}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ ex{\isacharunderscore}{\kern0pt}GS{\isacharunderscore}{\kern0pt}opt{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}{\isasymexists}d{\isachardot}{\kern0pt}\ {\isasymforall}s{\isachardot}{\kern0pt}\ is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max\ {\isacharparenleft}{\kern0pt}{\isasymlambda}d{\isachardot}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}step\ d\ v\ {\isachardollar}{\kern0pt}\ s{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymlambda}d{\isachardot}{\kern0pt}\ d\ {\isasymin}\ D\isactrlsub D{\isacharparenright}{\kern0pt}\ d{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ ex{\isacharunderscore}{\kern0pt}GS{\isacharunderscore}{\kern0pt}opt{\isacharunderscore}{\kern0pt}le{\isacharbrackleft}{\kern0pt}of\ {\isachardoublequoteopen}Max\ UNIV{\isachardoublequoteclose}{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}GS{\isacharunderscore}{\kern0pt}inv{\isacharprime}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}GS{\isacharunderscore}{\kern0pt}rec\ v\ {\isachardollar}{\kern0pt}\ s\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSqunion}d{\isasymin}D\isactrlsub D{\isachardot}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}inv\ d\ v\ {\isachardollar}{\kern0pt}\ s{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{obtain}\isamarkupfalse%
-\ d\ \isakeyword{where}\ d{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isasymAnd}s{\isachardot}{\kern0pt}\ is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max\ {\isacharparenleft}{\kern0pt}{\isasymlambda}d{\isachardot}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}step\ d\ v\ {\isachardollar}{\kern0pt}\ s{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymlambda}d{\isachardot}{\kern0pt}\ d\ {\isasymin}\ D\isactrlsub D{\isacharparenright}{\kern0pt}\ d{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ ex{\isacharunderscore}{\kern0pt}GS{\isacharunderscore}{\kern0pt}opt\ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isasymSqunion}d{\isasymin}D\isactrlsub D{\isachardot}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}step\ d\ v\ {\isachardollar}{\kern0pt}\ s{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}step\ d\ v\ {\isachardollar}{\kern0pt}\ s{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ d\ is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}GS{\isacharunderscore}{\kern0pt}rec\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}vec\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isasymSqunion}d{\isasymin}D\isactrlsub D{\isachardot}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}inv\ d\ v\ {\isachardollar}{\kern0pt}\ s{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}inv\ d\ v\ {\isachardollar}{\kern0pt}\ s{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}GS{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}splitting{\isacharbrackleft}{\kern0pt}OF\ d{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}subst\ arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}SUP{\isacharbrackleft}{\kern0pt}symmetric{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\ auto\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}GS{\isacharunderscore}{\kern0pt}inv\ d\ \isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ presburger\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}fun{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}GS{\isacharunderscore}{\kern0pt}inv{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}fun\ v\ s\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSqunion}d{\isasymin}D\isactrlsub D{\isachardot}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}inv\ d\ {\isacharparenleft}{\kern0pt}vec{\isacharunderscore}{\kern0pt}lambda\ v{\isacharparenright}{\kern0pt}\ {\isachardollar}{\kern0pt}\ s{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}GS{\isacharunderscore}{\kern0pt}inv{\isacharprime}{\kern0pt}{\isacharbrackleft}{\kern0pt}of\ {\isachardoublequoteopen}vec{\isacharunderscore}{\kern0pt}lambda\ v{\isachardoublequoteclose}{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \isacommand{unfolding}\isamarkupfalse%
-\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ vec{\isacharunderscore}{\kern0pt}lambda{\isacharunderscore}{\kern0pt}inverse{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ invertible{\isacharunderscore}{\kern0pt}Q{\isacharunderscore}{\kern0pt}GS{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}invertible\isactrlsub L\ {\isacharparenleft}{\kern0pt}Q{\isacharunderscore}{\kern0pt}GS\ d{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{for}\ d\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ Q{\isacharunderscore}{\kern0pt}mat{\isacharunderscore}{\kern0pt}invertible\ invertible{\isacharunderscore}{\kern0pt}invertible\isactrlsub L{\isacharunderscore}{\kern0pt}I{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ ex{\isacharunderscore}{\kern0pt}opt{\isacharunderscore}{\kern0pt}blinfun{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymexists}d{\isachardot}{\kern0pt}\ {\isasymforall}s{\isachardot}{\kern0pt}\ is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max\ {\isacharparenleft}{\kern0pt}{\isasymlambda}d{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}inv\isactrlsub L\ {\isacharparenleft}{\kern0pt}Q{\isacharunderscore}{\kern0pt}GS\ d{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}r{\isacharunderscore}{\kern0pt}det\isactrlsub b\ d\ {\isacharplus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}R{\isacharunderscore}{\kern0pt}GS\ d{\isacharparenright}{\kern0pt}\ v{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ s{\isacharparenright}{\kern0pt}\ is{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\ d{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}GS{\isacharunderscore}{\kern0pt}inv\ d\ {\isacharparenleft}{\kern0pt}vec{\isacharunderscore}{\kern0pt}lambda\ v{\isacharparenright}{\kern0pt}\ {\isachardollar}{\kern0pt}\ s\ {\isacharequal}{\kern0pt}\ inv\isactrlsub L\ {\isacharparenleft}{\kern0pt}Q{\isacharunderscore}{\kern0pt}GS\ d{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}r{\isacharunderscore}{\kern0pt}det\isactrlsub b\ d\ {\isacharplus}{\kern0pt}\ R{\isacharunderscore}{\kern0pt}GS\ d\ v{\isacharparenright}{\kern0pt}\ s{\isachardoublequoteclose}\ \isakeyword{for}\ d\ s\isanewline
-\ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ GS{\isacharunderscore}{\kern0pt}inv{\isacharunderscore}{\kern0pt}def\ plus{\isacharunderscore}{\kern0pt}bfun{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ invertible{\isacharunderscore}{\kern0pt}Q{\isacharunderscore}{\kern0pt}GS\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}mult{\isacharprime}{\kern0pt}\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}inverse{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isacharbrackleft}{\kern0pt}symmetric{\isacharbrackright}{\kern0pt}\ apply{\isacharunderscore}{\kern0pt}bfun{\isacharunderscore}{\kern0pt}inverse{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{obtain}\isamarkupfalse%
-\ d\ \isakeyword{where}\ {\isachardoublequoteopen}is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max\ {\isacharparenleft}{\kern0pt}{\isasymlambda}d{\isachardot}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}inv\ d\ {\isacharparenleft}{\kern0pt}vec{\isacharunderscore}{\kern0pt}lambda\ v{\isacharparenright}{\kern0pt}\ {\isachardollar}{\kern0pt}\ s{\isacharparenright}{\kern0pt}\ is{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\ d{\isachardoublequoteclose}\ \isakeyword{for}\ s\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ ex{\isacharunderscore}{\kern0pt}GS{\isacharunderscore}{\kern0pt}opt{\isacharbrackleft}{\kern0pt}of\ {\isachardoublequoteopen}vec{\isacharunderscore}{\kern0pt}lambda\ v{\isachardoublequoteclose}{\isacharbrackright}{\kern0pt}\ is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}GS{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}splitting\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\ \isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ GS{\isacharunderscore}{\kern0pt}inv{\isacharunderscore}{\kern0pt}blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}inv\isactrlsub L\ {\isacharparenleft}{\kern0pt}Q{\isacharunderscore}{\kern0pt}GS\ d{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}r{\isacharunderscore}{\kern0pt}det\isactrlsub b\ d\ {\isacharplus}{\kern0pt}\ R{\isacharunderscore}{\kern0pt}GS\ d\ v{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ Bfun\ {\isacharparenleft}{\kern0pt}vec{\isacharunderscore}{\kern0pt}nth\ {\isacharparenleft}{\kern0pt}GS{\isacharunderscore}{\kern0pt}inv\ d\ {\isacharparenleft}{\kern0pt}vec{\isacharunderscore}{\kern0pt}lambda\ v{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ GS{\isacharunderscore}{\kern0pt}inv{\isacharunderscore}{\kern0pt}def\ plus{\isacharunderscore}{\kern0pt}bfun{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ invertible{\isacharunderscore}{\kern0pt}Q{\isacharunderscore}{\kern0pt}GS\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}inverse{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isacharbrackleft}{\kern0pt}symmetric{\isacharbrackright}{\kern0pt}\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}mult{\isacharprime}{\kern0pt}{\isacharprime}{\kern0pt}\ apply{\isacharunderscore}{\kern0pt}bfun{\isacharunderscore}{\kern0pt}inverse\ {\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ norm{\isacharunderscore}{\kern0pt}GS{\isacharunderscore}{\kern0pt}QR{\isacharunderscore}{\kern0pt}le{\isacharunderscore}{\kern0pt}disc{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}norm\ {\isacharparenleft}{\kern0pt}inv\isactrlsub L\ {\isacharparenleft}{\kern0pt}Q{\isacharunderscore}{\kern0pt}GS\ d{\isacharparenright}{\kern0pt}\ o\isactrlsub L\ R{\isacharunderscore}{\kern0pt}GS\ d{\isacharparenright}{\kern0pt}\ {\isasymle}\ l{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}norm\ {\isacharparenleft}{\kern0pt}inv\isactrlsub L\ {\isacharparenleft}{\kern0pt}Q{\isacharunderscore}{\kern0pt}GS\ d{\isacharparenright}{\kern0pt}\ o\isactrlsub L\ R{\isacharunderscore}{\kern0pt}GS\ d{\isacharparenright}{\kern0pt}\ {\isasymle}\ norm\ {\isacharparenleft}{\kern0pt}inv\isactrlsub L\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}{\isasymlambda}{\isacharunderscore}{\kern0pt}{\isachardot}{\kern0pt}\ id{\isacharunderscore}{\kern0pt}blinfun{\isacharparenright}{\kern0pt}\ d{\isacharparenright}{\kern0pt}\ o\isactrlsub L\ {\isacharparenleft}{\kern0pt}l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isasymP}\isactrlsub {\isadigit{1}}\ {\isacharparenleft}{\kern0pt}mk{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\ d{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isachardoublequoteclose}\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}rule\ norm{\isacharunderscore}{\kern0pt}splitting{\isacharunderscore}{\kern0pt}le{\isacharbrackleft}{\kern0pt}of\ {\isachardoublequoteopen}mk{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\ d{\isachardoublequoteclose}{\isacharbrackright}{\kern0pt}{\isacharcomma}{\kern0pt}\ goal{\isacharunderscore}{\kern0pt}cases{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{case}\isamarkupfalse%
-\ {\isadigit{1}}\isanewline
-\ \ \ \ \isacommand{then}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}case\ \isanewline
-\ \ \ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ is{\isacharunderscore}{\kern0pt}splitting{\isacharunderscore}{\kern0pt}blin{\isacharunderscore}{\kern0pt}def{\isacharprime}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ nonneg{\isacharunderscore}{\kern0pt}id{\isacharunderscore}{\kern0pt}blinfun\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}scaleR\ nonneg{\isacharunderscore}{\kern0pt}{\isasymP}\isactrlsub {\isadigit{1}}\ scaleR{\isacharunderscore}{\kern0pt}nonneg{\isacharunderscore}{\kern0pt}nonneg{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{case}\isamarkupfalse%
-\ {\isadigit{3}}\isanewline
-\ \ \ \ \isacommand{then}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}case\ \isanewline
-\ \ \ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ R{\isacharunderscore}{\kern0pt}mat{\isacharunderscore}{\kern0pt}def\ P{\isacharunderscore}{\kern0pt}upper{\isacharunderscore}{\kern0pt}def\ Finite{\isacharunderscore}{\kern0pt}Cartesian{\isacharunderscore}{\kern0pt}Product{\isachardot}{\kern0pt}less{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}vec{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ nonneg{\isacharunderscore}{\kern0pt}P{\isacharunderscore}{\kern0pt}dec\ \isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ P{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}def\ nonneg{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}nonneg\ blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix{\isacharunderscore}{\kern0pt}scaleR{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ splitting{\isacharunderscore}{\kern0pt}gauss{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isacharequal}{\kern0pt}\ norm\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}l\ {\isacharasterisk}{\kern0pt}\isactrlsub R\ {\isasymP}\isactrlsub {\isadigit{1}}\ {\isacharparenleft}{\kern0pt}mk{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\ d{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isasymle}\ l{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{finally}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isacommand{{\isachardot}{\kern0pt}}\isamarkupfalse%
-\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{sublocale}\isamarkupfalse%
-\ GS{\isacharcolon}{\kern0pt}\ MDP{\isacharunderscore}{\kern0pt}QR\ A\ K\ r\ l\ Q{\isacharunderscore}{\kern0pt}GS\ R{\isacharunderscore}{\kern0pt}GS\isanewline
-\ \ \isakeyword{rewrites}\ {\isachardoublequoteopen}GS{\isachardot}{\kern0pt}{\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}split\ {\isacharequal}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}fun\isactrlsub b{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isasymSqunion}d{\isasymin}D\isactrlsub D{\isachardot}{\kern0pt}\ norm\ {\isacharparenleft}{\kern0pt}inv\isactrlsub L\ {\isacharparenleft}{\kern0pt}Q{\isacharunderscore}{\kern0pt}GS\ d{\isacharparenright}{\kern0pt}\ o\isactrlsub L\ R{\isacharunderscore}{\kern0pt}GS\ d{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharless}{\kern0pt}\ {\isadigit{1}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ norm{\isacharunderscore}{\kern0pt}GS{\isacharunderscore}{\kern0pt}QR{\isacharunderscore}{\kern0pt}le{\isacharunderscore}{\kern0pt}disc\ ex{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}fastforce\ intro{\isacharcolon}{\kern0pt}\ le{\isacharunderscore}{\kern0pt}less{\isacharunderscore}{\kern0pt}trans{\isacharbrackleft}{\kern0pt}of\ {\isacharunderscore}{\kern0pt}\ l\ {\isadigit{1}}{\isacharbrackright}{\kern0pt}\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ cSUP{\isacharunderscore}{\kern0pt}least{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}MDP{\isacharunderscore}{\kern0pt}QR\ A\ K\ r\ l\ Q{\isacharunderscore}{\kern0pt}GS\ R{\isacharunderscore}{\kern0pt}GS{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ unfold{\isacharunderscore}{\kern0pt}locales\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ splitting{\isacharunderscore}{\kern0pt}gauss\ ex{\isacharunderscore}{\kern0pt}opt{\isacharunderscore}{\kern0pt}blinfun{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}MDP{\isacharunderscore}{\kern0pt}QR{\isachardot}{\kern0pt}{\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}split\ A\ r\ Q{\isacharunderscore}{\kern0pt}GS\ R{\isacharunderscore}{\kern0pt}GS\ {\isacharequal}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}fun\isactrlsub b{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}fastforce\ simp{\isacharcolon}{\kern0pt}\ MDP{\isacharunderscore}{\kern0pt}QR{\isachardot}{\kern0pt}{\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}split{\isachardot}{\kern0pt}rep{\isacharunderscore}{\kern0pt}eq\ MDP{\isacharunderscore}{\kern0pt}QR{\isachardot}{\kern0pt}{\isasymL}{\isacharunderscore}{\kern0pt}split{\isacharunderscore}{\kern0pt}def\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}fun\isactrlsub b{\isachardot}{\kern0pt}rep{\isacharunderscore}{\kern0pt}eq\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}fun{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}GS{\isacharunderscore}{\kern0pt}inv\ GS{\isacharunderscore}{\kern0pt}inv{\isacharunderscore}{\kern0pt}blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{abbreviation}\isamarkupfalse%
-\ {\isachardoublequoteopen}gs{\isacharunderscore}{\kern0pt}measure\ {\isasymequiv}\ {\isacharparenleft}{\kern0pt}{\isasymlambda}{\isacharparenleft}{\kern0pt}eps{\isacharcomma}{\kern0pt}\ v{\isacharparenright}{\kern0pt}{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ if\ v\ {\isacharequal}{\kern0pt}\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt\ {\isasymor}\ l\ {\isacharequal}{\kern0pt}\ {\isadigit{0}}\isanewline
-\ \ \ \ then\ {\isadigit{0}}\isanewline
-\ \ \ \ else\ nat\ {\isacharparenleft}{\kern0pt}ceiling\ {\isacharparenleft}{\kern0pt}log\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharslash}{\kern0pt}l{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}dist\ v\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt{\isacharparenright}{\kern0pt}\ {\isacharminus}{\kern0pt}\ log\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharslash}{\kern0pt}l{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}eps\ {\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharminus}{\kern0pt}l{\isacharparenright}{\kern0pt}\ {\isacharslash}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isadigit{8}}\ {\isacharasterisk}{\kern0pt}\ l{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ dist{\isacharunderscore}{\kern0pt}{\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}split{\isacharunderscore}{\kern0pt}lt{\isacharunderscore}{\kern0pt}dist{\isacharunderscore}{\kern0pt}opt{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}dist\ v\ {\isacharparenleft}{\kern0pt}GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}fun\isactrlsub b\ v{\isacharparenright}{\kern0pt}\ {\isasymle}\ {\isadigit{2}}\ {\isacharasterisk}{\kern0pt}\ dist\ v\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ le{\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}dist\ v\ {\isacharparenleft}{\kern0pt}GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}fun\isactrlsub b\ v{\isacharparenright}{\kern0pt}\ {\isasymle}\ dist\ v\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt\ {\isacharplus}{\kern0pt}\ dist\ {\isacharparenleft}{\kern0pt}GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}fun\isactrlsub b\ v{\isacharparenright}{\kern0pt}\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ dist{\isacharunderscore}{\kern0pt}triangle\ dist{\isacharunderscore}{\kern0pt}commute{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ le{\isadigit{2}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}dist\ {\isacharparenleft}{\kern0pt}GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}fun\isactrlsub b\ v{\isacharparenright}{\kern0pt}\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt\ {\isasymle}\ GS{\isachardot}{\kern0pt}QR{\isacharunderscore}{\kern0pt}disc\ {\isacharasterisk}{\kern0pt}\ dist\ v\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ GS{\isachardot}{\kern0pt}{\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}split{\isacharunderscore}{\kern0pt}contraction\ GS{\isachardot}{\kern0pt}{\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}split{\isacharunderscore}{\kern0pt}fix\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharcomma}{\kern0pt}\ lifting{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ mult{\isacharunderscore}{\kern0pt}right{\isacharunderscore}{\kern0pt}mono{\isacharbrackleft}{\kern0pt}of\ GS{\isachardot}{\kern0pt}QR{\isacharunderscore}{\kern0pt}disc\ {\isadigit{1}}{\isacharbrackright}{\kern0pt}\ GS{\isachardot}{\kern0pt}QR{\isacharunderscore}{\kern0pt}contraction\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}fastforce\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ order{\isachardot}{\kern0pt}trans{\isacharbrackleft}{\kern0pt}OF\ le{\isadigit{2}}{\isacharbrackright}{\kern0pt}\ order{\isachardot}{\kern0pt}trans{\isacharbrackleft}{\kern0pt}OF\ le{\isadigit{1}}{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ GS{\isacharunderscore}{\kern0pt}QR{\isacharunderscore}{\kern0pt}disc{\isacharunderscore}{\kern0pt}le{\isacharunderscore}{\kern0pt}disc{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}GS{\isachardot}{\kern0pt}QR{\isacharunderscore}{\kern0pt}disc\ {\isasymle}\ l{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ norm{\isacharunderscore}{\kern0pt}GS{\isacharunderscore}{\kern0pt}QR{\isacharunderscore}{\kern0pt}le{\isacharunderscore}{\kern0pt}disc\ ex{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}fastforce\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ cSUP{\isacharunderscore}{\kern0pt}least{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ gs{\isacharunderscore}{\kern0pt}rel{\isacharunderscore}{\kern0pt}dec{\isacharcolon}{\kern0pt}\ \isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}l\ {\isasymnoteq}\ {\isadigit{0}}{\isachardoublequoteclose}\ {\isachardoublequoteopen}GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}fun\isactrlsub b\ v\ {\isasymnoteq}\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}{\isasymlceil}log\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}\ {\isacharslash}{\kern0pt}\ l{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}dist\ {\isacharparenleft}{\kern0pt}GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}fun\isactrlsub b\ v{\isacharparenright}{\kern0pt}\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt{\isacharparenright}{\kern0pt}\ {\isacharminus}{\kern0pt}\ c{\isasymrceil}\ {\isacharless}{\kern0pt}\ {\isasymlceil}log\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}\ {\isacharslash}{\kern0pt}\ l{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}dist\ v\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt{\isacharparenright}{\kern0pt}\ {\isacharminus}{\kern0pt}\ c{\isasymrceil}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}log\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}\ {\isacharslash}{\kern0pt}\ l{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}dist\ {\isacharparenleft}{\kern0pt}GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}fun\isactrlsub b\ v{\isacharparenright}{\kern0pt}\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt{\isacharparenright}{\kern0pt}\ {\isacharminus}{\kern0pt}\ c\ {\isasymle}\ log\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}\ {\isacharslash}{\kern0pt}\ l{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}l\ {\isacharasterisk}{\kern0pt}\ dist\ v\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt{\isacharparenright}{\kern0pt}\ {\isacharminus}{\kern0pt}\ c{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ GS{\isachardot}{\kern0pt}{\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}split{\isacharunderscore}{\kern0pt}contraction{\isacharbrackleft}{\kern0pt}of\ {\isacharunderscore}{\kern0pt}\ {\isachardoublequoteopen}{\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt{\isachardoublequoteclose}{\isacharbrackright}{\kern0pt}\ GS{\isachardot}{\kern0pt}QR{\isacharunderscore}{\kern0pt}contraction\ norm{\isacharunderscore}{\kern0pt}GS{\isacharunderscore}{\kern0pt}QR{\isacharunderscore}{\kern0pt}le{\isacharunderscore}{\kern0pt}disc\ disc{\isacharunderscore}{\kern0pt}lt{\isacharunderscore}{\kern0pt}one\ GS{\isacharunderscore}{\kern0pt}QR{\isacharunderscore}{\kern0pt}disc{\isacharunderscore}{\kern0pt}le{\isacharunderscore}{\kern0pt}disc\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}fastforce\ simp{\isacharcolon}{\kern0pt}\ assms\ less{\isacharunderscore}{\kern0pt}le\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ log{\isacharunderscore}{\kern0pt}le\ order{\isachardot}{\kern0pt}trans{\isacharbrackleft}{\kern0pt}OF\ GS{\isachardot}{\kern0pt}{\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}split{\isacharunderscore}{\kern0pt}contraction{\isacharbrackleft}{\kern0pt}of\ v\ {\isachardoublequoteopen}{\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt{\isachardoublequoteclose}{\isacharcomma}{\kern0pt}\ simplified{\isacharbrackright}{\kern0pt}{\isacharbrackright}{\kern0pt}\ mult{\isacharunderscore}{\kern0pt}right{\isacharunderscore}{\kern0pt}mono{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isacharequal}{\kern0pt}\ log\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}\ {\isacharslash}{\kern0pt}\ l{\isacharparenright}{\kern0pt}\ l\ {\isacharplus}{\kern0pt}\ log\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharslash}{\kern0pt}l{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}dist\ v\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt{\isacharparenright}{\kern0pt}\ {\isacharminus}{\kern0pt}\ c{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assms\ disc{\isacharunderscore}{\kern0pt}lt{\isacharunderscore}{\kern0pt}one\ \isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ less{\isacharunderscore}{\kern0pt}le\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ log{\isacharunderscore}{\kern0pt}mult{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isacharequal}{\kern0pt}\ {\isacharminus}{\kern0pt}{\isacharparenleft}{\kern0pt}log\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}\ {\isacharslash}{\kern0pt}\ l{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharslash}{\kern0pt}l{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharplus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}log\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharslash}{\kern0pt}l{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}dist\ v\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharminus}{\kern0pt}\ c{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assms\ disc{\isacharunderscore}{\kern0pt}lt{\isacharunderscore}{\kern0pt}one\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}subst\ log{\isacharunderscore}{\kern0pt}inverse{\isacharbrackleft}{\kern0pt}symmetric{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ less{\isacharunderscore}{\kern0pt}le\ right{\isacharunderscore}{\kern0pt}inverse{\isacharunderscore}{\kern0pt}eq{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}log\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharslash}{\kern0pt}l{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}dist\ v\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharminus}{\kern0pt}\ {\isadigit{1}}\ {\isacharminus}{\kern0pt}\ c{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assms\ order{\isachardot}{\kern0pt}strict{\isacharunderscore}{\kern0pt}implies{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}eq{\isacharbrackleft}{\kern0pt}OF\ disc{\isacharunderscore}{\kern0pt}lt{\isacharunderscore}{\kern0pt}one{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ log{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}one\ neq{\isacharunderscore}{\kern0pt}le{\isacharunderscore}{\kern0pt}trans{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{finally}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}log\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}\ {\isacharslash}{\kern0pt}\ l{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}dist\ {\isacharparenleft}{\kern0pt}GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}fun\isactrlsub b\ v{\isacharparenright}{\kern0pt}\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt{\isacharparenright}{\kern0pt}\ {\isacharminus}{\kern0pt}\ c\ {\isasymle}\ log\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}\ {\isacharslash}{\kern0pt}\ l{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}dist\ v\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt{\isacharparenright}{\kern0pt}\ {\isacharminus}{\kern0pt}\ {\isadigit{1}}\ {\isacharminus}{\kern0pt}\ c{\isachardoublequoteclose}\ \isacommand{{\isachardot}{\kern0pt}}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ linarith\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{function}\isamarkupfalse%
-\ gs{\isacharunderscore}{\kern0pt}iteration\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}real\ {\isasymRightarrow}\ {\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}s\ {\isasymRightarrow}\isactrlsub b\ real{\isacharparenright}{\kern0pt}\ {\isasymRightarrow}\ {\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}s\ {\isasymRightarrow}\isactrlsub b\ real{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}gs{\isacharunderscore}{\kern0pt}iteration\ eps\ v\ {\isacharequal}{\kern0pt}\isanewline
-\ \ {\isacharparenleft}{\kern0pt}if\ {\isadigit{2}}\ {\isacharasterisk}{\kern0pt}\ l\ {\isacharasterisk}{\kern0pt}\ dist\ v\ {\isacharparenleft}{\kern0pt}GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}fun\isactrlsub b\ v{\isacharparenright}{\kern0pt}\ {\isacharless}{\kern0pt}\ eps\ {\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharminus}{\kern0pt}l{\isacharparenright}{\kern0pt}\ {\isasymor}\ eps\ {\isasymle}\ {\isadigit{0}}\ then\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}fun\isactrlsub b\ v\ else\ gs{\isacharunderscore}{\kern0pt}iteration\ eps\ {\isacharparenleft}{\kern0pt}GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}fun\isactrlsub b\ v{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isacommand{termination}\isamarkupfalse%
-\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}relation\ {\isachardoublequoteopen}Wellfounded{\isachardot}{\kern0pt}measure\ gs{\isacharunderscore}{\kern0pt}measure{\isachardoublequoteclose}{\isacharcomma}{\kern0pt}\ {\isacharparenleft}{\kern0pt}simp{\isacharsemicolon}{\kern0pt}\ fail{\isacharparenright}{\kern0pt}{\isacharcomma}{\kern0pt}\ cases\ {\isachardoublequoteopen}l\ {\isacharequal}{\kern0pt}\ {\isadigit{0}}{\isachardoublequoteclose}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{case}\isamarkupfalse%
-\ False\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\ eps\ v\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\ h{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymnot}\ {\isacharparenleft}{\kern0pt}{\isadigit{2}}\ {\isacharasterisk}{\kern0pt}\ l\ {\isacharasterisk}{\kern0pt}\ dist\ v\ {\isacharparenleft}{\kern0pt}GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}fun\isactrlsub b\ v{\isacharparenright}{\kern0pt}\ {\isacharless}{\kern0pt}\ eps\ {\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}\ {\isacharminus}{\kern0pt}\ l{\isacharparenright}{\kern0pt}\ {\isasymor}\ eps\ {\isasymle}\ {\isadigit{0}}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}eps{\isacharcomma}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}fun\isactrlsub b\ v{\isacharparenright}{\kern0pt}{\isacharcomma}{\kern0pt}\ eps{\isacharcomma}{\kern0pt}\ v{\isacharparenright}{\kern0pt}\ {\isasymin}\ Wellfounded{\isachardot}{\kern0pt}measure\ gs{\isacharunderscore}{\kern0pt}measure{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ gt{\isacharunderscore}{\kern0pt}zero{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}l\ {\isasymnoteq}\ {\isadigit{0}}{\isachardoublequoteclose}\ {\isachardoublequoteopen}eps\ {\isachargreater}{\kern0pt}\ {\isadigit{0}}{\isachardoublequoteclose}\ \isakeyword{and}\ dist{\isacharunderscore}{\kern0pt}ge{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}eps\ {\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}\ {\isacharminus}{\kern0pt}\ l{\isacharparenright}{\kern0pt}\ {\isasymle}\ dist\ v\ {\isacharparenleft}{\kern0pt}GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}fun\isactrlsub b\ v{\isacharparenright}{\kern0pt}\ {\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isadigit{2}}\ {\isacharasterisk}{\kern0pt}\ l{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ h\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ algebra{\isacharunderscore}{\kern0pt}simps{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ v{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}opt{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}v\ {\isasymnoteq}\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ h\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}log\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}\ {\isacharslash}{\kern0pt}\ l{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}eps\ {\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}\ {\isacharminus}{\kern0pt}\ l{\isacharparenright}{\kern0pt}\ {\isacharslash}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isadigit{8}}\ {\isacharasterisk}{\kern0pt}\ l{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharless}{\kern0pt}\ log\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}\ {\isacharslash}{\kern0pt}\ l{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}dist\ v\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}intro\ log{\isacharunderscore}{\kern0pt}less{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isadigit{1}}\ {\isacharless}{\kern0pt}\ {\isadigit{1}}\ {\isacharslash}{\kern0pt}\ l{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ mult{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}less{\isacharunderscore}{\kern0pt}div{\isacharunderscore}{\kern0pt}pos\ intro{\isacharcolon}{\kern0pt}\ neq{\isacharunderscore}{\kern0pt}le{\isacharunderscore}{\kern0pt}trans{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isadigit{0}}\ {\isacharless}{\kern0pt}\ eps\ {\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}\ {\isacharminus}{\kern0pt}\ l{\isacharparenright}{\kern0pt}\ {\isacharslash}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isadigit{8}}\ {\isacharasterisk}{\kern0pt}\ l{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ mult{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}less{\isacharunderscore}{\kern0pt}div{\isacharunderscore}{\kern0pt}pos\ intro{\isacharcolon}{\kern0pt}\ neq{\isacharunderscore}{\kern0pt}le{\isacharunderscore}{\kern0pt}trans{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}eps\ {\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}\ {\isacharminus}{\kern0pt}\ l{\isacharparenright}{\kern0pt}\ {\isacharslash}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isadigit{8}}\ {\isacharasterisk}{\kern0pt}\ l{\isacharparenright}{\kern0pt}\ {\isacharless}{\kern0pt}\ dist\ v\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt{\isachardoublequoteclose}\ \isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ dist{\isacharunderscore}{\kern0pt}pos{\isacharunderscore}{\kern0pt}lt{\isacharbrackleft}{\kern0pt}OF\ v{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}opt{\isacharbrackright}{\kern0pt}\ dist{\isacharunderscore}{\kern0pt}{\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}split{\isacharunderscore}{\kern0pt}lt{\isacharunderscore}{\kern0pt}dist{\isacharunderscore}{\kern0pt}opt{\isacharbrackleft}{\kern0pt}of\ v{\isacharbrackright}{\kern0pt}\ gt{\isacharunderscore}{\kern0pt}zero\ zero{\isacharunderscore}{\kern0pt}le{\isacharunderscore}{\kern0pt}disc\ \isanewline
-\ \ \ \ \ \ \ \ \ \ mult{\isacharunderscore}{\kern0pt}strict{\isacharunderscore}{\kern0pt}left{\isacharunderscore}{\kern0pt}mono{\isacharbrackleft}{\kern0pt}of\ {\isachardoublequoteopen}dist\ v\ {\isacharparenleft}{\kern0pt}GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}fun\isactrlsub b\ v{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isadigit{4}}\ {\isacharasterisk}{\kern0pt}\ dist\ v\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ l{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}intro\ mult{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}div{\isacharunderscore}{\kern0pt}pos{\isacharunderscore}{\kern0pt}less\ le{\isacharunderscore}{\kern0pt}less{\isacharunderscore}{\kern0pt}trans{\isacharbrackleft}{\kern0pt}OF\ dist{\isacharunderscore}{\kern0pt}ge{\isacharbrackright}{\kern0pt}{\isacharcomma}{\kern0pt}\ argo{\isacharplus}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ gs{\isacharunderscore}{\kern0pt}rel{\isacharunderscore}{\kern0pt}dec\ h\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\isacommand{qed}\isamarkupfalse%
-\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ THE{\isacharunderscore}{\kern0pt}fix{\isacharunderscore}{\kern0pt}GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}fun\isactrlsub b{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}THE\ v{\isachardot}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}fun\isactrlsub b\ v\ {\isacharequal}{\kern0pt}\ v{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ GS{\isachardot}{\kern0pt}{\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}lim{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}\ GS{\isachardot}{\kern0pt}{\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}split{\isacharunderscore}{\kern0pt}fix\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ blast{\isacharplus}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\begin{isamarkuptext}%
-The distance between an estimate for the value and the optimal value can be bounded with respect to 
-the distance between the estimate and the result of applying it to \isa{{\isasymL}\isactrlsub b}%
-\end{isamarkuptext}\isamarkuptrue%
-\isacommand{lemma}\isamarkupfalse%
-\ contraction{\isacharunderscore}{\kern0pt}{\isasymL}{\isacharunderscore}{\kern0pt}split{\isacharunderscore}{\kern0pt}dist{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isadigit{1}}\ {\isacharminus}{\kern0pt}\ l{\isacharparenright}{\kern0pt}\ {\isacharasterisk}{\kern0pt}\ dist\ v\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt\ {\isasymle}\ dist\ v\ {\isacharparenleft}{\kern0pt}GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}fun\isactrlsub b\ v{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ GS{\isacharunderscore}{\kern0pt}QR{\isacharunderscore}{\kern0pt}disc{\isacharunderscore}{\kern0pt}le{\isacharunderscore}{\kern0pt}disc\ \isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}fastforce\ \isanewline
-\ \ \ \ \ \ simp{\isacharcolon}{\kern0pt}\ THE{\isacharunderscore}{\kern0pt}fix{\isacharunderscore}{\kern0pt}GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}fun\isactrlsub b\ \isanewline
-\ \ \ \ \ \ intro{\isacharcolon}{\kern0pt}\ order{\isachardot}{\kern0pt}trans{\isacharbrackleft}{\kern0pt}OF\ {\isacharunderscore}{\kern0pt}\ contraction{\isacharunderscore}{\kern0pt}dist{\isacharcomma}{\kern0pt}\ of\ {\isacharunderscore}{\kern0pt}\ l{\isacharbrackright}{\kern0pt}\ order{\isachardot}{\kern0pt}trans{\isacharbrackleft}{\kern0pt}OF\ GS{\isachardot}{\kern0pt}{\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}split{\isacharunderscore}{\kern0pt}contraction{\isacharbrackright}{\kern0pt}\ mult{\isacharunderscore}{\kern0pt}right{\isacharunderscore}{\kern0pt}mono{\isacharparenright}{\kern0pt}{\isacharplus}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ dist{\isacharunderscore}{\kern0pt}{\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}split{\isacharunderscore}{\kern0pt}opt{\isacharunderscore}{\kern0pt}eps{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}eps\ {\isachargreater}{\kern0pt}\ {\isadigit{0}}{\isachardoublequoteclose}\ {\isachardoublequoteopen}{\isadigit{2}}\ {\isacharasterisk}{\kern0pt}\ l\ {\isacharasterisk}{\kern0pt}\ dist\ v\ {\isacharparenleft}{\kern0pt}GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}fun\isactrlsub b\ v{\isacharparenright}{\kern0pt}\ {\isacharless}{\kern0pt}\ eps\ {\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharminus}{\kern0pt}l{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}dist\ {\isacharparenleft}{\kern0pt}GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}fun\isactrlsub b\ v{\isacharparenright}{\kern0pt}\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt\ {\isacharless}{\kern0pt}\ eps\ {\isacharslash}{\kern0pt}\ {\isadigit{2}}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}dist\ v\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt\ {\isasymle}\ dist\ v\ {\isacharparenleft}{\kern0pt}GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}fun\isactrlsub b\ v{\isacharparenright}{\kern0pt}\ {\isacharslash}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}\ {\isacharminus}{\kern0pt}\ l{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ contraction{\isacharunderscore}{\kern0pt}{\isasymL}{\isacharunderscore}{\kern0pt}split{\isacharunderscore}{\kern0pt}dist\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ mult{\isachardot}{\kern0pt}commute\ pos{\isacharunderscore}{\kern0pt}le{\isacharunderscore}{\kern0pt}divide{\isacharunderscore}{\kern0pt}eq{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isadigit{2}}\ {\isacharasterisk}{\kern0pt}\ l\ {\isacharasterisk}{\kern0pt}\ dist\ v\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt\ {\isasymle}\ {\isadigit{2}}\ {\isacharasterisk}{\kern0pt}\ l\ {\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}dist\ v\ {\isacharparenleft}{\kern0pt}GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}fun\isactrlsub b\ v{\isacharparenright}{\kern0pt}\ {\isacharslash}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}\ {\isacharminus}{\kern0pt}\ l{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ contraction{\isacharunderscore}{\kern0pt}{\isasymL}{\isacharunderscore}{\kern0pt}dist\ assms\ mult{\isacharunderscore}{\kern0pt}le{\isacharunderscore}{\kern0pt}cancel{\isacharunderscore}{\kern0pt}left{\isacharunderscore}{\kern0pt}pos{\isacharbrackleft}{\kern0pt}of\ {\isachardoublequoteopen}{\isadigit{2}}\ {\isacharasterisk}{\kern0pt}\ l{\isachardoublequoteclose}{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}fastforce\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ mult{\isacharunderscore}{\kern0pt}left{\isacharunderscore}{\kern0pt}mono{\isacharbrackleft}{\kern0pt}of\ {\isacharunderscore}{\kern0pt}\ {\isacharunderscore}{\kern0pt}\ {\isachardoublequoteopen}{\isadigit{2}}\ {\isacharasterisk}{\kern0pt}\ l{\isachardoublequoteclose}{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isadigit{2}}\ {\isacharasterisk}{\kern0pt}\ l\ {\isacharasterisk}{\kern0pt}\ dist\ v\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt\ {\isacharless}{\kern0pt}\ eps{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ assms{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}\ pos{\isacharunderscore}{\kern0pt}divide{\isacharunderscore}{\kern0pt}less{\isacharunderscore}{\kern0pt}eq\ intro{\isacharcolon}{\kern0pt}\ order{\isachardot}{\kern0pt}strict{\isacharunderscore}{\kern0pt}trans{\isadigit{1}}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}dist\ v\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt\ {\isacharasterisk}{\kern0pt}\ l\ {\isacharless}{\kern0pt}\ eps\ {\isacharslash}{\kern0pt}\ {\isadigit{2}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ argo\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isacharasterisk}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}l\ {\isacharasterisk}{\kern0pt}\ dist\ v\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt\ {\isacharless}{\kern0pt}\ eps\ {\isacharslash}{\kern0pt}\ {\isadigit{2}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ algebra{\isacharunderscore}{\kern0pt}simps{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}dist\ {\isacharparenleft}{\kern0pt}GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}fun\isactrlsub b\ v{\isacharparenright}{\kern0pt}\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt\ {\isacharless}{\kern0pt}\ eps\ {\isacharslash}{\kern0pt}\ {\isadigit{2}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ GS{\isachardot}{\kern0pt}{\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}split{\isacharunderscore}{\kern0pt}contraction{\isacharbrackleft}{\kern0pt}of\ v\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt{\isacharbrackright}{\kern0pt}\ order{\isachardot}{\kern0pt}trans\ mult{\isacharunderscore}{\kern0pt}right{\isacharunderscore}{\kern0pt}mono{\isacharbrackleft}{\kern0pt}OF\ GS{\isacharunderscore}{\kern0pt}QR{\isacharunderscore}{\kern0pt}disc{\isacharunderscore}{\kern0pt}le{\isacharunderscore}{\kern0pt}disc\ zero{\isacharunderscore}{\kern0pt}le{\isacharunderscore}{\kern0pt}dist{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}fastforce\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ le{\isacharunderscore}{\kern0pt}less{\isacharunderscore}{\kern0pt}trans{\isacharbrackleft}{\kern0pt}OF\ {\isacharunderscore}{\kern0pt}\ {\isacharasterisk}{\kern0pt}{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isacommand{end}\isamarkupfalse%
-\isanewline
-\isanewline
-\isacommand{context}\isamarkupfalse%
-\ MDP{\isacharunderscore}{\kern0pt}ord\ \isanewline
-\isakeyword{begin}\ \ \ \ \ \ \ \isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ is{\isacharunderscore}{\kern0pt}am{\isacharunderscore}{\kern0pt}GS{\isacharunderscore}{\kern0pt}inv{\isacharunderscore}{\kern0pt}extend{\isacharprime}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isasymAnd}s{\isachardot}{\kern0pt}\ s\ {\isacharless}{\kern0pt}\ x\ {\isasymLongrightarrow}\ is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max\ {\isacharparenleft}{\kern0pt}{\isasymlambda}d{\isachardot}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}inv\ d\ v\ {\isachardollar}{\kern0pt}\ s{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymlambda}d{\isachardot}{\kern0pt}\ d\ {\isasymin}\ D\isactrlsub D{\isacharparenright}{\kern0pt}\ d{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max\ {\isacharparenleft}{\kern0pt}{\isasymlambda}d{\isachardot}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}step\ d\ v\ {\isachardollar}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymlambda}d{\isachardot}{\kern0pt}\ d\ {\isasymin}\ D\isactrlsub D{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}d{\isacharparenleft}{\kern0pt}x\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ a{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}s\ {\isasymle}\ x{\isachardoublequoteclose}\ {\isachardoublequoteopen}d\ {\isasymin}\ D\isactrlsub D{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max\ {\isacharparenleft}{\kern0pt}{\isasymlambda}d{\isachardot}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}inv\ d\ v\ {\isachardollar}{\kern0pt}\ s{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymlambda}d{\isachardot}{\kern0pt}\ d\ {\isasymin}\ D\isactrlsub D{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}d{\isacharparenleft}{\kern0pt}x\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ a{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ a{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}a\ {\isasymin}\ A\ x{\isachardoublequoteclose}\ \isacommand{using}\isamarkupfalse%
-\ assms{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}\ \isacommand{unfolding}\isamarkupfalse%
-\ is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}linorder\ is{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det{\isacharunderscore}{\kern0pt}def\ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ split{\isacharcolon}{\kern0pt}\ if{\isacharunderscore}{\kern0pt}splits{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isacharasterisk}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymexists}y{\isachardot}{\kern0pt}\ y\ {\isacharless}{\kern0pt}\ x\ {\isasymLongrightarrow}\ s{\isasymle}Max\ {\isacharbraceleft}{\kern0pt}y{\isachardot}{\kern0pt}\ y\ {\isacharless}{\kern0pt}\ x{\isacharbraceright}{\kern0pt}\ {\isasymlongleftrightarrow}\ s\ {\isacharless}{\kern0pt}\ x{\isachardoublequoteclose}\ \isakeyword{for}\ x\ s\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isacharprime}{\kern0pt}s\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ linorder{\isacharunderscore}{\kern0pt}class{\isachardot}{\kern0pt}Max{\isacharunderscore}{\kern0pt}ge{\isacharunderscore}{\kern0pt}iff{\isacharbrackleft}{\kern0pt}OF\ finite{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isasymAnd}s{\isachardot}{\kern0pt}\ s\ {\isacharless}{\kern0pt}\ x\ {\isasymLongrightarrow}\ is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max\ {\isacharparenleft}{\kern0pt}{\isasymlambda}d{\isachardot}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}step\ d\ v\ {\isachardollar}{\kern0pt}\ s{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymlambda}d{\isachardot}{\kern0pt}\ d\ {\isasymin}\ D\isactrlsub D{\isacharparenright}{\kern0pt}\ d{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}gs{\isacharunderscore}{\kern0pt}iff{\isacharbrackleft}{\kern0pt}OF\ assms{\isacharparenleft}{\kern0pt}{\isadigit{4}}{\isacharparenright}{\kern0pt}{\isacharcomma}{\kern0pt}\ of\ {\isachardoublequoteopen}Max\ {\isacharbraceleft}{\kern0pt}y{\isachardot}{\kern0pt}\ y\ {\isacharless}{\kern0pt}\ x{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}{\isacharbrackright}{\kern0pt}\ assms{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}cases\ {\isachardoublequoteopen}{\isasymexists}y{\isachardot}{\kern0pt}\ y\ {\isacharless}{\kern0pt}\ x{\isachardoublequoteclose}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ {\isacharasterisk}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isasymAnd}s{\isachardot}{\kern0pt}\ s\ {\isacharless}{\kern0pt}\ x\ {\isasymLongrightarrow}\ is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max\ {\isacharparenleft}{\kern0pt}{\isasymlambda}d{\isachardot}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}step\ d\ v\ {\isachardollar}{\kern0pt}\ s{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymlambda}d{\isachardot}{\kern0pt}\ d\ {\isasymin}\ D\isactrlsub D{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}d{\isacharparenleft}{\kern0pt}x\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ a{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ GS{\isacharunderscore}{\kern0pt}opt{\isacharunderscore}{\kern0pt}indep{\isacharunderscore}{\kern0pt}high\ a\ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isasymAnd}s{\isachardot}{\kern0pt}\ s\ {\isasymle}\ x\ {\isasymLongrightarrow}\ is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max\ {\isacharparenleft}{\kern0pt}{\isasymlambda}d{\isachardot}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}step\ d\ v\ {\isachardollar}{\kern0pt}\ s{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymlambda}d{\isachardot}{\kern0pt}\ d\ {\isasymin}\ D\isactrlsub D{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}d{\isacharparenleft}{\kern0pt}x\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ a{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assms{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}\ antisym{\isacharunderscore}{\kern0pt}conv{\isadigit{1}}\ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}gs{\isacharunderscore}{\kern0pt}iff{\isacharbrackleft}{\kern0pt}of\ {\isachardoublequoteopen}d{\isacharparenleft}{\kern0pt}x\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ a{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ s{\isacharbrackright}{\kern0pt}\ assms{\isacharparenleft}{\kern0pt}{\isadigit{4}}{\isacharparenright}{\kern0pt}\ assms\ a\ \isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}intro\ is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}GS{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}splitting{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}\ auto\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ {\isachardoublequoteopen}opt{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}gs{\isacharprime}{\kern0pt}\ d\ v\ s\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}LEAST\ a{\isachardot}{\kern0pt}\ is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max\ {\isacharparenleft}{\kern0pt}{\isasymlambda}a{\isachardot}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}step\ {\isacharparenleft}{\kern0pt}d{\isacharparenleft}{\kern0pt}s\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ a{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ v\ {\isachardollar}{\kern0pt}\ s{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymlambda}a{\isachardot}{\kern0pt}\ a\ {\isasymin}\ A\ s{\isacharparenright}{\kern0pt}\ a{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ {\isachardoublequoteopen}GS{\isacharunderscore}{\kern0pt}iter\ a\ v\ s\ {\isacharequal}{\kern0pt}\ r\ {\isacharparenleft}{\kern0pt}s{\isacharcomma}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isacharplus}{\kern0pt}\ l\ {\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSum}s{\isacharprime}{\kern0pt}\ {\isasymin}\ UNIV{\isachardot}{\kern0pt}\ pmf\ {\isacharparenleft}{\kern0pt}K{\isacharparenleft}{\kern0pt}s{\isacharcomma}{\kern0pt}a{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ s{\isacharprime}{\kern0pt}\ {\isacharasterisk}{\kern0pt}\ v\ {\isachardollar}{\kern0pt}\ s{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ {\isachardoublequoteopen}GS{\isacharunderscore}{\kern0pt}iter{\isacharunderscore}{\kern0pt}max\ v\ s\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSqunion}a\ {\isasymin}\ A\ s{\isachardot}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}iter\ a\ v\ s{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}iter{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}{\isasymAnd}s{\isachardot}{\kern0pt}\ s\ {\isacharless}{\kern0pt}\ k\ {\isasymLongrightarrow}\ v{\isacharprime}{\kern0pt}\ {\isachardollar}{\kern0pt}\ s\ {\isacharequal}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}rec\ v\ {\isachardollar}{\kern0pt}\ s{\isachardoublequoteclose}\ {\isachardoublequoteopen}{\isasymAnd}s{\isachardot}{\kern0pt}\ k\ {\isasymle}\ s\ {\isasymLongrightarrow}\ v{\isacharprime}{\kern0pt}\ {\isachardollar}{\kern0pt}\ s\ {\isacharequal}{\kern0pt}\ v\ {\isachardollar}{\kern0pt}\ s{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}step\ {\isacharparenleft}{\kern0pt}d{\isacharparenleft}{\kern0pt}k\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ a{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ v\ {\isachardollar}{\kern0pt}\ k\ {\isacharequal}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}iter\ a\ v{\isacharprime}{\kern0pt}\ k{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}P{\isacharunderscore}{\kern0pt}lower\ d\ {\isacharasterisk}{\kern0pt}v\ GS{\isacharunderscore}{\kern0pt}rec\ v{\isacharparenright}{\kern0pt}\ {\isachardollar}{\kern0pt}\ k\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}P{\isacharunderscore}{\kern0pt}lower\ d\ {\isacharasterisk}{\kern0pt}v\ v{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isachardollar}{\kern0pt}\ k{\isachardoublequoteclose}\ \isakeyword{for}\ d\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ slt{\isacharunderscore}{\kern0pt}P{\isacharunderscore}{\kern0pt}lower\ assms\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ strict{\isacharunderscore}{\kern0pt}lower{\isacharunderscore}{\kern0pt}triangular{\isacharunderscore}{\kern0pt}mat{\isacharunderscore}{\kern0pt}mult{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}P{\isacharunderscore}{\kern0pt}upper\ d\ {\isacharasterisk}{\kern0pt}v\ v{\isacharparenright}{\kern0pt}\ {\isachardollar}{\kern0pt}\ k\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}P{\isacharunderscore}{\kern0pt}upper\ d\ {\isacharasterisk}{\kern0pt}v\ v{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isachardollar}{\kern0pt}\ k{\isachardoublequoteclose}\ \isakeyword{for}\ d\isanewline
-\ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ P{\isacharunderscore}{\kern0pt}upper{\isacharunderscore}{\kern0pt}def\ \isacommand{using}\isamarkupfalse%
-\ assms\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ matrix{\isacharunderscore}{\kern0pt}vector{\isacharunderscore}{\kern0pt}mult{\isacharunderscore}{\kern0pt}def\ if{\isacharunderscore}{\kern0pt}distrib{\isacharbrackleft}{\kern0pt}of\ {\isachardoublequoteopen}{\isasymlambda}x{\isachardot}{\kern0pt}\ x\ {\isacharasterisk}{\kern0pt}\ {\isacharunderscore}{\kern0pt}\ {\isachardollar}{\kern0pt}\ {\isacharunderscore}{\kern0pt}{\isachardoublequoteclose}{\isacharbrackright}{\kern0pt}\ cong{\isacharcolon}{\kern0pt}\ if{\isacharunderscore}{\kern0pt}cong{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}P{\isacharunderscore}{\kern0pt}lower\ d\ {\isacharplus}{\kern0pt}\ P{\isacharunderscore}{\kern0pt}upper\ d\ {\isacharequal}{\kern0pt}\ P{\isacharunderscore}{\kern0pt}dec\ d{\isachardoublequoteclose}\ \isakeyword{for}\ d\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ P{\isacharunderscore}{\kern0pt}lower{\isacharunderscore}{\kern0pt}def\ P{\isacharunderscore}{\kern0pt}upper{\isacharunderscore}{\kern0pt}def\ Finite{\isacharunderscore}{\kern0pt}Cartesian{\isacharunderscore}{\kern0pt}Product{\isachardot}{\kern0pt}vec{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}iff{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ vector{\isacharunderscore}{\kern0pt}add{\isacharunderscore}{\kern0pt}component{\isacharbrackleft}{\kern0pt}symmetric{\isacharbrackright}{\kern0pt}\ matrix{\isacharunderscore}{\kern0pt}vector{\isacharunderscore}{\kern0pt}mult{\isacharunderscore}{\kern0pt}diff{\isacharunderscore}{\kern0pt}rdistrib{\isacharbrackleft}{\kern0pt}symmetric{\isacharbrackright}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}step{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ matrix{\isacharunderscore}{\kern0pt}vector{\isacharunderscore}{\kern0pt}mult{\isacharunderscore}{\kern0pt}def\ P{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}elem\ P{\isacharunderscore}{\kern0pt}lower{\isacharunderscore}{\kern0pt}def\ P{\isacharunderscore}{\kern0pt}upper{\isacharunderscore}{\kern0pt}def\ GS{\isacharunderscore}{\kern0pt}iter{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}fastforce\ simp{\isacharcolon}{\kern0pt}\ sum{\isachardot}{\kern0pt}distrib{\isacharbrackleft}{\kern0pt}symmetric{\isacharbrackright}{\kern0pt}\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ sum{\isachardot}{\kern0pt}cong{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}iter{\isacharunderscore}{\kern0pt}max{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}{\isasymAnd}s{\isachardot}{\kern0pt}\ s\ {\isacharless}{\kern0pt}\ k\ {\isasymLongrightarrow}\ v{\isacharprime}{\kern0pt}\ {\isachardollar}{\kern0pt}\ s\ {\isacharequal}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}rec\ v\ {\isachardollar}{\kern0pt}\ s{\isachardoublequoteclose}\ {\isachardoublequoteopen}{\isasymAnd}s{\isachardot}{\kern0pt}\ k\ {\isasymle}\ s\ {\isasymLongrightarrow}\ v{\isacharprime}{\kern0pt}\ {\isachardollar}{\kern0pt}\ s\ {\isacharequal}{\kern0pt}\ v\ {\isachardollar}{\kern0pt}\ s{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}GS{\isacharunderscore}{\kern0pt}rec\ v\ {\isachardollar}{\kern0pt}\ k\ {\isacharequal}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}iter{\isacharunderscore}{\kern0pt}max\ v{\isacharprime}{\kern0pt}\ k{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}iter{\isacharbrackleft}{\kern0pt}OF\ assms{\isacharbrackright}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}eq{\isacharprime}{\kern0pt}{\isacharbrackleft}{\kern0pt}of\ {\isacharunderscore}{\kern0pt}\ {\isacharunderscore}{\kern0pt}\ undefined{\isacharbrackright}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}iter{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}def\ \isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ {\isachardoublequoteopen}GS{\isacharunderscore}{\kern0pt}iter{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max\ v\ s\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}LEAST\ a{\isachardot}{\kern0pt}\ is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max\ {\isacharparenleft}{\kern0pt}{\isasymlambda}a{\isachardot}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}iter\ a\ v\ s{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymlambda}a{\isachardot}{\kern0pt}\ a\ {\isasymin}\ A\ s{\isacharparenright}{\kern0pt}\ a{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ {\isachardoublequoteopen}GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}am{\isacharunderscore}{\kern0pt}code\ v\ d\ s\ {\isacharequal}{\kern0pt}\ foldl\ {\isacharparenleft}{\kern0pt}{\isasymlambda}vd\ s{\isachardot}{\kern0pt}\ vd{\isacharparenleft}{\kern0pt}s\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}GS{\isacharunderscore}{\kern0pt}iter{\isacharunderscore}{\kern0pt}max\ {\isacharparenleft}{\kern0pt}{\isasymchi}\ s{\isachardot}{\kern0pt}\ fst\ {\isacharparenleft}{\kern0pt}vd\ s{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ s{\isacharcomma}{\kern0pt}\ \ GS{\isacharunderscore}{\kern0pt}iter{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max\ {\isacharparenleft}{\kern0pt}{\isasymchi}\ s{\isachardot}{\kern0pt}\ fst\ {\isacharparenleft}{\kern0pt}vd\ s{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ s{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymlambda}s{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}v\ {\isachardollar}{\kern0pt}\ s{\isacharcomma}{\kern0pt}\ d\ s{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}sorted{\isacharunderscore}{\kern0pt}list{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}set\ {\isacharbraceleft}{\kern0pt}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}s{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ s{\isachardoublequoteclose}\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ {\isachardoublequoteopen}GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}am{\isacharunderscore}{\kern0pt}code{\isacharprime}{\kern0pt}\ v\ d\ s\ {\isacharequal}{\kern0pt}\ foldl\ {\isacharparenleft}{\kern0pt}{\isasymlambda}vd\ s{\isachardot}{\kern0pt}\ vd{\isacharparenleft}{\kern0pt}s\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}GS{\isacharunderscore}{\kern0pt}iter{\isacharunderscore}{\kern0pt}max\ {\isacharparenleft}{\kern0pt}{\isasymchi}\ s{\isachardot}{\kern0pt}\ fst\ {\isacharparenleft}{\kern0pt}vd\ s{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ s{\isacharcomma}{\kern0pt}\ \ GS{\isacharunderscore}{\kern0pt}iter{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max\ {\isacharparenleft}{\kern0pt}{\isasymchi}\ s{\isachardot}{\kern0pt}\ fst\ {\isacharparenleft}{\kern0pt}vd\ s{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ s{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymlambda}s{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}v\ {\isachardollar}{\kern0pt}\ s{\isacharcomma}{\kern0pt}\ d\ s{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}sorted{\isacharunderscore}{\kern0pt}list{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}set\ UNIV{\isacharparenright}{\kern0pt}\ s{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}am{\isacharunderscore}{\kern0pt}code{\isacharprime}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}am{\isacharunderscore}{\kern0pt}code\ {\isacharequal}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}am{\isacharunderscore}{\kern0pt}code{\isacharprime}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isacharasterisk}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}sorted{\isacharunderscore}{\kern0pt}list{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}set\ UNIV\ {\isacharequal}{\kern0pt}\ sorted{\isacharunderscore}{\kern0pt}list{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}set{\isacharbraceleft}{\kern0pt}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}s{\isacharbraceright}{\kern0pt}\ {\isacharat}{\kern0pt}\ sorted{\isacharunderscore}{\kern0pt}list{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}set{\isacharbraceleft}{\kern0pt}s{\isacharless}{\kern0pt}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{for}\ s\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isacharprime}{\kern0pt}s\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ sorted{\isacharunderscore}{\kern0pt}list{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}set{\isacharunderscore}{\kern0pt}split{\isacharprime}{\kern0pt}{\isacharbrackleft}{\kern0pt}OF\ finite{\isacharcomma}{\kern0pt}\ of\ UNIV\ s{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ atMost{\isacharunderscore}{\kern0pt}def\ greaterThan{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}am{\isacharunderscore}{\kern0pt}code\ v\ d\ s\ {\isacharequal}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}am{\isacharunderscore}{\kern0pt}code{\isacharprime}{\kern0pt}\ v\ d\ s{\isachardoublequoteclose}\ \isakeyword{for}\ v\ d\ s\isanewline
-\ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}am{\isacharunderscore}{\kern0pt}code{\isacharunderscore}{\kern0pt}def\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}am{\isacharunderscore}{\kern0pt}code{\isacharprime}{\kern0pt}{\isacharunderscore}{\kern0pt}def\ {\isacharasterisk}{\kern0pt}{\isacharbrackleft}{\kern0pt}of\ s{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}fastforce\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ foldl{\isacharunderscore}{\kern0pt}upd{\isacharunderscore}{\kern0pt}notin{\isacharprime}{\kern0pt}{\isacharbrackleft}{\kern0pt}symmetric{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\ \isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ opt{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}gs{\isacharprime}{\kern0pt}{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}GS{\isacharunderscore}{\kern0pt}iter{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}{\isasymAnd}s{\isachardot}{\kern0pt}\ s\ {\isacharless}{\kern0pt}\ k\ {\isasymLongrightarrow}\ v{\isacharprime}{\kern0pt}\ {\isachardollar}{\kern0pt}\ s\ {\isacharequal}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}rec\ v\ {\isachardollar}{\kern0pt}\ s{\isachardoublequoteclose}\ {\isachardoublequoteopen}{\isasymAnd}s{\isachardot}{\kern0pt}\ k\ {\isasymle}\ s\ {\isasymLongrightarrow}\ v{\isacharprime}{\kern0pt}\ {\isachardollar}{\kern0pt}\ s\ {\isacharequal}{\kern0pt}\ v\ {\isachardollar}{\kern0pt}\ s{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}opt{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}gs{\isacharprime}{\kern0pt}\ d\ v\ k\ {\isacharequal}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}iter{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max\ v{\isacharprime}{\kern0pt}\ k{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ opt{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}gs{\isacharprime}{\kern0pt}{\isacharunderscore}{\kern0pt}def\ GS{\isacharunderscore}{\kern0pt}iter{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}subst\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}iter{\isacharbrackleft}{\kern0pt}OF\ assms{\isacharcomma}{\kern0pt}\ of\ k\ d{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ opt{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}gs{\isacharprime}{\kern0pt}{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}GS{\isacharunderscore}{\kern0pt}iter{\isacharprime}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ {\isachardoublequoteopen}opt{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}gs{\isacharprime}{\kern0pt}\ d\ v\ k\ {\isacharequal}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}iter{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max\ {\isacharparenleft}{\kern0pt}{\isasymchi}\ s{\isachardot}{\kern0pt}\ if\ s\ {\isacharless}{\kern0pt}\ k\ then\ GS{\isacharunderscore}{\kern0pt}rec\ v\ {\isachardollar}{\kern0pt}\ s\ else\ v\ {\isachardollar}{\kern0pt}\ s{\isacharparenright}{\kern0pt}\ k{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ opt{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}gs{\isacharprime}{\kern0pt}{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}GS{\isacharunderscore}{\kern0pt}iter\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ leD{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ opt{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}gs{\isacharprime}{\kern0pt}{\isacharunderscore}{\kern0pt}is{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}opt{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}gs{\isacharprime}{\kern0pt}\ d\ v\ {\isasymin}\ D\isactrlsub D{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ opt{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}gs{\isacharprime}{\kern0pt}{\isacharunderscore}{\kern0pt}def\ is{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \isacommand{using}\isamarkupfalse%
-\ finite{\isacharunderscore}{\kern0pt}is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharbrackleft}{\kern0pt}OF\ finite\ A{\isacharunderscore}{\kern0pt}ne{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ intro{\isacharcolon}{\kern0pt}\ LeastI{\isadigit{2}}{\isacharunderscore}{\kern0pt}ex{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ opt{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}gs{\isacharprime}{\kern0pt}{\isacharunderscore}{\kern0pt}is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max\ {\isacharparenleft}{\kern0pt}{\isasymlambda}d{\isachardot}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}inv\ d\ v\ {\isachardollar}{\kern0pt}\ s{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymlambda}d{\isachardot}{\kern0pt}\ d\ {\isasymin}\ D\isactrlsub D{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}opt{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}gs{\isacharprime}{\kern0pt}\ d\ v{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}induction\ arbitrary{\isacharcolon}{\kern0pt}\ d\ rule{\isacharcolon}{\kern0pt}\ less{\isacharunderscore}{\kern0pt}induct{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{case}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}less\ x{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}s\ {\isacharless}{\kern0pt}\ x\ {\isasymLongrightarrow}\ is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max\ {\isacharparenleft}{\kern0pt}{\isasymlambda}d{\isachardot}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}inv\ d\ v\ {\isachardollar}{\kern0pt}\ s{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymlambda}d{\isachardot}{\kern0pt}\ d\ {\isasymin}\ D\isactrlsub D{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}opt{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}gs{\isacharprime}{\kern0pt}\ d\ v{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{for}\ d\ s\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ less\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isacharasterisk}{\kern0pt}{\isacharcolon}{\kern0pt}{\isachardoublequoteopen}s\ {\isacharless}{\kern0pt}\ x\ {\isasymLongrightarrow}\ is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max\ {\isacharparenleft}{\kern0pt}{\isasymlambda}d{\isachardot}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}step\ d\ v\ {\isachardollar}{\kern0pt}\ s{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymlambda}d{\isachardot}{\kern0pt}\ d\ {\isasymin}\ D\isactrlsub D{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}opt{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}gs{\isacharprime}{\kern0pt}\ d\ v{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{for}\ d\ s\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}intro\ is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}GS{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}splitting{\isacharprime}{\kern0pt}{\isacharprime}{\kern0pt}{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}\ auto\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max\ {\isacharparenleft}{\kern0pt}{\isasymlambda}a{\isachardot}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}step\ {\isacharparenleft}{\kern0pt}d{\isacharparenleft}{\kern0pt}x\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ a{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ v\ {\isachardollar}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymlambda}a{\isachardot}{\kern0pt}\ a\ {\isasymin}\ A\ x{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}opt{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}gs{\isacharprime}{\kern0pt}\ d\ v\ x{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{for}\ d\isanewline
-\ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ opt{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}gs{\isacharprime}{\kern0pt}{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ finite{\isacharunderscore}{\kern0pt}is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharbrackleft}{\kern0pt}OF\ {\isacharunderscore}{\kern0pt}\ A{\isacharunderscore}{\kern0pt}ne{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ intro{\isacharcolon}{\kern0pt}\ LeastI{\isacharunderscore}{\kern0pt}ex{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max\ {\isacharparenleft}{\kern0pt}{\isasymlambda}d{\isachardot}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}step\ d\ v\ {\isachardollar}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymlambda}d{\isachardot}{\kern0pt}\ d\ {\isasymin}\ D\isactrlsub D{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}opt{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}gs{\isacharprime}{\kern0pt}\ d\ v{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{for}\ d\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ opt{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}gs{\isacharprime}{\kern0pt}{\isacharunderscore}{\kern0pt}is{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}intro\ is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}step{\isacharunderscore}{\kern0pt}act{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}\ auto\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}s\ {\isasymle}\ x\ {\isasymLongrightarrow}\ is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max\ {\isacharparenleft}{\kern0pt}{\isasymlambda}d{\isachardot}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}step\ d\ v\ {\isachardollar}{\kern0pt}\ s{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymlambda}d{\isachardot}{\kern0pt}\ d\ {\isasymin}\ D\isactrlsub D{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}opt{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}gs{\isacharprime}{\kern0pt}\ d\ v{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{for}\ d\ s\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isacharasterisk}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ order{\isachardot}{\kern0pt}order{\isacharunderscore}{\kern0pt}iff{\isacharunderscore}{\kern0pt}strict{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}s\ {\isasymle}\ x\ {\isasymLongrightarrow}\ is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max\ {\isacharparenleft}{\kern0pt}{\isasymlambda}d{\isachardot}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}inv\ d\ v\ {\isachardollar}{\kern0pt}\ s{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymlambda}d{\isachardot}{\kern0pt}\ d\ {\isasymin}\ D\isactrlsub D{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}opt{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}gs{\isacharprime}{\kern0pt}\ d\ v{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{for}\ d\ s\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}GS{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}splitting{\isacharprime}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}case\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ {\isachardoublequoteopen}GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}am{\isacharunderscore}{\kern0pt}code\ v\ d\ s\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}GS{\isacharunderscore}{\kern0pt}rec\ v\ {\isachardollar}{\kern0pt}\ s{\isacharcomma}{\kern0pt}\ opt{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}gs{\isacharprime}{\kern0pt}\ d\ v\ s{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}induction\ s\ arbitrary{\isacharcolon}{\kern0pt}\ d\ rule{\isacharcolon}{\kern0pt}\ less{\isacharunderscore}{\kern0pt}induct{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{case}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}less\ x{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}case\ \isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}cases\ {\isachardoublequoteopen}{\isasymexists}x{\isacharprime}{\kern0pt}{\isachardot}{\kern0pt}\ x{\isacharprime}{\kern0pt}\ {\isacharless}{\kern0pt}\ x{\isachardoublequoteclose}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{case}\isamarkupfalse%
-\ True\isanewline
-\ \ \ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}f\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isasymlambda}vd\ s{\isachardot}{\kern0pt}\ vd{\isacharparenleft}{\kern0pt}s\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}GS{\isacharunderscore}{\kern0pt}iter{\isacharunderscore}{\kern0pt}max\ {\isacharparenleft}{\kern0pt}{\isasymchi}\ s{\isachardot}{\kern0pt}\ fst\ {\isacharparenleft}{\kern0pt}vd\ s{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ s{\isacharcomma}{\kern0pt}\ \ GS{\isacharunderscore}{\kern0pt}iter{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max\ {\isacharparenleft}{\kern0pt}{\isasymchi}\ s{\isachardot}{\kern0pt}\ fst\ {\isacharparenleft}{\kern0pt}vd\ s{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ s{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{define}\isamarkupfalse%
-\ x{\isacharprime}{\kern0pt}\ \isakeyword{where}\ {\isachardoublequoteopen}x{\isacharprime}{\kern0pt}\ {\isacharequal}{\kern0pt}\ Max\ {\isacharbraceleft}{\kern0pt}x{\isacharprime}{\kern0pt}{\isachardot}{\kern0pt}\ x{\isacharprime}{\kern0pt}\ {\isacharless}{\kern0pt}\ x{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}old\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}foldl\ {\isacharquery}{\kern0pt}f\ {\isacharparenleft}{\kern0pt}{\isasymlambda}s{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}v\ {\isachardollar}{\kern0pt}\ s{\isacharcomma}{\kern0pt}\ d\ s{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}sorted{\isacharunderscore}{\kern0pt}list{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}set\ {\isacharbraceleft}{\kern0pt}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}x{\isacharprime}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}s\ {\isacharless}{\kern0pt}\ x\ {\isasymLongrightarrow}\ {\isacharparenleft}{\kern0pt}s\ {\isasymnotin}\ set\ {\isacharparenleft}{\kern0pt}sorted{\isacharunderscore}{\kern0pt}list{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}set\ {\isacharbraceleft}{\kern0pt}s{\isacharprime}{\kern0pt}\ {\isasymin}\ {\isacharbraceleft}{\kern0pt}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}x{\isacharprime}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ s\ {\isacharless}{\kern0pt}\ s{\isacharprime}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{for}\ s\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isacharprime}{\kern0pt}s\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}s\ {\isacharless}{\kern0pt}\ x\ {\isasymLongrightarrow}\ foldl\ {\isacharquery}{\kern0pt}f\ {\isacharparenleft}{\kern0pt}{\isasymlambda}s{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}v\ {\isachardollar}{\kern0pt}\ s{\isacharcomma}{\kern0pt}\ d\ s{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}sorted{\isacharunderscore}{\kern0pt}list{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}set\ {\isacharbraceleft}{\kern0pt}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}x{\isacharprime}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ s\ {\isacharequal}{\kern0pt}\ foldl\ {\isacharquery}{\kern0pt}f\ {\isacharparenleft}{\kern0pt}{\isasymlambda}s{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}v\ {\isachardollar}{\kern0pt}\ s{\isacharcomma}{\kern0pt}\ d\ s{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}sorted{\isacharunderscore}{\kern0pt}list{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}set\ {\isacharbraceleft}{\kern0pt}s{\isacharprime}{\kern0pt}\ {\isasymin}\ {\isacharbraceleft}{\kern0pt}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}x{\isacharprime}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ s{\isacharprime}{\kern0pt}\ {\isasymle}\ s{\isacharbraceright}{\kern0pt}\ {\isacharat}{\kern0pt}\ sorted{\isacharunderscore}{\kern0pt}list{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}set\ {\isacharbraceleft}{\kern0pt}s{\isacharprime}{\kern0pt}\ {\isasymin}\ {\isacharbraceleft}{\kern0pt}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}x{\isacharprime}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ s\ {\isacharless}{\kern0pt}\ s{\isacharprime}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ s{\isachardoublequoteclose}\ \isakeyword{for}\ s\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}subst\ sorted{\isacharunderscore}{\kern0pt}list{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}set{\isacharunderscore}{\kern0pt}split{\isacharprime}{\kern0pt}{\isacharbrackleft}{\kern0pt}symmetric{\isacharcomma}{\kern0pt}\ OF\ finite{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\ blast\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}s\ {\isacharless}{\kern0pt}\ x\ {\isasymLongrightarrow}\ foldl\ {\isacharquery}{\kern0pt}f\ {\isacharparenleft}{\kern0pt}{\isasymlambda}s{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}v\ {\isachardollar}{\kern0pt}\ s{\isacharcomma}{\kern0pt}\ d\ s{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}sorted{\isacharunderscore}{\kern0pt}list{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}set\ {\isacharbraceleft}{\kern0pt}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}x{\isacharprime}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ s\ {\isacharequal}{\kern0pt}\ foldl\ {\isacharquery}{\kern0pt}f\ {\isacharparenleft}{\kern0pt}{\isasymlambda}s{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}v\ {\isachardollar}{\kern0pt}\ s{\isacharcomma}{\kern0pt}\ d\ s{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}sorted{\isacharunderscore}{\kern0pt}list{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}set\ {\isacharbraceleft}{\kern0pt}s{\isacharprime}{\kern0pt}\ {\isasymin}\ {\isacharbraceleft}{\kern0pt}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}x{\isacharprime}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ s{\isacharprime}{\kern0pt}\ {\isasymle}\ s{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ s{\isachardoublequoteclose}\ \isakeyword{for}\ s\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ foldl{\isacharunderscore}{\kern0pt}upd{\isacharunderscore}{\kern0pt}notin{\isacharprime}{\kern0pt}{\isacharbrackleft}{\kern0pt}OF\ {\isadigit{1}}{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ fastforce\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ {\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}s\ {\isacharless}{\kern0pt}\ x\ {\isasymLongrightarrow}\ foldl\ {\isacharquery}{\kern0pt}f\ {\isacharparenleft}{\kern0pt}{\isasymlambda}s{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}v\ {\isachardollar}{\kern0pt}\ s{\isacharcomma}{\kern0pt}\ d\ s{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}sorted{\isacharunderscore}{\kern0pt}list{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}set\ {\isacharbraceleft}{\kern0pt}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}x{\isacharprime}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ s\ {\isacharequal}{\kern0pt}\ foldl\ {\isacharquery}{\kern0pt}f\ {\isacharparenleft}{\kern0pt}{\isasymlambda}s{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}v\ {\isachardollar}{\kern0pt}\ s{\isacharcomma}{\kern0pt}\ d\ s{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}sorted{\isacharunderscore}{\kern0pt}list{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}set\ {\isacharbraceleft}{\kern0pt}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}s{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ s{\isachardoublequoteclose}\ \isakeyword{for}\ s\isanewline
-\ \ \ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ x{\isacharprime}{\kern0pt}{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ True\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ atMost{\isacharunderscore}{\kern0pt}def\ Max{\isacharunderscore}{\kern0pt}ge{\isacharunderscore}{\kern0pt}iff{\isacharbrackleft}{\kern0pt}OF\ finite{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\ meson\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ fst{\isacharunderscore}{\kern0pt}IH{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}fst\ {\isacharparenleft}{\kern0pt}{\isacharquery}{\kern0pt}old\ s{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}rec\ v\ {\isachardollar}{\kern0pt}\ s{\isachardoublequoteclose}\ \isakeyword{if}\ {\isachardoublequoteopen}s\ {\isacharless}{\kern0pt}\ x{\isachardoublequoteclose}\ \isakeyword{for}\ s\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isadigit{1}}{\isacharbrackleft}{\kern0pt}OF\ that{\isacharbrackright}{\kern0pt}\ less{\isacharbrackleft}{\kern0pt}unfolded\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}am{\isacharunderscore}{\kern0pt}code{\isacharunderscore}{\kern0pt}def{\isacharbrackright}{\kern0pt}\ that\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ fst{\isacharunderscore}{\kern0pt}IH{\isacharprime}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}fst\ {\isacharparenleft}{\kern0pt}{\isacharquery}{\kern0pt}old\ s{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ v\ {\isachardollar}{\kern0pt}\ s{\isachardoublequoteclose}\ \isakeyword{if}\ {\isachardoublequoteopen}x\ {\isasymle}\ s{\isachardoublequoteclose}\ \isakeyword{for}\ s\ \isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ True\ that\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}subst\ foldl{\isacharunderscore}{\kern0pt}upd{\isacharunderscore}{\kern0pt}notin{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ x{\isacharprime}{\kern0pt}{\isacharunderscore}{\kern0pt}def\ \ Max{\isacharunderscore}{\kern0pt}ge{\isacharunderscore}{\kern0pt}iff{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ fst{\isacharunderscore}{\kern0pt}IH{\isacharprime}{\kern0pt}{\isacharprime}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}fst\ {\isacharparenleft}{\kern0pt}{\isacharquery}{\kern0pt}old\ s{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}if\ s\ {\isacharless}{\kern0pt}\ x\ then\ GS{\isacharunderscore}{\kern0pt}rec\ v\ {\isachardollar}{\kern0pt}\ s\ else\ v\ {\isachardollar}{\kern0pt}\ s{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{for}\ s\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ fst{\isacharunderscore}{\kern0pt}IH\ fst{\isacharunderscore}{\kern0pt}IH{\isacharprime}{\kern0pt}\ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}foldl\ {\isacharquery}{\kern0pt}f\ {\isacharparenleft}{\kern0pt}{\isasymlambda}s{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}v\ {\isachardollar}{\kern0pt}\ s{\isacharcomma}{\kern0pt}\ d\ s{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}sorted{\isacharunderscore}{\kern0pt}list{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}set\ {\isacharbraceleft}{\kern0pt}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}x{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ foldl\ {\isacharquery}{\kern0pt}f\ {\isacharparenleft}{\kern0pt}{\isasymlambda}s{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}v\ {\isachardollar}{\kern0pt}\ s{\isacharcomma}{\kern0pt}\ d\ s{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}sorted{\isacharunderscore}{\kern0pt}list{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}set\ {\isacharbraceleft}{\kern0pt}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}x{\isacharprime}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharat}{\kern0pt}\ sorted{\isacharunderscore}{\kern0pt}list{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}set\ {\isacharbraceleft}{\kern0pt}x{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isacharasterisk}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}x{\isacharprime}{\kern0pt}{\isachardot}{\kern0pt}\ x{\isacharprime}{\kern0pt}\ {\isacharless}{\kern0pt}\ x{\isacharbraceright}{\kern0pt}\ {\isasymnoteq}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\ \isacommand{using}\isamarkupfalse%
-\ True\ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ {\isacharasterisk}{\kern0pt}{\isacharasterisk}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}x{\isacharprime}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}y\ {\isasymin}\ {\isacharbraceleft}{\kern0pt}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}x{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ y\ {\isacharless}{\kern0pt}\ x{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}x{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}y\ {\isasymin}\ {\isacharbraceleft}{\kern0pt}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}x{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ x\ {\isasymle}\ y{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ x{\isacharprime}{\kern0pt}{\isacharunderscore}{\kern0pt}def\ Max{\isacharunderscore}{\kern0pt}ge{\isacharunderscore}{\kern0pt}iff{\isacharbrackleft}{\kern0pt}OF\ finite\ {\isacharasterisk}{\kern0pt}{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ {\isacharasterisk}{\kern0pt}{\isacharasterisk}{\kern0pt}\ sorted{\isacharunderscore}{\kern0pt}list{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}set{\isacharunderscore}{\kern0pt}split{\isacharbrackleft}{\kern0pt}symmetric{\isacharcomma}{\kern0pt}\ OF\ finite{\isacharbrackright}{\kern0pt}\ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ \ {\isacharequal}{\kern0pt}\ {\isacharquery}{\kern0pt}f\ {\isacharparenleft}{\kern0pt}foldl\ {\isacharquery}{\kern0pt}f\ {\isacharparenleft}{\kern0pt}{\isasymlambda}s{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}v\ {\isachardollar}{\kern0pt}\ s{\isacharcomma}{\kern0pt}\ d\ s{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}sorted{\isacharunderscore}{\kern0pt}list{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}set\ {\isacharbraceleft}{\kern0pt}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}x{\isacharprime}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharquery}{\kern0pt}old\ {\isacharparenleft}{\kern0pt}x\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}GS{\isacharunderscore}{\kern0pt}rec\ v\ {\isachardollar}{\kern0pt}\ x{\isacharcomma}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}iter{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max\ {\isacharparenleft}{\kern0pt}{\isasymchi}\ s{\isachardot}{\kern0pt}\ fst\ {\isacharparenleft}{\kern0pt}{\isacharquery}{\kern0pt}old\ s{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ x{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}subst\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}iter{\isacharunderscore}{\kern0pt}max{\isacharbrackleft}{\kern0pt}of\ {\isacharunderscore}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isasymchi}\ s{\isachardot}{\kern0pt}\ fst\ {\isacharparenleft}{\kern0pt}{\isacharquery}{\kern0pt}old\ s{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}{\isacharbrackright}{\kern0pt}{\isacharcomma}{\kern0pt}\ goal{\isacharunderscore}{\kern0pt}cases{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{case}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}\ s{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{then}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}case\ \isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ fst{\isacharunderscore}{\kern0pt}IH\ \isacommand{by}\isamarkupfalse%
-\ auto\ \isanewline
-\ \ \ \ \isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \isacommand{case}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}{\isadigit{2}}\ s{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{then}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}case\isanewline
-\ \ \ \ \ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ vec{\isacharunderscore}{\kern0pt}lambda{\isacharunderscore}{\kern0pt}inverse{\isacharbrackleft}{\kern0pt}OF\ UNIV{\isacharunderscore}{\kern0pt}I{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ True\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}subst\ foldl{\isacharunderscore}{\kern0pt}upd{\isacharunderscore}{\kern0pt}notin{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ x{\isacharprime}{\kern0pt}{\isacharunderscore}{\kern0pt}def\ Max{\isacharunderscore}{\kern0pt}ge{\isacharunderscore}{\kern0pt}iff{\isacharbrackleft}{\kern0pt}OF\ finite{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{qed}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharquery}{\kern0pt}old\ {\isacharparenleft}{\kern0pt}x\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}GS{\isacharunderscore}{\kern0pt}rec\ v\ {\isachardollar}{\kern0pt}\ x{\isacharcomma}{\kern0pt}\ opt{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}gs{\isacharprime}{\kern0pt}\ d\ v\ x{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ fst{\isacharunderscore}{\kern0pt}IH{\isacharprime}{\kern0pt}{\isacharprime}{\kern0pt}\ opt{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}gs{\isacharprime}{\kern0pt}{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}GS{\isacharunderscore}{\kern0pt}iter{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{finally}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}foldl\ {\isacharquery}{\kern0pt}f\ {\isacharparenleft}{\kern0pt}{\isasymlambda}s{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}v\ {\isachardollar}{\kern0pt}\ s{\isacharcomma}{\kern0pt}\ d\ s{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}sorted{\isacharunderscore}{\kern0pt}list{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}set\ {\isacharbraceleft}{\kern0pt}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}x{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharquery}{\kern0pt}old\ {\isacharparenleft}{\kern0pt}x\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}GS{\isacharunderscore}{\kern0pt}rec\ v\ {\isachardollar}{\kern0pt}\ x{\isacharcomma}{\kern0pt}\ opt{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}gs{\isacharprime}{\kern0pt}\ d\ v\ x{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isacommand{{\isachardot}{\kern0pt}}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\ \ \ \ \isanewline
-\ \ \ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}am{\isacharunderscore}{\kern0pt}code{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{case}\isamarkupfalse%
-\ False\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}x{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}x{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ not{\isacharunderscore}{\kern0pt}less\ antisym{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}am{\isacharunderscore}{\kern0pt}code{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ opt{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}gs{\isacharprime}{\kern0pt}{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}GS{\isacharunderscore}{\kern0pt}iter{\isacharbrackleft}{\kern0pt}of\ x\ v{\isacharbrackright}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}iter{\isacharunderscore}{\kern0pt}max{\isacharbrackleft}{\kern0pt}of\ x\ v{\isacharbrackright}{\kern0pt}\ False\ \isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ fastforce\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}am{\isacharunderscore}{\kern0pt}code{\isacharunderscore}{\kern0pt}eq{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}am{\isacharunderscore}{\kern0pt}code\ v\ d\ s\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}GS{\isacharunderscore}{\kern0pt}rec\ v\ {\isachardollar}{\kern0pt}\ s{\isacharcomma}{\kern0pt}\ opt{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}gs{\isacharprime}{\kern0pt}\ d\ v\ s{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}induction\ s\ arbitrary{\isacharcolon}{\kern0pt}\ d\ rule{\isacharcolon}{\kern0pt}\ less{\isacharunderscore}{\kern0pt}induct{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{case}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}less\ x{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}case\ \isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}cases\ {\isachardoublequoteopen}{\isasymexists}x{\isacharprime}{\kern0pt}{\isachardot}{\kern0pt}\ x{\isacharprime}{\kern0pt}\ {\isacharless}{\kern0pt}\ x{\isachardoublequoteclose}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{case}\isamarkupfalse%
-\ True\isanewline
-\ \ \ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}f\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isasymlambda}vd\ s{\isachardot}{\kern0pt}\ vd{\isacharparenleft}{\kern0pt}s\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}GS{\isacharunderscore}{\kern0pt}iter{\isacharunderscore}{\kern0pt}max\ {\isacharparenleft}{\kern0pt}{\isasymchi}\ s{\isachardot}{\kern0pt}\ fst\ {\isacharparenleft}{\kern0pt}vd\ s{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ s{\isacharcomma}{\kern0pt}\ \ GS{\isacharunderscore}{\kern0pt}iter{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max\ {\isacharparenleft}{\kern0pt}{\isasymchi}\ s{\isachardot}{\kern0pt}\ fst\ {\isacharparenleft}{\kern0pt}vd\ s{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ s{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{define}\isamarkupfalse%
-\ x{\isacharprime}{\kern0pt}\ \isakeyword{where}\ {\isachardoublequoteopen}x{\isacharprime}{\kern0pt}\ {\isacharequal}{\kern0pt}\ Max\ {\isacharbraceleft}{\kern0pt}x{\isacharprime}{\kern0pt}{\isachardot}{\kern0pt}\ x{\isacharprime}{\kern0pt}\ {\isacharless}{\kern0pt}\ x{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}old\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}foldl\ {\isacharquery}{\kern0pt}f\ {\isacharparenleft}{\kern0pt}{\isasymlambda}s{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}v\ {\isachardollar}{\kern0pt}\ s{\isacharcomma}{\kern0pt}\ d\ s{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}sorted{\isacharunderscore}{\kern0pt}list{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}set\ {\isacharbraceleft}{\kern0pt}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}x{\isacharprime}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}s\ {\isacharless}{\kern0pt}\ x\ {\isasymLongrightarrow}\ {\isacharparenleft}{\kern0pt}s\ {\isasymnotin}\ set\ {\isacharparenleft}{\kern0pt}sorted{\isacharunderscore}{\kern0pt}list{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}set\ {\isacharbraceleft}{\kern0pt}s{\isacharprime}{\kern0pt}\ {\isasymin}\ {\isacharbraceleft}{\kern0pt}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}x{\isacharprime}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ s\ {\isacharless}{\kern0pt}\ s{\isacharprime}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{for}\ s\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isacharprime}{\kern0pt}s\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}s\ {\isacharless}{\kern0pt}\ x\ {\isasymLongrightarrow}\ foldl\ {\isacharquery}{\kern0pt}f\ {\isacharparenleft}{\kern0pt}{\isasymlambda}s{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}v\ {\isachardollar}{\kern0pt}\ s{\isacharcomma}{\kern0pt}\ d\ s{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}sorted{\isacharunderscore}{\kern0pt}list{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}set\ {\isacharbraceleft}{\kern0pt}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}x{\isacharprime}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ s\ {\isacharequal}{\kern0pt}\ \ foldl\ {\isacharquery}{\kern0pt}f\ {\isacharparenleft}{\kern0pt}{\isasymlambda}s{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}v\ {\isachardollar}{\kern0pt}\ s{\isacharcomma}{\kern0pt}\ d\ s{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}sorted{\isacharunderscore}{\kern0pt}list{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}set\ {\isacharbraceleft}{\kern0pt}s{\isacharprime}{\kern0pt}\ {\isasymin}\ {\isacharbraceleft}{\kern0pt}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}x{\isacharprime}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ s{\isacharprime}{\kern0pt}\ {\isasymle}\ s{\isacharbraceright}{\kern0pt}\ {\isacharat}{\kern0pt}\ sorted{\isacharunderscore}{\kern0pt}list{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}set\ {\isacharbraceleft}{\kern0pt}s{\isacharprime}{\kern0pt}\ {\isasymin}\ {\isacharbraceleft}{\kern0pt}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}x{\isacharprime}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ s\ {\isacharless}{\kern0pt}\ s{\isacharprime}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ s{\isachardoublequoteclose}\ \isakeyword{for}\ s\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}subst\ sorted{\isacharunderscore}{\kern0pt}list{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}set{\isacharunderscore}{\kern0pt}split{\isacharprime}{\kern0pt}{\isacharbrackleft}{\kern0pt}symmetric{\isacharcomma}{\kern0pt}\ OF\ finite{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\ blast\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}s\ {\isacharless}{\kern0pt}\ x\ {\isasymLongrightarrow}\ foldl\ {\isacharquery}{\kern0pt}f\ {\isacharparenleft}{\kern0pt}{\isasymlambda}s{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}v\ {\isachardollar}{\kern0pt}\ s{\isacharcomma}{\kern0pt}\ d\ s{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}sorted{\isacharunderscore}{\kern0pt}list{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}set\ {\isacharbraceleft}{\kern0pt}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}x{\isacharprime}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ s\ {\isacharequal}{\kern0pt}\ \ foldl\ {\isacharquery}{\kern0pt}f\ {\isacharparenleft}{\kern0pt}{\isasymlambda}s{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}v\ {\isachardollar}{\kern0pt}\ s{\isacharcomma}{\kern0pt}\ d\ s{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}sorted{\isacharunderscore}{\kern0pt}list{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}set\ {\isacharbraceleft}{\kern0pt}s{\isacharprime}{\kern0pt}\ {\isasymin}\ {\isacharbraceleft}{\kern0pt}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}x{\isacharprime}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ s{\isacharprime}{\kern0pt}\ {\isasymle}\ s{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ s{\isachardoublequoteclose}\ \isakeyword{for}\ s\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ foldl{\isacharunderscore}{\kern0pt}upd{\isacharunderscore}{\kern0pt}notin{\isacharprime}{\kern0pt}{\isacharbrackleft}{\kern0pt}OF\ {\isadigit{1}}{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ fastforce\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ {\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}s\ {\isacharless}{\kern0pt}\ x\ {\isasymLongrightarrow}\ foldl\ {\isacharquery}{\kern0pt}f\ {\isacharparenleft}{\kern0pt}{\isasymlambda}s{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}v\ {\isachardollar}{\kern0pt}\ s{\isacharcomma}{\kern0pt}\ d\ s{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}sorted{\isacharunderscore}{\kern0pt}list{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}set\ {\isacharbraceleft}{\kern0pt}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}x{\isacharprime}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ s\ {\isacharequal}{\kern0pt}\ \ foldl\ {\isacharquery}{\kern0pt}f\ {\isacharparenleft}{\kern0pt}{\isasymlambda}s{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}v\ {\isachardollar}{\kern0pt}\ s{\isacharcomma}{\kern0pt}\ d\ s{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}sorted{\isacharunderscore}{\kern0pt}list{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}set\ {\isacharbraceleft}{\kern0pt}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}s{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ s{\isachardoublequoteclose}\ \isakeyword{for}\ s\isanewline
-\ \ \ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ x{\isacharprime}{\kern0pt}{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ True\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ atMost{\isacharunderscore}{\kern0pt}def\ Max{\isacharunderscore}{\kern0pt}ge{\isacharunderscore}{\kern0pt}iff{\isacharbrackleft}{\kern0pt}OF\ finite{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\ meson\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ fst{\isacharunderscore}{\kern0pt}IH{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}fst\ {\isacharparenleft}{\kern0pt}{\isacharquery}{\kern0pt}old\ s{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}rec\ v\ {\isachardollar}{\kern0pt}\ s{\isachardoublequoteclose}\ \isakeyword{if}\ {\isachardoublequoteopen}s\ {\isacharless}{\kern0pt}\ x{\isachardoublequoteclose}\ \isakeyword{for}\ s\isanewline
-\ \ \ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ {\isadigit{1}}{\isacharbrackleft}{\kern0pt}OF\ that{\isacharbrackright}{\kern0pt}\ less{\isacharbrackleft}{\kern0pt}unfolded\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}am{\isacharunderscore}{\kern0pt}code{\isacharunderscore}{\kern0pt}def{\isacharcomma}{\kern0pt}\ OF\ that{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ fst{\isacharunderscore}{\kern0pt}IH{\isacharprime}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}fst\ {\isacharparenleft}{\kern0pt}{\isacharquery}{\kern0pt}old\ s{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ v\ {\isachardollar}{\kern0pt}\ s{\isachardoublequoteclose}\ \isakeyword{if}\ {\isachardoublequoteopen}x\ {\isasymle}\ s{\isachardoublequoteclose}\ \isakeyword{for}\ s\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ True\ that\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}subst\ foldl{\isacharunderscore}{\kern0pt}upd{\isacharunderscore}{\kern0pt}notin{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ x{\isacharprime}{\kern0pt}{\isacharunderscore}{\kern0pt}def\ atMost{\isacharunderscore}{\kern0pt}def\ Max{\isacharunderscore}{\kern0pt}ge{\isacharunderscore}{\kern0pt}iff{\isacharbrackleft}{\kern0pt}OF\ finite{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ fst{\isacharunderscore}{\kern0pt}IH{\isacharprime}{\kern0pt}{\isacharprime}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}fst\ {\isacharparenleft}{\kern0pt}{\isacharquery}{\kern0pt}old\ s{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}if\ s\ {\isacharless}{\kern0pt}\ x\ then\ GS{\isacharunderscore}{\kern0pt}rec\ v\ {\isachardollar}{\kern0pt}\ s\ else\ v\ {\isachardollar}{\kern0pt}\ s{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{for}\ s\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ fst{\isacharunderscore}{\kern0pt}IH\ fst{\isacharunderscore}{\kern0pt}IH{\isacharprime}{\kern0pt}\ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}foldl\ {\isacharquery}{\kern0pt}f\ {\isacharparenleft}{\kern0pt}{\isasymlambda}s{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}v\ {\isachardollar}{\kern0pt}\ s{\isacharcomma}{\kern0pt}\ d\ s{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}sorted{\isacharunderscore}{\kern0pt}list{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}set\ {\isacharbraceleft}{\kern0pt}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}x{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ foldl\ {\isacharquery}{\kern0pt}f\ {\isacharparenleft}{\kern0pt}{\isasymlambda}s{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}v\ {\isachardollar}{\kern0pt}\ s{\isacharcomma}{\kern0pt}\ d\ s{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}sorted{\isacharunderscore}{\kern0pt}list{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}set\ {\isacharbraceleft}{\kern0pt}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}x{\isacharprime}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharat}{\kern0pt}\ sorted{\isacharunderscore}{\kern0pt}list{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}set\ {\isacharbraceleft}{\kern0pt}x{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isacharasterisk}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}x{\isacharprime}{\kern0pt}{\isachardot}{\kern0pt}\ x{\isacharprime}{\kern0pt}\ {\isacharless}{\kern0pt}\ x{\isacharbraceright}{\kern0pt}\ {\isasymnoteq}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\ \isacommand{using}\isamarkupfalse%
-\ True\ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ {\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}x{\isacharprime}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}y\ {\isasymin}\ {\isacharbraceleft}{\kern0pt}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}x{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ y\ {\isacharless}{\kern0pt}\ x{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ x{\isacharprime}{\kern0pt}{\isacharunderscore}{\kern0pt}def\ Max{\isacharunderscore}{\kern0pt}ge{\isacharunderscore}{\kern0pt}iff{\isacharbrackleft}{\kern0pt}OF\ finite\ {\isacharasterisk}{\kern0pt}{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{2}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}x{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}y\ {\isasymin}\ {\isacharbraceleft}{\kern0pt}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}x{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ x\ {\isasymle}\ y{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ {\isadigit{1}}\ {\isadigit{2}}\ sorted{\isacharunderscore}{\kern0pt}list{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}set{\isacharunderscore}{\kern0pt}split{\isacharbrackleft}{\kern0pt}symmetric{\isacharcomma}{\kern0pt}\ OF\ finite{\isacharbrackright}{\kern0pt}\ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ \ {\isacharequal}{\kern0pt}\ {\isacharquery}{\kern0pt}f\ {\isacharparenleft}{\kern0pt}foldl\ {\isacharquery}{\kern0pt}f\ {\isacharparenleft}{\kern0pt}{\isasymlambda}s{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}v\ {\isachardollar}{\kern0pt}\ s{\isacharcomma}{\kern0pt}\ d\ s{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}sorted{\isacharunderscore}{\kern0pt}list{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}set\ {\isacharbraceleft}{\kern0pt}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}x{\isacharprime}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharquery}{\kern0pt}old\ {\isacharparenleft}{\kern0pt}x\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}GS{\isacharunderscore}{\kern0pt}rec\ v\ {\isachardollar}{\kern0pt}\ x{\isacharcomma}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}iter{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max\ {\isacharparenleft}{\kern0pt}{\isasymchi}\ s{\isachardot}{\kern0pt}\ fst\ {\isacharparenleft}{\kern0pt}{\isacharquery}{\kern0pt}old\ s{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ x{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}subst\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}iter{\isacharunderscore}{\kern0pt}max{\isacharbrackleft}{\kern0pt}of\ {\isacharunderscore}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isasymchi}\ s{\isachardot}{\kern0pt}\ fst\ {\isacharparenleft}{\kern0pt}{\isacharquery}{\kern0pt}old\ s{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}{\isacharbrackright}{\kern0pt}{\isacharcomma}{\kern0pt}\ goal{\isacharunderscore}{\kern0pt}cases{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{case}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}{\isadigit{2}}\ s{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{then}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}case\isanewline
-\ \ \ \ \ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ vec{\isacharunderscore}{\kern0pt}lambda{\isacharunderscore}{\kern0pt}inverse{\isacharbrackleft}{\kern0pt}OF\ UNIV{\isacharunderscore}{\kern0pt}I{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ True\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}subst\ foldl{\isacharunderscore}{\kern0pt}upd{\isacharunderscore}{\kern0pt}notin{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ x{\isacharprime}{\kern0pt}{\isacharunderscore}{\kern0pt}def\ Max{\isacharunderscore}{\kern0pt}ge{\isacharunderscore}{\kern0pt}iff{\isacharbrackleft}{\kern0pt}OF\ finite{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{qed}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ fst{\isacharunderscore}{\kern0pt}IH{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharquery}{\kern0pt}old\ {\isacharparenleft}{\kern0pt}x\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}GS{\isacharunderscore}{\kern0pt}rec\ v\ {\isachardollar}{\kern0pt}\ x{\isacharcomma}{\kern0pt}\ opt{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}gs{\isacharprime}{\kern0pt}\ d\ v\ x{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ fst{\isacharunderscore}{\kern0pt}IH{\isacharprime}{\kern0pt}{\isacharprime}{\kern0pt}\ opt{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}gs{\isacharprime}{\kern0pt}{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}GS{\isacharunderscore}{\kern0pt}iter{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{finally}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}foldl\ {\isacharquery}{\kern0pt}f\ {\isacharparenleft}{\kern0pt}{\isasymlambda}s{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}v\ {\isachardollar}{\kern0pt}\ s{\isacharcomma}{\kern0pt}\ d\ s{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}sorted{\isacharunderscore}{\kern0pt}list{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}set\ {\isacharbraceleft}{\kern0pt}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}x{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharquery}{\kern0pt}old\ {\isacharparenleft}{\kern0pt}x\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}GS{\isacharunderscore}{\kern0pt}rec\ v\ {\isachardollar}{\kern0pt}\ x{\isacharcomma}{\kern0pt}\ opt{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}gs{\isacharprime}{\kern0pt}\ d\ v\ x{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isacommand{{\isachardot}{\kern0pt}}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\ \ \ \ \isanewline
-\ \ \ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}am{\isacharunderscore}{\kern0pt}code{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{case}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}False{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}x{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}x{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ not{\isacharunderscore}{\kern0pt}less\ antisym{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ {\isacharasterisk}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}sorted{\isacharunderscore}{\kern0pt}list{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}set\ {\isacharbraceleft}{\kern0pt}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}x{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharbrackleft}{\kern0pt}x{\isacharbrackright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}am{\isacharunderscore}{\kern0pt}code{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ opt{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}gs{\isacharprime}{\kern0pt}{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}GS{\isacharunderscore}{\kern0pt}iter{\isacharbrackleft}{\kern0pt}of\ x\ v{\isacharbrackright}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}iter{\isacharunderscore}{\kern0pt}max{\isacharbrackleft}{\kern0pt}of\ x\ v{\isacharbrackright}{\kern0pt}\ False\ \isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}fastforce\ simp{\isacharcolon}{\kern0pt}\ {\isacharasterisk}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}iter{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}iter{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max\ v\ s\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}LEAST\ a{\isachardot}{\kern0pt}\ is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max\ {\isacharparenleft}{\kern0pt}{\isasymlambda}a{\isachardot}{\kern0pt}\ r\ {\isacharparenleft}{\kern0pt}s{\isacharcomma}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isacharplus}{\kern0pt}\ l\ {\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSum}s{\isacharprime}{\kern0pt}\ {\isasymin}\ UNIV{\isachardot}{\kern0pt}\ pmf\ {\isacharparenleft}{\kern0pt}K\ {\isacharparenleft}{\kern0pt}s{\isacharcomma}{\kern0pt}a{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ s{\isacharprime}{\kern0pt}\ {\isacharasterisk}{\kern0pt}\ v\ s{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymlambda}a{\isachardot}{\kern0pt}\ a\ {\isasymin}\ A\ s{\isacharparenright}{\kern0pt}\ a{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ {\isachardoublequoteopen}opt{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}gs\ v\ s\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}LEAST\ a{\isachardot}{\kern0pt}\ is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max\ {\isacharparenleft}{\kern0pt}{\isasymlambda}a{\isachardot}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}fun{\isacharunderscore}{\kern0pt}inner\ v\ s\ a{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymlambda}a{\isachardot}{\kern0pt}\ a\ {\isasymin}\ A\ s{\isacharparenright}{\kern0pt}\ a{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ opt{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}gs{\isacharunderscore}{\kern0pt}eq{\isacharprime}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}opt{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}gs\ v\ {\isacharequal}{\kern0pt}\ opt{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}gs{\isacharprime}{\kern0pt}\ d\ {\isacharparenleft}{\kern0pt}vec{\isacharunderscore}{\kern0pt}lambda\ v{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ opt{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}gs{\isacharunderscore}{\kern0pt}def\ opt{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}gs{\isacharprime}{\kern0pt}{\isacharunderscore}{\kern0pt}def\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}fun{\isacharunderscore}{\kern0pt}inner{\isacharunderscore}{\kern0pt}def\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}step{\isacharunderscore}{\kern0pt}elem\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}fun\isactrlsub b{\isachardot}{\kern0pt}rep{\isacharunderscore}{\kern0pt}eq\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}def\ vec{\isacharunderscore}{\kern0pt}lambda{\isacharunderscore}{\kern0pt}inverse{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{declare}\isamarkupfalse%
-\ gs{\isacharunderscore}{\kern0pt}iteration{\isachardot}{\kern0pt}simps{\isacharbrackleft}{\kern0pt}simp\ del{\isacharbrackright}{\kern0pt}\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ gs{\isacharunderscore}{\kern0pt}iteration{\isacharunderscore}{\kern0pt}error{\isacharcolon}{\kern0pt}\ \isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}eps\ {\isachargreater}{\kern0pt}\ {\isadigit{0}}{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}dist\ {\isacharparenleft}{\kern0pt}gs{\isacharunderscore}{\kern0pt}iteration\ eps\ v{\isacharparenright}{\kern0pt}\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt\ {\isacharless}{\kern0pt}\ eps\ {\isacharslash}{\kern0pt}\ {\isadigit{2}}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ assms\ dist{\isacharunderscore}{\kern0pt}{\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}split{\isacharunderscore}{\kern0pt}opt{\isacharunderscore}{\kern0pt}eps\ gs{\isacharunderscore}{\kern0pt}iteration{\isachardot}{\kern0pt}simps\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}induction\ eps\ v\ rule{\isacharcolon}{\kern0pt}\ gs{\isacharunderscore}{\kern0pt}iteration{\isachardot}{\kern0pt}induct{\isacharparenright}{\kern0pt}\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}fun{\isacharunderscore}{\kern0pt}inner{\isacharunderscore}{\kern0pt}vec{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}fun{\isacharunderscore}{\kern0pt}inner\ v\ s\ a\ {\isacharequal}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}step\ {\isacharparenleft}{\kern0pt}d{\isacharparenleft}{\kern0pt}s\ {\isacharcolon}{\kern0pt}{\isacharequal}{\kern0pt}\ a{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}vec{\isacharunderscore}{\kern0pt}lambda\ v{\isacharparenright}{\kern0pt}\ {\isachardollar}{\kern0pt}\ s{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}step{\isacharunderscore}{\kern0pt}elem\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}fun{\isacharunderscore}{\kern0pt}inner{\isacharunderscore}{\kern0pt}def\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}def\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}fun\isactrlsub b{\isachardot}{\kern0pt}rep{\isacharunderscore}{\kern0pt}eq\ vec{\isacharunderscore}{\kern0pt}lambda{\isacharunderscore}{\kern0pt}inverse{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ find{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}error{\isacharunderscore}{\kern0pt}bound{\isacharunderscore}{\kern0pt}gs{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}eps\ {\isachargreater}{\kern0pt}\ {\isadigit{0}}{\isachardoublequoteclose}\ {\isachardoublequoteopen}{\isadigit{2}}\ {\isacharasterisk}{\kern0pt}\ l\ {\isacharasterisk}{\kern0pt}\ dist\ v\ {\isacharparenleft}{\kern0pt}GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}fun\isactrlsub b\ v{\isacharparenright}{\kern0pt}\ {\isacharless}{\kern0pt}\ eps\ {\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharminus}{\kern0pt}l{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}dist\ {\isacharparenleft}{\kern0pt}{\isasymnu}\isactrlsub b\ {\isacharparenleft}{\kern0pt}mk{\isacharunderscore}{\kern0pt}stationary{\isacharunderscore}{\kern0pt}det\ {\isacharparenleft}{\kern0pt}opt{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}gs\ {\isacharparenleft}{\kern0pt}GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}fun\isactrlsub b\ v{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt\ {\isacharless}{\kern0pt}\ eps{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}rule\ GS{\isachardot}{\kern0pt}find{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}QR{\isacharunderscore}{\kern0pt}error{\isacharunderscore}{\kern0pt}bound{\isacharbrackleft}{\kern0pt}OF\ assms{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isadigit{2}}\ {\isacharasterisk}{\kern0pt}\ GS{\isachardot}{\kern0pt}QR{\isacharunderscore}{\kern0pt}disc\ {\isacharasterisk}{\kern0pt}\ dist\ v\ {\isacharparenleft}{\kern0pt}GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}fun\isactrlsub b\ v{\isacharparenright}{\kern0pt}\ {\isasymle}\ {\isadigit{2}}\ {\isacharasterisk}{\kern0pt}\ l\ {\isacharasterisk}{\kern0pt}\ dist\ v\ {\isacharparenleft}{\kern0pt}GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}fun\isactrlsub b\ v{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ GS{\isacharunderscore}{\kern0pt}QR{\isacharunderscore}{\kern0pt}disc{\isacharunderscore}{\kern0pt}le{\isacharunderscore}{\kern0pt}disc\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ mult{\isacharunderscore}{\kern0pt}right{\isacharunderscore}{\kern0pt}mono{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isacharless}{\kern0pt}\ \ eps\ {\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharminus}{\kern0pt}l{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isacommand{using}\isamarkupfalse%
-\ assms\ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isasymle}\ eps\ {\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}\ {\isacharminus}{\kern0pt}\ GS{\isachardot}{\kern0pt}QR{\isacharunderscore}{\kern0pt}disc{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assms\ GS{\isacharunderscore}{\kern0pt}QR{\isacharunderscore}{\kern0pt}disc{\isacharunderscore}{\kern0pt}le{\isacharunderscore}{\kern0pt}disc\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ mult{\isacharunderscore}{\kern0pt}left{\isacharunderscore}{\kern0pt}mono{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{finally}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isadigit{2}}\ {\isacharasterisk}{\kern0pt}\ GS{\isachardot}{\kern0pt}QR{\isacharunderscore}{\kern0pt}disc\ {\isacharasterisk}{\kern0pt}\ dist\ v\ {\isacharparenleft}{\kern0pt}GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}fun\isactrlsub b\ v{\isacharparenright}{\kern0pt}\ {\isacharless}{\kern0pt}\ eps\ {\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}\ {\isacharminus}{\kern0pt}\ GS{\isachardot}{\kern0pt}QR{\isacharunderscore}{\kern0pt}disc{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isacommand{{\isachardot}{\kern0pt}}\isamarkupfalse%
-\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{obtain}\isamarkupfalse%
-\ d\ \isakeyword{where}\ d{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}is{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\ d{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ ex{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max\ {\isacharparenleft}{\kern0pt}{\isasymlambda}d{\isachardot}{\kern0pt}\ apply{\isacharunderscore}{\kern0pt}bfun\ {\isacharparenleft}{\kern0pt}GS{\isachardot}{\kern0pt}L{\isacharunderscore}{\kern0pt}split\ d\ {\isacharparenleft}{\kern0pt}GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}fun\isactrlsub b\ v{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ s{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymlambda}d{\isachardot}{\kern0pt}\ d\ {\isasymin}\ D\isactrlsub D{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}opt{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}gs\ {\isacharparenleft}{\kern0pt}GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}fun\isactrlsub b\ v{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{for}\ s\isanewline
-\ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ opt{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}gs{\isacharunderscore}{\kern0pt}eq{\isacharprime}{\kern0pt}{\isacharbrackleft}{\kern0pt}of\ {\isacharunderscore}{\kern0pt}\ d{\isacharbrackright}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}inv{\isacharunderscore}{\kern0pt}blinfun{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}matrix\ \isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ opt{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}gs{\isacharprime}{\kern0pt}{\isacharunderscore}{\kern0pt}is{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ {\isachardoublequoteopen}vi{\isacharunderscore}{\kern0pt}gs{\isacharunderscore}{\kern0pt}policy\ eps\ v\ {\isacharequal}{\kern0pt}\ opt{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}gs\ {\isacharparenleft}{\kern0pt}gs{\isacharunderscore}{\kern0pt}iteration\ eps\ v{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ vi{\isacharunderscore}{\kern0pt}gs{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}opt{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}{\isadigit{0}}\ {\isacharless}{\kern0pt}\ eps{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}dist\ {\isacharparenleft}{\kern0pt}{\isasymnu}\isactrlsub b\ {\isacharparenleft}{\kern0pt}mk{\isacharunderscore}{\kern0pt}stationary{\isacharunderscore}{\kern0pt}det\ {\isacharparenleft}{\kern0pt}vi{\isacharunderscore}{\kern0pt}gs{\isacharunderscore}{\kern0pt}policy\ eps\ v{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt\ {\isacharless}{\kern0pt}\ eps{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ vi{\isacharunderscore}{\kern0pt}gs{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}def\ \isanewline
-\ \ \isacommand{using}\isamarkupfalse%
-\ assms\isanewline
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}induction\ eps\ v\ rule{\isacharcolon}{\kern0pt}\ gs{\isacharunderscore}{\kern0pt}iteration{\isachardot}{\kern0pt}induct{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{case}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}\ v{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{then}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}case\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ find{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}error{\isacharunderscore}{\kern0pt}bound{\isacharunderscore}{\kern0pt}gs\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}subst\ gs{\isacharunderscore}{\kern0pt}iteration{\isachardot}{\kern0pt}simps{\isacharparenright}{\kern0pt}\ auto\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}iter{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}iter{\isacharunderscore}{\kern0pt}max{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}iter\ v\ {\isacharequal}{\kern0pt}\ GS{\isacharunderscore}{\kern0pt}iter{\isacharunderscore}{\kern0pt}max\ {\isacharparenleft}{\kern0pt}vec{\isacharunderscore}{\kern0pt}lambda\ v{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ GS{\isacharunderscore}{\kern0pt}rec{\isacharunderscore}{\kern0pt}iter{\isacharunderscore}{\kern0pt}def\ GS{\isacharunderscore}{\kern0pt}iter{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}def\ GS{\isacharunderscore}{\kern0pt}iter{\isacharunderscore}{\kern0pt}def\ \isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isacommand{end}\isamarkupfalse%
-\isanewline
-%
-\isadelimtheory
-\isanewline
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{end}\isamarkupfalse%
-%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\end{isabellebody}%
-\endinput
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diff --git a/thys/MDP-Algorithms/output/outline/Value_Iteration.tex b/thys/MDP-Algorithms/output/outline/Value_Iteration.tex
deleted file mode 100644
--- a/thys/MDP-Algorithms/output/outline/Value_Iteration.tex
+++ /dev/null
@@ -1,1507 +0,0 @@
-%
-\begin{isabellebody}%
-\setisabellecontext{Value{\isacharunderscore}{\kern0pt}Iteration}%
-%
-\isadelimtheory
-\isanewline
-\isanewline
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{theory}\isamarkupfalse%
-\ Value{\isacharunderscore}{\kern0pt}Iteration\isanewline
-\ \ \isakeyword{imports}\ {\isachardoublequoteopen}MDP{\isacharminus}{\kern0pt}Rewards{\isachardot}{\kern0pt}MDP{\isacharunderscore}{\kern0pt}reward{\isachardoublequoteclose}\isanewline
-\isakeyword{begin}%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-\isanewline
-%
-\endisadelimtheory
-\isanewline
-\isacommand{context}\isamarkupfalse%
-\ MDP{\isacharunderscore}{\kern0pt}att{\isacharunderscore}{\kern0pt}{\isasymL}\isanewline
-\isakeyword{begin}%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsection{Value Iteration%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\begin{isamarkuptext}%
-In the previous sections we derived that repeated application of \isa{{\isasymL}\isactrlsub b} to any bounded 
-function from states to the reals converges to the optimal value of the MDP \isa{{\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt}.
-
-We can turn this procedure into an algorithm that computes not only an approximation of 
-\isa{{\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt} but also a policy that is arbitrarily close to optimal.
-
-Most of the proofs rely on the assumption that the supremum in \isa{{\isasymL}\isactrlsub b} can always be attained.%
-\end{isamarkuptext}\isamarkuptrue%
-%
-\begin{isamarkuptext}%
-The following lemma shows that the relation we use to prove termination of the value iteration 
-algorithm decreases in each step.
-In essence, the distance of the estimate to the optimal value decreases by a factor of at 
-least \isa{l} per iteration.%
-\end{isamarkuptext}\isamarkuptrue%
-\isacommand{lemma}\isamarkupfalse%
-\ vi{\isacharunderscore}{\kern0pt}rel{\isacharunderscore}{\kern0pt}dec{\isacharcolon}{\kern0pt}\ \isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}l\ {\isasymnoteq}\ {\isadigit{0}}{\isachardoublequoteclose}\ {\isachardoublequoteopen}{\isasymL}\isactrlsub b\ v\ {\isasymnoteq}\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}{\isasymlceil}log\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}\ {\isacharslash}{\kern0pt}\ l{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}dist\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ v{\isacharparenright}{\kern0pt}\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt{\isacharparenright}{\kern0pt}\ {\isacharminus}{\kern0pt}\ c{\isasymrceil}\ {\isacharless}{\kern0pt}\ {\isasymlceil}log\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}\ {\isacharslash}{\kern0pt}\ l{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}dist\ v\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt{\isacharparenright}{\kern0pt}\ {\isacharminus}{\kern0pt}\ c{\isasymrceil}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}log\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}\ {\isacharslash}{\kern0pt}\ l{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}dist\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ v{\isacharparenright}{\kern0pt}\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt{\isacharparenright}{\kern0pt}\ {\isacharminus}{\kern0pt}\ c\ {\isasymle}\ log\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}\ {\isacharslash}{\kern0pt}\ l{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}l\ {\isacharasterisk}{\kern0pt}\ dist\ v\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt{\isacharparenright}{\kern0pt}\ {\isacharminus}{\kern0pt}\ c{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ contraction{\isacharunderscore}{\kern0pt}{\isasymL}{\isacharbrackleft}{\kern0pt}of\ {\isacharunderscore}{\kern0pt}\ {\isachardoublequoteopen}{\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt{\isachardoublequoteclose}{\isacharbrackright}{\kern0pt}\ disc{\isacharunderscore}{\kern0pt}lt{\isacharunderscore}{\kern0pt}one\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ assms\ less{\isacharunderscore}{\kern0pt}le\ intro{\isacharcolon}{\kern0pt}\ log{\isacharunderscore}{\kern0pt}le{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isacharequal}{\kern0pt}\ log\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}\ {\isacharslash}{\kern0pt}\ l{\isacharparenright}{\kern0pt}\ l\ {\isacharplus}{\kern0pt}\ log\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharslash}{\kern0pt}l{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}dist\ v\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt{\isacharparenright}{\kern0pt}\ {\isacharminus}{\kern0pt}\ c{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assms\ disc{\isacharunderscore}{\kern0pt}lt{\isacharunderscore}{\kern0pt}one\ \isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ less{\isacharunderscore}{\kern0pt}le\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ log{\isacharunderscore}{\kern0pt}mult{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isacharequal}{\kern0pt}\ {\isacharminus}{\kern0pt}{\isacharparenleft}{\kern0pt}log\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}\ {\isacharslash}{\kern0pt}\ l{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharslash}{\kern0pt}l{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharplus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}log\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharslash}{\kern0pt}l{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}dist\ v\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharminus}{\kern0pt}\ c{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assms\ disc{\isacharunderscore}{\kern0pt}lt{\isacharunderscore}{\kern0pt}one\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}subst\ log{\isacharunderscore}{\kern0pt}inverse{\isacharbrackleft}{\kern0pt}symmetric{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ less{\isacharunderscore}{\kern0pt}le\ right{\isacharunderscore}{\kern0pt}inverse{\isacharunderscore}{\kern0pt}eq{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}log\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharslash}{\kern0pt}l{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}dist\ v\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharminus}{\kern0pt}\ {\isadigit{1}}\ {\isacharminus}{\kern0pt}\ c{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assms\ order{\isachardot}{\kern0pt}strict{\isacharunderscore}{\kern0pt}implies{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}eq{\isacharbrackleft}{\kern0pt}OF\ disc{\isacharunderscore}{\kern0pt}lt{\isacharunderscore}{\kern0pt}one{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ log{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}one\ neq{\isacharunderscore}{\kern0pt}le{\isacharunderscore}{\kern0pt}trans{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{finally}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}log\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}\ {\isacharslash}{\kern0pt}\ l{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}dist\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ v{\isacharparenright}{\kern0pt}\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt{\isacharparenright}{\kern0pt}\ {\isacharminus}{\kern0pt}\ c\ {\isasymle}\ log\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}\ {\isacharslash}{\kern0pt}\ l{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}dist\ v\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt{\isacharparenright}{\kern0pt}\ {\isacharminus}{\kern0pt}\ {\isadigit{1}}\ {\isacharminus}{\kern0pt}\ c{\isachardoublequoteclose}\ \isacommand{{\isachardot}{\kern0pt}}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ linarith\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ dist{\isacharunderscore}{\kern0pt}{\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}lt{\isacharunderscore}{\kern0pt}dist{\isacharunderscore}{\kern0pt}opt{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}dist\ v\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ v{\isacharparenright}{\kern0pt}\ {\isasymle}\ {\isadigit{2}}\ {\isacharasterisk}{\kern0pt}\ dist\ v\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ le{\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}dist\ v\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ v{\isacharparenright}{\kern0pt}\ {\isasymle}\ dist\ v\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt\ {\isacharplus}{\kern0pt}\ dist\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ v{\isacharparenright}{\kern0pt}\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ dist{\isacharunderscore}{\kern0pt}triangle\ dist{\isacharunderscore}{\kern0pt}commute{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ le{\isadigit{2}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}dist\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ v{\isacharparenright}{\kern0pt}\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt\ {\isasymle}\ l\ {\isacharasterisk}{\kern0pt}\ dist\ v\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}opt\ contraction{\isacharunderscore}{\kern0pt}{\isasymL}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ mult{\isacharunderscore}{\kern0pt}right{\isacharunderscore}{\kern0pt}mono{\isacharbrackleft}{\kern0pt}of\ l\ {\isadigit{1}}{\isacharbrackright}{\kern0pt}\ disc{\isacharunderscore}{\kern0pt}lt{\isacharunderscore}{\kern0pt}one\ \isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}fastforce\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ order{\isachardot}{\kern0pt}trans{\isacharbrackleft}{\kern0pt}OF\ le{\isadigit{2}}{\isacharbrackright}{\kern0pt}\ order{\isachardot}{\kern0pt}trans{\isacharbrackleft}{\kern0pt}OF\ le{\isadigit{1}}{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{abbreviation}\isamarkupfalse%
-\ {\isachardoublequoteopen}term{\isacharunderscore}{\kern0pt}measure\ {\isasymequiv}\ {\isacharparenleft}{\kern0pt}{\isasymlambda}{\isacharparenleft}{\kern0pt}eps{\isacharcomma}{\kern0pt}\ v{\isacharparenright}{\kern0pt}{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ if\ v\ {\isacharequal}{\kern0pt}\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt\ {\isasymor}\ l\ {\isacharequal}{\kern0pt}\ {\isadigit{0}}\isanewline
-\ \ \ \ then\ {\isadigit{0}}\isanewline
-\ \ \ \ else\ nat\ {\isacharparenleft}{\kern0pt}ceiling\ {\isacharparenleft}{\kern0pt}log\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharslash}{\kern0pt}l{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}dist\ v\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt{\isacharparenright}{\kern0pt}\ {\isacharminus}{\kern0pt}\ log\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharslash}{\kern0pt}l{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}eps\ {\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharminus}{\kern0pt}l{\isacharparenright}{\kern0pt}\ {\isacharslash}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isadigit{8}}\ {\isacharasterisk}{\kern0pt}\ l{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{function}\isamarkupfalse%
-\ value{\isacharunderscore}{\kern0pt}iteration\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}real\ {\isasymRightarrow}\ {\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}s\ {\isasymRightarrow}\isactrlsub b\ real{\isacharparenright}{\kern0pt}\ {\isasymRightarrow}\ {\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}s\ {\isasymRightarrow}\isactrlsub b\ real{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}value{\isacharunderscore}{\kern0pt}iteration\ eps\ v\ {\isacharequal}{\kern0pt}\isanewline
-\ \ {\isacharparenleft}{\kern0pt}if\ {\isadigit{2}}\ {\isacharasterisk}{\kern0pt}\ l\ {\isacharasterisk}{\kern0pt}\ dist\ v\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ v{\isacharparenright}{\kern0pt}\ {\isacharless}{\kern0pt}\ eps\ {\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharminus}{\kern0pt}l{\isacharparenright}{\kern0pt}\ {\isasymor}\ eps\ {\isasymle}\ {\isadigit{0}}\ then\ {\isasymL}\isactrlsub b\ v\ else\ value{\isacharunderscore}{\kern0pt}iteration\ eps\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ v{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{termination}\isamarkupfalse%
-\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}relation\ {\isachardoublequoteopen}Wellfounded{\isachardot}{\kern0pt}measure\ term{\isacharunderscore}{\kern0pt}measure{\isachardoublequoteclose}{\isacharcomma}{\kern0pt}\ {\isacharparenleft}{\kern0pt}simp{\isacharsemicolon}{\kern0pt}\ fail{\isacharparenright}{\kern0pt}{\isacharcomma}{\kern0pt}\ cases\ {\isachardoublequoteopen}l\ {\isacharequal}{\kern0pt}\ {\isadigit{0}}{\isachardoublequoteclose}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{case}\isamarkupfalse%
-\ False\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\ eps\ v\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\ h{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymnot}\ {\isacharparenleft}{\kern0pt}{\isadigit{2}}\ {\isacharasterisk}{\kern0pt}\ l\ {\isacharasterisk}{\kern0pt}\ dist\ v\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ v{\isacharparenright}{\kern0pt}\ {\isacharless}{\kern0pt}\ eps\ {\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}\ {\isacharminus}{\kern0pt}\ l{\isacharparenright}{\kern0pt}\ {\isasymor}\ eps\ {\isasymle}\ {\isadigit{0}}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}eps{\isacharcomma}{\kern0pt}\ {\isasymL}\isactrlsub b\ v{\isacharparenright}{\kern0pt}{\isacharcomma}{\kern0pt}\ eps{\isacharcomma}{\kern0pt}\ v{\isacharparenright}{\kern0pt}\ {\isasymin}\ Wellfounded{\isachardot}{\kern0pt}measure\ term{\isacharunderscore}{\kern0pt}measure{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ gt{\isacharunderscore}{\kern0pt}zero{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}l\ {\isasymnoteq}\ {\isadigit{0}}{\isachardoublequoteclose}\ {\isachardoublequoteopen}eps\ {\isachargreater}{\kern0pt}\ {\isadigit{0}}{\isachardoublequoteclose}\ \isakeyword{and}\ dist{\isacharunderscore}{\kern0pt}ge{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}eps\ {\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}\ {\isacharminus}{\kern0pt}\ l{\isacharparenright}{\kern0pt}\ {\isasymle}\ dist\ v\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ v{\isacharparenright}{\kern0pt}\ {\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isadigit{2}}\ {\isacharasterisk}{\kern0pt}\ l{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ h\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ algebra{\isacharunderscore}{\kern0pt}simps{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ v{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}opt{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}v\ {\isasymnoteq}\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ h\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ force\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}log\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}\ {\isacharslash}{\kern0pt}\ l{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}eps\ {\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}\ {\isacharminus}{\kern0pt}\ l{\isacharparenright}{\kern0pt}\ {\isacharslash}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isadigit{8}}\ {\isacharasterisk}{\kern0pt}\ l{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharless}{\kern0pt}\ log\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}\ {\isacharslash}{\kern0pt}\ l{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}dist\ v\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}intro\ log{\isacharunderscore}{\kern0pt}less{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isadigit{1}}\ {\isacharless}{\kern0pt}\ {\isadigit{1}}\ {\isacharslash}{\kern0pt}\ l{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ mult{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}less{\isacharunderscore}{\kern0pt}div{\isacharunderscore}{\kern0pt}pos\ intro{\isacharcolon}{\kern0pt}\ neq{\isacharunderscore}{\kern0pt}le{\isacharunderscore}{\kern0pt}trans{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isadigit{0}}\ {\isacharless}{\kern0pt}\ eps\ {\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}\ {\isacharminus}{\kern0pt}\ l{\isacharparenright}{\kern0pt}\ {\isacharslash}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isadigit{8}}\ {\isacharasterisk}{\kern0pt}\ l{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ mult{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}less{\isacharunderscore}{\kern0pt}div{\isacharunderscore}{\kern0pt}pos\ intro{\isacharcolon}{\kern0pt}\ neq{\isacharunderscore}{\kern0pt}le{\isacharunderscore}{\kern0pt}trans{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}eps\ {\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}\ {\isacharminus}{\kern0pt}\ l{\isacharparenright}{\kern0pt}\ {\isacharslash}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isadigit{8}}\ {\isacharasterisk}{\kern0pt}\ l{\isacharparenright}{\kern0pt}\ {\isacharless}{\kern0pt}\ dist\ v\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt{\isachardoublequoteclose}\ \isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ dist{\isacharunderscore}{\kern0pt}pos{\isacharunderscore}{\kern0pt}lt{\isacharbrackleft}{\kern0pt}OF\ v{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}opt{\isacharbrackright}{\kern0pt}\ dist{\isacharunderscore}{\kern0pt}{\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}lt{\isacharunderscore}{\kern0pt}dist{\isacharunderscore}{\kern0pt}opt{\isacharbrackleft}{\kern0pt}of\ v{\isacharbrackright}{\kern0pt}\ gt{\isacharunderscore}{\kern0pt}zero\ zero{\isacharunderscore}{\kern0pt}le{\isacharunderscore}{\kern0pt}disc\ \isanewline
-\ \ \ \ \ \ \ \ \ \ mult{\isacharunderscore}{\kern0pt}strict{\isacharunderscore}{\kern0pt}left{\isacharunderscore}{\kern0pt}mono{\isacharbrackleft}{\kern0pt}of\ {\isachardoublequoteopen}dist\ v\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ v{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isadigit{4}}\ {\isacharasterisk}{\kern0pt}\ dist\ v\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ l{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}intro\ mult{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}div{\isacharunderscore}{\kern0pt}pos{\isacharunderscore}{\kern0pt}less\ le{\isacharunderscore}{\kern0pt}less{\isacharunderscore}{\kern0pt}trans{\isacharbrackleft}{\kern0pt}OF\ dist{\isacharunderscore}{\kern0pt}ge{\isacharbrackright}{\kern0pt}{\isacharcomma}{\kern0pt}\ argo{\isacharplus}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ vi{\isacharunderscore}{\kern0pt}rel{\isacharunderscore}{\kern0pt}dec\ h\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\isacommand{qed}\isamarkupfalse%
-\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\begin{isamarkuptext}%
-The distance between an estimate for the value and the optimal value can be bounded with respect to 
-the distance between the estimate and the result of applying it to \isa{{\isasymL}\isactrlsub b}%
-\end{isamarkuptext}\isamarkuptrue%
-\isacommand{lemma}\isamarkupfalse%
-\ contraction{\isacharunderscore}{\kern0pt}{\isasymL}{\isacharunderscore}{\kern0pt}dist{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isadigit{1}}\ {\isacharminus}{\kern0pt}\ l{\isacharparenright}{\kern0pt}\ {\isacharasterisk}{\kern0pt}\ dist\ v\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt\ {\isasymle}\ dist\ v\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ v{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ contraction{\isacharunderscore}{\kern0pt}dist\ contraction{\isacharunderscore}{\kern0pt}{\isasymL}\ disc{\isacharunderscore}{\kern0pt}lt{\isacharunderscore}{\kern0pt}one\ zero{\isacharunderscore}{\kern0pt}le{\isacharunderscore}{\kern0pt}disc\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ fastforce%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ dist{\isacharunderscore}{\kern0pt}{\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}opt{\isacharunderscore}{\kern0pt}eps{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}eps\ {\isachargreater}{\kern0pt}\ {\isadigit{0}}{\isachardoublequoteclose}\ {\isachardoublequoteopen}{\isadigit{2}}\ {\isacharasterisk}{\kern0pt}\ l\ {\isacharasterisk}{\kern0pt}\ dist\ v\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ v{\isacharparenright}{\kern0pt}\ {\isacharless}{\kern0pt}\ eps\ {\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharminus}{\kern0pt}l{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}dist\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ v{\isacharparenright}{\kern0pt}\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt\ {\isacharless}{\kern0pt}\ eps\ {\isacharslash}{\kern0pt}\ {\isadigit{2}}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}dist\ v\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt\ {\isasymle}\ dist\ v\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ v{\isacharparenright}{\kern0pt}\ {\isacharslash}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}\ {\isacharminus}{\kern0pt}\ l{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ contraction{\isacharunderscore}{\kern0pt}{\isasymL}{\isacharunderscore}{\kern0pt}dist\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ mult{\isachardot}{\kern0pt}commute\ pos{\isacharunderscore}{\kern0pt}le{\isacharunderscore}{\kern0pt}divide{\isacharunderscore}{\kern0pt}eq{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isadigit{2}}\ {\isacharasterisk}{\kern0pt}\ l\ {\isacharasterisk}{\kern0pt}\ dist\ v\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt\ {\isasymle}\ {\isadigit{2}}\ {\isacharasterisk}{\kern0pt}\ l\ {\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}dist\ v\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ v{\isacharparenright}{\kern0pt}\ {\isacharslash}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}\ {\isacharminus}{\kern0pt}\ l{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ contraction{\isacharunderscore}{\kern0pt}{\isasymL}{\isacharunderscore}{\kern0pt}dist\ assms\ mult{\isacharunderscore}{\kern0pt}le{\isacharunderscore}{\kern0pt}cancel{\isacharunderscore}{\kern0pt}left{\isacharunderscore}{\kern0pt}pos{\isacharbrackleft}{\kern0pt}of\ {\isachardoublequoteopen}{\isadigit{2}}\ {\isacharasterisk}{\kern0pt}\ l{\isachardoublequoteclose}{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}fastforce\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ mult{\isacharunderscore}{\kern0pt}left{\isacharunderscore}{\kern0pt}mono{\isacharbrackleft}{\kern0pt}of\ {\isacharunderscore}{\kern0pt}\ {\isacharunderscore}{\kern0pt}\ {\isachardoublequoteopen}{\isadigit{2}}\ {\isacharasterisk}{\kern0pt}\ l{\isachardoublequoteclose}{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isadigit{2}}\ {\isacharasterisk}{\kern0pt}\ l\ {\isacharasterisk}{\kern0pt}\ dist\ v\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt\ {\isacharless}{\kern0pt}\ eps{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ assms{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}\ pos{\isacharunderscore}{\kern0pt}divide{\isacharunderscore}{\kern0pt}less{\isacharunderscore}{\kern0pt}eq\ intro{\isacharcolon}{\kern0pt}\ order{\isachardot}{\kern0pt}strict{\isacharunderscore}{\kern0pt}trans{\isadigit{1}}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}dist\ v\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt\ {\isacharasterisk}{\kern0pt}\ l\ {\isacharless}{\kern0pt}\ eps\ {\isacharslash}{\kern0pt}\ {\isadigit{2}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ argo\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}l\ {\isacharasterisk}{\kern0pt}\ dist\ v\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt\ {\isacharless}{\kern0pt}\ eps\ {\isacharslash}{\kern0pt}\ {\isadigit{2}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ algebra{\isacharunderscore}{\kern0pt}simps{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}dist\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ v{\isacharparenright}{\kern0pt}\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt\ {\isacharless}{\kern0pt}\ eps\ {\isacharslash}{\kern0pt}\ {\isadigit{2}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ contraction{\isacharunderscore}{\kern0pt}{\isasymL}{\isacharbrackleft}{\kern0pt}of\ v\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt{\isacharbrackright}{\kern0pt}\ \isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\begin{isamarkuptext}%
-The estimates above allow to give a bound on the error of \isa{value{\isacharunderscore}{\kern0pt}iteration}.%
-\end{isamarkuptext}\isamarkuptrue%
-\isacommand{declare}\isamarkupfalse%
-\ value{\isacharunderscore}{\kern0pt}iteration{\isachardot}{\kern0pt}simps{\isacharbrackleft}{\kern0pt}simp\ del{\isacharbrackright}{\kern0pt}\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ value{\isacharunderscore}{\kern0pt}iteration{\isacharunderscore}{\kern0pt}error{\isacharcolon}{\kern0pt}\ \isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}eps\ {\isachargreater}{\kern0pt}\ {\isadigit{0}}{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}dist\ {\isacharparenleft}{\kern0pt}value{\isacharunderscore}{\kern0pt}iteration\ eps\ v{\isacharparenright}{\kern0pt}\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt\ {\isacharless}{\kern0pt}\ eps\ {\isacharslash}{\kern0pt}\ {\isadigit{2}}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ assms\ dist{\isacharunderscore}{\kern0pt}{\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}opt{\isacharunderscore}{\kern0pt}eps\ value{\isacharunderscore}{\kern0pt}iteration{\isachardot}{\kern0pt}simps\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}induction\ eps\ v\ rule{\isacharcolon}{\kern0pt}\ value{\isacharunderscore}{\kern0pt}iteration{\isachardot}{\kern0pt}induct{\isacharparenright}{\kern0pt}\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\begin{isamarkuptext}%
-After the value iteration terminates, one can easily obtain a stationary deterministic 
-epsilon-optimal policy.
-
-Such a policy does not exist in general, attainment of the supremum in \isa{{\isasymL}\isactrlsub b} is required.%
-\end{isamarkuptext}\isamarkuptrue%
-\isacommand{definition}\isamarkupfalse%
-\ {\isachardoublequoteopen}find{\isacharunderscore}{\kern0pt}policy\ {\isacharparenleft}{\kern0pt}v\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isacharprime}{\kern0pt}s\ {\isasymRightarrow}\isactrlsub b\ real{\isacharparenright}{\kern0pt}\ s\ {\isacharequal}{\kern0pt}\ arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}on\ {\isacharparenleft}{\kern0pt}{\isasymlambda}a{\isachardot}{\kern0pt}\ L\isactrlsub a\ a\ v\ s{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}A\ s{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ {\isachardoublequoteopen}vi{\isacharunderscore}{\kern0pt}policy\ eps\ v\ {\isacharequal}{\kern0pt}\ find{\isacharunderscore}{\kern0pt}policy\ {\isacharparenleft}{\kern0pt}value{\isacharunderscore}{\kern0pt}iteration\ eps\ v{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}%
-\begin{isamarkuptext}%
-We formalize the attainment of the supremum using a predicate \isa{has{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max}.%
-\end{isamarkuptext}\isamarkuptrue%
-\isacommand{abbreviation}\isamarkupfalse%
-\ {\isachardoublequoteopen}vi\ u\ n\ {\isasymequiv}\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ {\isacharcircum}{\kern0pt}{\isacharcircum}{\kern0pt}n{\isacharparenright}{\kern0pt}\ u{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ {\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}iter{\isacharunderscore}{\kern0pt}mono{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}u\ {\isasymle}\ v{\isachardoublequoteclose}\ \isakeyword{shows}\ {\isachardoublequoteopen}vi\ u\ n\ {\isasymle}\ vi\ v\ n{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ assms\ {\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}mono\ \isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}induction\ n{\isacharparenright}{\kern0pt}\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ \isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}vi\ v\ {\isacharparenleft}{\kern0pt}Suc\ n{\isacharparenright}{\kern0pt}\ {\isasymle}\ vi\ v\ n{\isachardoublequoteclose}\ \isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}vi\ v\ {\isacharparenleft}{\kern0pt}Suc\ n\ {\isacharplus}{\kern0pt}\ m{\isacharparenright}{\kern0pt}\ {\isasymle}\ vi\ v\ {\isacharparenleft}{\kern0pt}n\ {\isacharplus}{\kern0pt}\ m{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}vi\ v\ {\isacharparenleft}{\kern0pt}Suc\ n\ {\isacharplus}{\kern0pt}\ m{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ vi\ {\isacharparenleft}{\kern0pt}vi\ v\ {\isacharparenleft}{\kern0pt}Suc\ n{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ m{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ Groups{\isachardot}{\kern0pt}add{\isacharunderscore}{\kern0pt}ac{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}\ funpow{\isacharunderscore}{\kern0pt}add\ funpow{\isacharunderscore}{\kern0pt}swap{\isadigit{1}}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}\ {\isasymle}\ vi\ {\isacharparenleft}{\kern0pt}vi\ v\ n{\isacharparenright}{\kern0pt}\ m{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}iter{\isacharunderscore}{\kern0pt}mono{\isacharbrackleft}{\kern0pt}OF\ assms{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}\ {\isacharequal}{\kern0pt}\ vi\ v\ {\isacharparenleft}{\kern0pt}n\ {\isacharplus}{\kern0pt}\ m{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ add{\isachardot}{\kern0pt}commute\ funpow{\isacharunderscore}{\kern0pt}add{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{finally}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\ \isacommand{{\isachardot}{\kern0pt}}\isamarkupfalse%
-\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ \isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}vi\ v\ n\ {\isasymle}\ vi\ v\ {\isacharparenleft}{\kern0pt}Suc\ n{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}vi\ v\ {\isacharparenleft}{\kern0pt}n\ {\isacharplus}{\kern0pt}\ m{\isacharparenright}{\kern0pt}\ {\isasymle}\ vi\ v\ {\isacharparenleft}{\kern0pt}Suc\ n\ {\isacharplus}{\kern0pt}\ m{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}vi\ v\ {\isacharparenleft}{\kern0pt}n\ {\isacharplus}{\kern0pt}\ m{\isacharparenright}{\kern0pt}\ {\isasymle}\ vi\ {\isacharparenleft}{\kern0pt}vi\ v\ n{\isacharparenright}{\kern0pt}\ m{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ Groups{\isachardot}{\kern0pt}add{\isacharunderscore}{\kern0pt}ac{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}\ funpow{\isacharunderscore}{\kern0pt}add\ funpow{\isacharunderscore}{\kern0pt}swap{\isadigit{1}}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isasymle}\ vi\ v\ {\isacharparenleft}{\kern0pt}Suc\ n\ {\isacharplus}{\kern0pt}\ m{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}iter{\isacharunderscore}{\kern0pt}mono{\isacharbrackleft}{\kern0pt}OF\ assms{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp\ only{\isacharcolon}{\kern0pt}\ add{\isachardot}{\kern0pt}commute\ funpow{\isacharunderscore}{\kern0pt}add\ o{\isacharunderscore}{\kern0pt}apply{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{finally}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\ \isacommand{{\isachardot}{\kern0pt}}\isamarkupfalse%
-\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ {\isachardoublequoteopen}vi\ v\ {\isasymlonglonglongrightarrow}\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ {\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}lim\isacommand{{\isachardot}{\kern0pt}}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isasymlambda}n{\isachardot}{\kern0pt}\ dist\ {\isacharparenleft}{\kern0pt}vi\ v\ {\isacharparenleft}{\kern0pt}Suc\ n{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}vi\ v\ n{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymlonglonglongrightarrow}\ {\isadigit{0}}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ thm{\isacharunderscore}{\kern0pt}{\isadigit{6}}{\isacharunderscore}{\kern0pt}{\isadigit{3}}{\isacharunderscore}{\kern0pt}{\isadigit{1}}{\isacharunderscore}{\kern0pt}b{\isacharunderscore}{\kern0pt}aux{\isacharbrackleft}{\kern0pt}of\ v{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp\ only{\isacharcolon}{\kern0pt}\ dist{\isacharunderscore}{\kern0pt}commute{\isacharbrackleft}{\kern0pt}of\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ {\isacharcircum}{\kern0pt}{\isacharcircum}{\kern0pt}\ Suc\ {\isacharunderscore}{\kern0pt}{\isacharparenright}{\kern0pt}\ v{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isanewline
-\isacommand{end}\isamarkupfalse%
-\isanewline
-\isanewline
-\isacommand{context}\isamarkupfalse%
-\ MDP{\isacharunderscore}{\kern0pt}att{\isacharunderscore}{\kern0pt}{\isasymL}\ \isanewline
-\isakeyword{begin}%
-\begin{isamarkuptext}%
-The error of the resulting policy is bounded by the distance from its value to the value computed 
-by the value iteration plus the error in the value iteration itself.
-We show that both are less than \isa{eps\ {\isacharslash}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}{\isacharprime}{\kern0pt}b{\isacharparenright}{\kern0pt}} when the algorithm terminates.%
-\end{isamarkuptext}\isamarkuptrue%
-\isacommand{lemma}\isamarkupfalse%
-\ find{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}error{\isacharunderscore}{\kern0pt}bound{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}eps\ {\isachargreater}{\kern0pt}\ {\isadigit{0}}{\isachardoublequoteclose}\ {\isachardoublequoteopen}{\isadigit{2}}\ {\isacharasterisk}{\kern0pt}\ l\ {\isacharasterisk}{\kern0pt}\ dist\ v\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ v{\isacharparenright}{\kern0pt}\ {\isacharless}{\kern0pt}\ eps\ {\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharminus}{\kern0pt}l{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}dist\ {\isacharparenleft}{\kern0pt}{\isasymnu}\isactrlsub b\ {\isacharparenleft}{\kern0pt}mk{\isacharunderscore}{\kern0pt}stationary{\isacharunderscore}{\kern0pt}det\ {\isacharparenleft}{\kern0pt}find{\isacharunderscore}{\kern0pt}policy\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ v{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt\ {\isacharless}{\kern0pt}\ eps{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}d\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}mk{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\ {\isacharparenleft}{\kern0pt}find{\isacharunderscore}{\kern0pt}policy\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ v{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}p\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}mk{\isacharunderscore}{\kern0pt}stationary\ {\isacharquery}{\kern0pt}d{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ L{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}{\isasymL}\isactrlsub b{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}L\ {\isacharparenleft}{\kern0pt}mk{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\ {\isacharparenleft}{\kern0pt}find{\isacharunderscore}{\kern0pt}policy\ v{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ v\ {\isacharequal}{\kern0pt}\ {\isasymL}\isactrlsub b\ v{\isachardoublequoteclose}\ \isakeyword{for}\ v\isanewline
-\ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ find{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}intro\ antisym{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}L\ {\isacharparenleft}{\kern0pt}mk{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\ {\isacharparenleft}{\kern0pt}{\isasymlambda}s{\isachardot}{\kern0pt}\ arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}on\ {\isacharparenleft}{\kern0pt}{\isasymlambda}a{\isachardot}{\kern0pt}\ L\isactrlsub a\ a\ v\ s{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}A\ s{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ v\ {\isasymle}\ {\isasymL}\isactrlsub b\ v{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ Sup{\isacharunderscore}{\kern0pt}att\ has{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max\ abs{\isacharunderscore}{\kern0pt}L{\isacharunderscore}{\kern0pt}le\isanewline
-\ \ \ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ {\isasymL}\isactrlsub b{\isachardot}{\kern0pt}rep{\isacharunderscore}{\kern0pt}eq\ {\isasymL}{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}SUP{\isacharunderscore}{\kern0pt}det\ less{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}bfun{\isacharunderscore}{\kern0pt}def\ arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}on{\isacharunderscore}{\kern0pt}def\ is{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det{\isacharunderscore}{\kern0pt}def\ max{\isacharunderscore}{\kern0pt}L{\isacharunderscore}{\kern0pt}ex{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ cSUP{\isacharunderscore}{\kern0pt}upper\ bounded{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}bdd{\isacharunderscore}{\kern0pt}above\ boundedI{\isacharbrackleft}{\kern0pt}of\ {\isacharunderscore}{\kern0pt}\ {\isachardoublequoteopen}r\isactrlsub M\ {\isacharplus}{\kern0pt}\ l\ {\isacharasterisk}{\kern0pt}\ norm\ v{\isachardoublequoteclose}{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymL}\isactrlsub b\ v\ {\isasymle}\ L\ {\isacharparenleft}{\kern0pt}mk{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\ {\isacharparenleft}{\kern0pt}{\isasymlambda}s{\isachardot}{\kern0pt}\ arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}on\ {\isacharparenleft}{\kern0pt}{\isasymlambda}a{\isachardot}{\kern0pt}\ L\isactrlsub a\ a\ v\ s{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}A\ s{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ v{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ less{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}bfun{\isacharunderscore}{\kern0pt}def\ {\isasymL}\isactrlsub b{\isachardot}{\kern0pt}rep{\isacharunderscore}{\kern0pt}eq\ {\isasymL}{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}SUP{\isacharunderscore}{\kern0pt}det\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ Sup{\isacharunderscore}{\kern0pt}att\ ex{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ cSUP{\isacharunderscore}{\kern0pt}least\ app{\isacharunderscore}{\kern0pt}arg{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}ge\ simp{\isacharcolon}{\kern0pt}\ L{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}L\isactrlsub a{\isacharunderscore}{\kern0pt}det\ max{\isacharunderscore}{\kern0pt}L{\isacharunderscore}{\kern0pt}ex{\isacharunderscore}{\kern0pt}def\ is{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}dist\ {\isacharparenleft}{\kern0pt}{\isasymnu}\isactrlsub b\ {\isacharquery}{\kern0pt}p{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ v{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ dist\ {\isacharparenleft}{\kern0pt}L\ {\isacharquery}{\kern0pt}d\ {\isacharparenleft}{\kern0pt}{\isasymnu}\isactrlsub b\ {\isacharquery}{\kern0pt}p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ v{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ L{\isacharunderscore}{\kern0pt}{\isasymnu}{\isacharunderscore}{\kern0pt}fix\ \isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ force\isanewline
-\ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isasymle}\ dist\ {\isacharparenleft}{\kern0pt}L\ {\isacharquery}{\kern0pt}d\ {\isacharparenleft}{\kern0pt}{\isasymnu}\isactrlsub b\ {\isacharquery}{\kern0pt}p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ v{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharplus}{\kern0pt}\ dist\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ v{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ v{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ dist{\isacharunderscore}{\kern0pt}triangle\ \isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isacharequal}{\kern0pt}\ dist\ {\isacharparenleft}{\kern0pt}L\ {\isacharquery}{\kern0pt}d\ {\isacharparenleft}{\kern0pt}{\isasymnu}\isactrlsub b\ {\isacharquery}{\kern0pt}p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}L\ {\isacharquery}{\kern0pt}d\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ v{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharplus}{\kern0pt}\ dist\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ v{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ v{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ L{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}{\isasymL}\isactrlsub b{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isasymle}\ l\ {\isacharasterisk}{\kern0pt}\ \ dist\ {\isacharparenleft}{\kern0pt}{\isasymnu}\isactrlsub b\ {\isacharquery}{\kern0pt}p{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ v{\isacharparenright}{\kern0pt}\ {\isacharplus}{\kern0pt}\ l\ {\isacharasterisk}{\kern0pt}\ dist\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ v{\isacharparenright}{\kern0pt}\ v{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ contraction{\isacharunderscore}{\kern0pt}{\isasymL}\ contraction{\isacharunderscore}{\kern0pt}L\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}fastforce\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ add{\isacharunderscore}{\kern0pt}mono{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{finally}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ aux{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}dist\ {\isacharparenleft}{\kern0pt}{\isasymnu}\isactrlsub b\ {\isacharquery}{\kern0pt}p{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ v{\isacharparenright}{\kern0pt}\ {\isasymle}\ l\ {\isacharasterisk}{\kern0pt}\ dist\ {\isacharparenleft}{\kern0pt}{\isasymnu}\isactrlsub b\ {\isacharquery}{\kern0pt}p{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ v{\isacharparenright}{\kern0pt}\ {\isacharplus}{\kern0pt}\ l\ {\isacharasterisk}{\kern0pt}\ dist\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ v{\isacharparenright}{\kern0pt}\ v{\isachardoublequoteclose}\ \isacommand{{\isachardot}{\kern0pt}}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}dist\ {\isacharparenleft}{\kern0pt}{\isasymnu}\isactrlsub b\ {\isacharquery}{\kern0pt}p{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ v{\isacharparenright}{\kern0pt}\ {\isacharminus}{\kern0pt}\ l\ {\isacharasterisk}{\kern0pt}\ dist\ {\isacharparenleft}{\kern0pt}{\isasymnu}\isactrlsub b\ {\isacharquery}{\kern0pt}p{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ v{\isacharparenright}{\kern0pt}\ {\isasymle}\ l\ {\isacharasterisk}{\kern0pt}\ dist\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ v{\isacharparenright}{\kern0pt}\ v{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}dist\ {\isacharparenleft}{\kern0pt}{\isasymnu}\isactrlsub b\ {\isacharquery}{\kern0pt}p{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ v{\isacharparenright}{\kern0pt}\ {\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}\ {\isacharminus}{\kern0pt}\ l{\isacharparenright}{\kern0pt}\ {\isasymle}\ l\ {\isacharasterisk}{\kern0pt}\ dist\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ v{\isacharparenright}{\kern0pt}\ v{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ argo\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ \ {\isachardoublequoteopen}{\isadigit{2}}\ {\isacharasterisk}{\kern0pt}\ dist\ {\isacharparenleft}{\kern0pt}{\isasymnu}\isactrlsub b\ {\isacharquery}{\kern0pt}p{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ v{\isacharparenright}{\kern0pt}\ {\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharminus}{\kern0pt}l{\isacharparenright}{\kern0pt}\ {\isasymle}\ {\isadigit{2}}\ {\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}l\ {\isacharasterisk}{\kern0pt}\ dist\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ v{\isacharparenright}{\kern0pt}\ v{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ zero{\isacharunderscore}{\kern0pt}le{\isacharunderscore}{\kern0pt}disc\ mult{\isacharunderscore}{\kern0pt}left{\isacharunderscore}{\kern0pt}mono\ \isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isasymle}\ eps\ {\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharminus}{\kern0pt}l{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assms\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ mult{\isacharunderscore}{\kern0pt}left{\isacharunderscore}{\kern0pt}mono\ simp{\isacharcolon}{\kern0pt}\ dist{\isacharunderscore}{\kern0pt}commute\ pos{\isacharunderscore}{\kern0pt}divide{\isacharunderscore}{\kern0pt}le{\isacharunderscore}{\kern0pt}eq{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{finally}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isadigit{2}}\ {\isacharasterisk}{\kern0pt}\ dist\ {\isacharparenleft}{\kern0pt}{\isasymnu}\isactrlsub b\ {\isacharquery}{\kern0pt}p{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ v{\isacharparenright}{\kern0pt}\ {\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}\ {\isacharminus}{\kern0pt}\ l{\isacharparenright}{\kern0pt}\ {\isasymle}\ eps\ {\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}\ {\isacharminus}{\kern0pt}\ l{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isacommand{{\isachardot}{\kern0pt}}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isadigit{2}}\ {\isacharasterisk}{\kern0pt}\ dist\ {\isacharparenleft}{\kern0pt}{\isasymnu}\isactrlsub b\ {\isacharquery}{\kern0pt}p{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ v{\isacharparenright}{\kern0pt}\ {\isasymle}\ eps{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ disc{\isacharunderscore}{\kern0pt}lt{\isacharunderscore}{\kern0pt}one\ mult{\isacharunderscore}{\kern0pt}right{\isacharunderscore}{\kern0pt}le{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}le\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isadigit{2}}\ {\isacharasterisk}{\kern0pt}\ dist\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ v{\isacharparenright}{\kern0pt}\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt\ {\isacharless}{\kern0pt}\ eps{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ dist{\isacharunderscore}{\kern0pt}{\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}opt{\isacharunderscore}{\kern0pt}eps\ assms\ \isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ fastforce\isanewline
-\ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}dist\ {\isacharparenleft}{\kern0pt}{\isasymnu}\isactrlsub b\ {\isacharquery}{\kern0pt}p{\isacharparenright}{\kern0pt}\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt\ {\isasymle}\ dist\ {\isacharparenleft}{\kern0pt}{\isasymnu}\isactrlsub b\ {\isacharquery}{\kern0pt}p{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ v{\isacharparenright}{\kern0pt}\ {\isacharplus}{\kern0pt}\ dist\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ v{\isacharparenright}{\kern0pt}\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ dist{\isacharunderscore}{\kern0pt}triangle\ \isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\ \ \isanewline
-\ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\ \isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ vi{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}opt{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}{\isadigit{0}}\ {\isacharless}{\kern0pt}\ eps{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}dist\ {\isacharparenleft}{\kern0pt}{\isasymnu}\isactrlsub b\ {\isacharparenleft}{\kern0pt}mk{\isacharunderscore}{\kern0pt}stationary{\isacharunderscore}{\kern0pt}det\ {\isacharparenleft}{\kern0pt}vi{\isacharunderscore}{\kern0pt}policy\ eps\ v{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt\ {\isacharless}{\kern0pt}\ eps{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ vi{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}def\ \isanewline
-\ \ \isacommand{using}\isamarkupfalse%
-\ assms\isanewline
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}induction\ eps\ v\ rule{\isacharcolon}{\kern0pt}\ value{\isacharunderscore}{\kern0pt}iteration{\isachardot}{\kern0pt}induct{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{case}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}\ v{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{then}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}case\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ find{\isacharunderscore}{\kern0pt}policy{\isacharunderscore}{\kern0pt}error{\isacharunderscore}{\kern0pt}bound\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}subst\ value{\isacharunderscore}{\kern0pt}iteration{\isachardot}{\kern0pt}simps{\isacharparenright}{\kern0pt}\ auto\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ lemma{\isacharunderscore}{\kern0pt}{\isadigit{6}}{\isacharunderscore}{\kern0pt}{\isadigit{3}}{\isacharunderscore}{\kern0pt}{\isadigit{1}}{\isacharunderscore}{\kern0pt}d{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}eps\ {\isachargreater}{\kern0pt}\ {\isadigit{0}}{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}{\isadigit{2}}\ {\isacharasterisk}{\kern0pt}\ l\ {\isacharasterisk}{\kern0pt}\ dist\ {\isacharparenleft}{\kern0pt}vi\ v\ {\isacharparenleft}{\kern0pt}Suc\ n{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}vi\ v\ n{\isacharparenright}{\kern0pt}\ {\isacharless}{\kern0pt}\ eps\ {\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharminus}{\kern0pt}l{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}dist\ {\isacharparenleft}{\kern0pt}vi\ v\ {\isacharparenleft}{\kern0pt}Suc\ n{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt\ {\isacharless}{\kern0pt}\ eps\ {\isacharslash}{\kern0pt}\ {\isadigit{2}}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ dist{\isacharunderscore}{\kern0pt}{\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}opt{\isacharunderscore}{\kern0pt}eps\ assms\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ dist{\isacharunderscore}{\kern0pt}commute{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{end}\isamarkupfalse%
-\isanewline
-\isanewline
-\isacommand{context}\isamarkupfalse%
-\ MDP{\isacharunderscore}{\kern0pt}act\ \isakeyword{begin}\isanewline
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ {\isachardoublequoteopen}find{\isacharunderscore}{\kern0pt}policy{\isacharprime}{\kern0pt}\ {\isacharparenleft}{\kern0pt}v\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isacharprime}{\kern0pt}s\ {\isasymRightarrow}\isactrlsub b\ real{\isacharparenright}{\kern0pt}\ s\ {\isacharequal}{\kern0pt}\ arb{\isacharunderscore}{\kern0pt}act\ {\isacharparenleft}{\kern0pt}opt{\isacharunderscore}{\kern0pt}acts\ v\ s{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ {\isachardoublequoteopen}vi{\isacharunderscore}{\kern0pt}policy{\isacharprime}{\kern0pt}\ eps\ v\ {\isacharequal}{\kern0pt}\ find{\isacharunderscore}{\kern0pt}policy{\isacharprime}{\kern0pt}\ {\isacharparenleft}{\kern0pt}value{\isacharunderscore}{\kern0pt}iteration\ eps\ v{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ find{\isacharunderscore}{\kern0pt}policy{\isacharprime}{\kern0pt}{\isacharunderscore}{\kern0pt}error{\isacharunderscore}{\kern0pt}bound{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}eps\ {\isachargreater}{\kern0pt}\ {\isadigit{0}}{\isachardoublequoteclose}\ {\isachardoublequoteopen}{\isadigit{2}}\ {\isacharasterisk}{\kern0pt}\ l\ {\isacharasterisk}{\kern0pt}\ dist\ v\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ v{\isacharparenright}{\kern0pt}\ {\isacharless}{\kern0pt}\ eps\ {\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharminus}{\kern0pt}l{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}dist\ {\isacharparenleft}{\kern0pt}{\isasymnu}\isactrlsub b\ {\isacharparenleft}{\kern0pt}mk{\isacharunderscore}{\kern0pt}stationary{\isacharunderscore}{\kern0pt}det\ {\isacharparenleft}{\kern0pt}find{\isacharunderscore}{\kern0pt}policy{\isacharprime}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ v{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt\ {\isacharless}{\kern0pt}\ eps{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}d\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}mk{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\ {\isacharparenleft}{\kern0pt}find{\isacharunderscore}{\kern0pt}policy{\isacharprime}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ v{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}p\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}mk{\isacharunderscore}{\kern0pt}stationary\ {\isacharquery}{\kern0pt}d{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ L{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}{\isasymL}\isactrlsub b{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}L\ {\isacharparenleft}{\kern0pt}mk{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}det\ {\isacharparenleft}{\kern0pt}find{\isacharunderscore}{\kern0pt}policy{\isacharprime}{\kern0pt}\ v{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ v\ {\isacharequal}{\kern0pt}\ {\isasymL}\isactrlsub b\ v{\isachardoublequoteclose}\ \isakeyword{for}\ v\isanewline
-\ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ find{\isacharunderscore}{\kern0pt}policy{\isacharprime}{\kern0pt}{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isasymnu}{\isacharunderscore}{\kern0pt}improving{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}{\isasymL}\isactrlsub b\ {\isasymnu}{\isacharunderscore}{\kern0pt}improving{\isacharunderscore}{\kern0pt}opt{\isacharunderscore}{\kern0pt}acts{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}dist\ {\isacharparenleft}{\kern0pt}{\isasymnu}\isactrlsub b\ {\isacharquery}{\kern0pt}p{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ v{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ dist\ {\isacharparenleft}{\kern0pt}L\ {\isacharquery}{\kern0pt}d\ {\isacharparenleft}{\kern0pt}{\isasymnu}\isactrlsub b\ {\isacharquery}{\kern0pt}p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ v{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ L{\isacharunderscore}{\kern0pt}{\isasymnu}{\isacharunderscore}{\kern0pt}fix\ \isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ force\isanewline
-\ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isasymle}\ dist\ {\isacharparenleft}{\kern0pt}L\ {\isacharquery}{\kern0pt}d\ {\isacharparenleft}{\kern0pt}{\isasymnu}\isactrlsub b\ {\isacharquery}{\kern0pt}p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ v{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharplus}{\kern0pt}\ dist\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ v{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ v{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ dist{\isacharunderscore}{\kern0pt}triangle\ \isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isacharequal}{\kern0pt}\ dist\ {\isacharparenleft}{\kern0pt}L\ {\isacharquery}{\kern0pt}d\ {\isacharparenleft}{\kern0pt}{\isasymnu}\isactrlsub b\ {\isacharquery}{\kern0pt}p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}L\ {\isacharquery}{\kern0pt}d\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ v{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharplus}{\kern0pt}\ dist\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ v{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ v{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ simp{\isacharcolon}{\kern0pt}\ L{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}{\isasymL}\isactrlsub b{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isasymle}\ l\ {\isacharasterisk}{\kern0pt}\ \ dist\ {\isacharparenleft}{\kern0pt}{\isasymnu}\isactrlsub b\ {\isacharquery}{\kern0pt}p{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ v{\isacharparenright}{\kern0pt}\ {\isacharplus}{\kern0pt}\ l\ {\isacharasterisk}{\kern0pt}\ dist\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ v{\isacharparenright}{\kern0pt}\ v{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ contraction{\isacharunderscore}{\kern0pt}{\isasymL}\ contraction{\isacharunderscore}{\kern0pt}L\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}fastforce\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ add{\isacharunderscore}{\kern0pt}mono{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{finally}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ aux{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}dist\ {\isacharparenleft}{\kern0pt}{\isasymnu}\isactrlsub b\ {\isacharquery}{\kern0pt}p{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ v{\isacharparenright}{\kern0pt}\ {\isasymle}\ l\ {\isacharasterisk}{\kern0pt}\ dist\ {\isacharparenleft}{\kern0pt}{\isasymnu}\isactrlsub b\ {\isacharquery}{\kern0pt}p{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ v{\isacharparenright}{\kern0pt}\ {\isacharplus}{\kern0pt}\ l\ {\isacharasterisk}{\kern0pt}\ dist\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ v{\isacharparenright}{\kern0pt}\ v{\isachardoublequoteclose}\ \isacommand{{\isachardot}{\kern0pt}}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}dist\ {\isacharparenleft}{\kern0pt}{\isasymnu}\isactrlsub b\ {\isacharquery}{\kern0pt}p{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ v{\isacharparenright}{\kern0pt}\ {\isacharminus}{\kern0pt}\ l\ {\isacharasterisk}{\kern0pt}\ dist\ {\isacharparenleft}{\kern0pt}{\isasymnu}\isactrlsub b\ {\isacharquery}{\kern0pt}p{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ v{\isacharparenright}{\kern0pt}\ {\isasymle}\ l\ {\isacharasterisk}{\kern0pt}\ dist\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ v{\isacharparenright}{\kern0pt}\ v{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}dist\ {\isacharparenleft}{\kern0pt}{\isasymnu}\isactrlsub b\ {\isacharquery}{\kern0pt}p{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ v{\isacharparenright}{\kern0pt}\ {\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}\ {\isacharminus}{\kern0pt}\ l{\isacharparenright}{\kern0pt}\ {\isasymle}\ l\ {\isacharasterisk}{\kern0pt}\ dist\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ v{\isacharparenright}{\kern0pt}\ v{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ argo\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ \ {\isachardoublequoteopen}{\isadigit{2}}\ {\isacharasterisk}{\kern0pt}\ dist\ {\isacharparenleft}{\kern0pt}{\isasymnu}\isactrlsub b\ {\isacharquery}{\kern0pt}p{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ v{\isacharparenright}{\kern0pt}\ {\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharminus}{\kern0pt}l{\isacharparenright}{\kern0pt}\ {\isasymle}\ {\isadigit{2}}\ {\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}l\ {\isacharasterisk}{\kern0pt}\ dist\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ v{\isacharparenright}{\kern0pt}\ v{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ zero{\isacharunderscore}{\kern0pt}le{\isacharunderscore}{\kern0pt}disc\ mult{\isacharunderscore}{\kern0pt}left{\isacharunderscore}{\kern0pt}mono\ \isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymdots}\ {\isasymle}\ eps\ {\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharminus}{\kern0pt}l{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assms\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto\ intro{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ mult{\isacharunderscore}{\kern0pt}left{\isacharunderscore}{\kern0pt}mono\ simp{\isacharcolon}{\kern0pt}\ dist{\isacharunderscore}{\kern0pt}commute\ pos{\isacharunderscore}{\kern0pt}divide{\isacharunderscore}{\kern0pt}le{\isacharunderscore}{\kern0pt}eq{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{finally}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isadigit{2}}\ {\isacharasterisk}{\kern0pt}\ dist\ {\isacharparenleft}{\kern0pt}{\isasymnu}\isactrlsub b\ {\isacharquery}{\kern0pt}p{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ v{\isacharparenright}{\kern0pt}\ {\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}\ {\isacharminus}{\kern0pt}\ l{\isacharparenright}{\kern0pt}\ {\isasymle}\ eps\ {\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}\ {\isacharminus}{\kern0pt}\ l{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isacommand{{\isachardot}{\kern0pt}}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isadigit{2}}\ {\isacharasterisk}{\kern0pt}\ dist\ {\isacharparenleft}{\kern0pt}{\isasymnu}\isactrlsub b\ {\isacharquery}{\kern0pt}p{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ v{\isacharparenright}{\kern0pt}\ {\isasymle}\ eps{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ disc{\isacharunderscore}{\kern0pt}lt{\isacharunderscore}{\kern0pt}one\ mult{\isacharunderscore}{\kern0pt}right{\isacharunderscore}{\kern0pt}le{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}le\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isadigit{2}}\ {\isacharasterisk}{\kern0pt}\ dist\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ v{\isacharparenright}{\kern0pt}\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt\ {\isacharless}{\kern0pt}\ eps{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ dist{\isacharunderscore}{\kern0pt}{\isasymL}\isactrlsub b{\isacharunderscore}{\kern0pt}opt{\isacharunderscore}{\kern0pt}eps\ assms\ \isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ fastforce\isanewline
-\ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}dist\ {\isacharparenleft}{\kern0pt}{\isasymnu}\isactrlsub b\ {\isacharquery}{\kern0pt}p{\isacharparenright}{\kern0pt}\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt\ {\isasymle}\ dist\ {\isacharparenleft}{\kern0pt}{\isasymnu}\isactrlsub b\ {\isacharquery}{\kern0pt}p{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ v{\isacharparenright}{\kern0pt}\ {\isacharplus}{\kern0pt}\ dist\ {\isacharparenleft}{\kern0pt}{\isasymL}\isactrlsub b\ v{\isacharparenright}{\kern0pt}\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ dist{\isacharunderscore}{\kern0pt}triangle\ \isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\ \ \isanewline
-\ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\ \isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ vi{\isacharunderscore}{\kern0pt}policy{\isacharprime}{\kern0pt}{\isacharunderscore}{\kern0pt}opt{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}eps\ {\isachargreater}{\kern0pt}\ {\isadigit{0}}{\isachardoublequoteclose}\ {\isachardoublequoteopen}l\ {\isachargreater}{\kern0pt}\ {\isadigit{0}}{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}dist\ {\isacharparenleft}{\kern0pt}{\isasymnu}\isactrlsub b\ {\isacharparenleft}{\kern0pt}mk{\isacharunderscore}{\kern0pt}stationary{\isacharunderscore}{\kern0pt}det\ {\isacharparenleft}{\kern0pt}vi{\isacharunderscore}{\kern0pt}policy{\isacharprime}{\kern0pt}\ eps\ v{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymnu}\isactrlsub b{\isacharunderscore}{\kern0pt}opt\ {\isacharless}{\kern0pt}\ eps{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ vi{\isacharunderscore}{\kern0pt}policy{\isacharprime}{\kern0pt}{\isacharunderscore}{\kern0pt}def\ \isanewline
-\ \ \isacommand{using}\isamarkupfalse%
-\ assms\isanewline
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}induction\ eps\ v\ rule{\isacharcolon}{\kern0pt}\ value{\isacharunderscore}{\kern0pt}iteration{\isachardot}{\kern0pt}induct{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{case}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}{\isadigit{1}}\ v{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{then}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}case\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ find{\isacharunderscore}{\kern0pt}policy{\isacharprime}{\kern0pt}{\isacharunderscore}{\kern0pt}error{\isacharunderscore}{\kern0pt}bound\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}subst\ value{\isacharunderscore}{\kern0pt}iteration{\isachardot}{\kern0pt}simps{\isacharparenright}{\kern0pt}\ auto\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{end}\isamarkupfalse%
-\isanewline
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{end}\isamarkupfalse%
-%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\end{isabellebody}%
-\endinput
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diff --git a/thys/MDP-Algorithms/output/outline/comment.sty b/thys/MDP-Algorithms/output/outline/comment.sty
deleted file mode 100644
--- a/thys/MDP-Algorithms/output/outline/comment.sty
+++ /dev/null
@@ -1,278 +0,0 @@
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-% Comment.sty   version 3.6, October 1999
-%
-% Purpose:
-% selectively in/exclude pieces of text: the user can define new
-% comment versions, and each is controlled separately.
-% Special comments can be defined where the user specifies the
-% action that is to be taken with each comment line.
-%
-% Author
-%    Victor Eijkhout
-%    Department of Computer Science
-%    University of Tennessee
-%    107 Ayres Hall
-%    Knoxville TN 37996
-%    USA
-%
-%    victor@eijkhout.net
-%
-% This program is free software; you can redistribute it and/or
-% modify it under the terms of the GNU General Public License
-% as published by the Free Software Foundation; either version 2
-% of the License, or (at your option) any later version.
-% 
-% This program is distributed in the hope that it will be useful,
-% but WITHOUT ANY WARRANTY; without even the implied warranty of
-% MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
-% GNU General Public License for more details.
-%
-% For a copy of the GNU General Public License, write to the 
-% Free Software Foundation, Inc.,
-% 59 Temple Place - Suite 330, Boston, MA  02111-1307, USA,
-% or find it on the net, for instance at
-% http://www.gnu.org/copyleft/gpl.html
-%
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-% This style can be used with plain TeX or LaTeX, and probably
-% most other packages too.
-%
-% Usage: all text included between
-%    \comment ... \endcomment
-% or \begin{comment} ... \end{comment}
-% is discarded. 
-%
-% The opening and closing commands should appear on a line
-% of their own. No starting spaces, nothing after it.
-% This environment should work with arbitrary amounts
-% of comment, and the comment can be arbitrary text.
-%
-% Other `comment' environments are defined by
-% and are selected/deselected with
-% \includecomment{versiona}
-% \excludecoment{versionb}
-%
-% These environments are used as
-% \versiona ... \endversiona
-% or \begin{versiona} ... \end{versiona}
-% with the opening and closing commands again on a line of 
-% their own.
-%
-% LaTeX users note: for an included comment, the
-% \begin and \end lines act as if they don't exist.
-% In particular, they don't imply grouping, so assignments 
-% &c are not local.
-%
-% Special comments are defined as
-% \specialcomment{name}{before commands}{after commands}
-% where the second and third arguments are executed before
-% and after each comment block. You can use this for global
-% formatting commands.
-% To keep definitions &c local, you can include \begingroup
-% in the `before commands' and \endgroup in the `after commands'.
-% ex:
-% \specialcomment{smalltt}
-%     {\begingroup\ttfamily\footnotesize}{\endgroup}
-% You do *not* have to do an additional
-% \includecomment{smalltt}
-% To remove 'smalltt' blocks, give \excludecomment{smalltt}
-% after the definition.
-%
-% Processing comments can apply processing to each line.
-% \processcomment{name}{each-line commands}%
-%    {before commands}{after commands}
-% By defining a control sequence 
-% \def\Thiscomment##1{...} in the before commands the user can
-% specify what is to be done with each comment line.
-% BUG this does not work quite yet BUG
-%
-% Trick for short in/exclude macros (such as \maybe{this snippet}):
-%\includecomment{cond}
-%\newcommand{\maybe}[1]{}
-%\begin{cond}
-%\renewcommand{\maybe}[1]{#1}
-%\end{cond}
-%
-% Basic approach of the implementation:
-% to comment something out, scoop up  every line in verbatim mode
-% as macro argument, then throw it away.
-% For inclusions, in LaTeX the block is written out to
-% a file \CommentCutFile (default "comment.cut"), which is
-% then included.
-% In plain TeX (and other formats) both the opening and
-% closing comands are defined as noop.
-%
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-% Changes in version 3.1
-% - updated author's address
-% - cleaned up some code
-% - trailing contents on \begin{env} line is always discarded
-%  even if you've done \includecomment{env}
-% - comments no longer define grouping!! you can even
-%   \includecomment{env}
-%   \begin{env}
-%   \begin{itemize}
-%   \end{env}
-%  Isn't that something ...
-% - included comments are written to file and input again.
-% Changes in 3.2
-% - \specialcomment brought up to date (thanks to Ivo Welch).
-% Changes in 3.3
-% - updated author's address again
-% - parametrised \CommentCutFile
-% Changes in 3.4
-% - added GNU public license
-% - added \processcomment, because Ivo's fix (above) brought an
-%   inconsistency to light.
-% Changes in 3.5
-% - corrected typo in header.
-% - changed author email
-% - corrected \specialcomment yet again.
-% - fixed excludecomment of an earlier defined environment.
-% Changes in 3.6
-% - The 'cut' file is now written more verbatim, using \meaning;
-%   some people reported having trouble with ISO latin 1, or umlaute.sty.
-% - removed some \newif statements.
-%   Has this suddenly become \outer again?
-%
-% Known bugs:
-% - excludecomment leads to one superfluous space
-% - processcomment leads to a superfluous line break
-%
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-
-\def\makeinnocent#1{\catcode`#1=12 }
-\def\csarg#1#2{\expandafter#1\csname#2\endcsname}
-\def\latexname{lplain}\def\latexename{LaTeX2e}
-\newwrite\CommentStream
-\def\CommentCutFile{comment.cut}
-
-\def\ProcessComment#1% start it all of
-   {\begingroup
-    \def\CurrentComment{#1}%
-    \let\do\makeinnocent \dospecials 
-    \makeinnocent\^^L% and whatever other special cases
-    \endlinechar`\^^M \catcode`\^^M=12 \xComment}
-%\def\ProcessCommentWithArg#1#2% to be used in \leveledcomment
-%   {\begingroup
-%    \def\CurrentComment{#1}%
-%    \let\do\makeinnocent \dospecials 
-%    \makeinnocent\^^L% and whatever other special cases
-%    \endlinechar`\^^M \catcode`\^^M=12 \xComment}
-{\catcode`\^^M=12 \endlinechar=-1 %
- \gdef\xComment#1^^M{%
-    \expandafter\ProcessCommentLine}
- \gdef\ProcessCommentLine#1^^M{\def\test{#1}
-      \csarg\ifx{End\CurrentComment Test}\test
-          \edef\next{\noexpand\EndOfComment{\CurrentComment}}%
-      \else \ThisComment{#1}\let\next\ProcessCommentLine
-      \fi \next}
-}
-
-\def\CSstringmeaning#1{\expandafter\CSgobblearrow\meaning#1}
-\def\CSstringcsnoescape#1{\expandafter\CSgobbleescape\string#1}
-{\escapechar-1
-\expandafter\expandafter\expandafter\gdef
-  \expandafter\expandafter\expandafter\CSgobblearrow
-    \expandafter\string\csname macro:->\endcsname{}
-}
-\def\CSgobbleescape#1{\ifnum`\\=`#1 \else #1\fi}
-\def\WriteCommentLine#1{\def\CStmp{#1}%
-    \immediate\write\CommentStream{\CSstringmeaning\CStmp}}
-
-% 3.1 change: in LaTeX and LaTeX2e prevent grouping
-\if 0%
-\ifx\fmtname\latexename 
-    0%
-\else \ifx\fmtname\latexname 
-          0%
-      \else 
-          1%
-\fi   \fi
-%%%%
-%%%% definitions for LaTeX
-%%%%
-\def\AfterIncludedComment
-   {\immediate\closeout\CommentStream
-    \input{\CommentCutFile}\relax
-    }%
-\def\TossComment{\immediate\closeout\CommentStream}
-\def\BeforeIncludedComment
-   {\immediate\openout\CommentStream=\CommentCutFile
-    \let\ThisComment\WriteCommentLine}
-\def\includecomment
- #1{\message{Include comment '#1'}%
-    \csarg\let{After#1Comment}\AfterIncludedComment
-    \csarg\def{#1}{\BeforeIncludedComment
-        \ProcessComment{#1}}%
-    \CommentEndDef{#1}}
-\long\def\specialcomment
- #1#2#3{\message{Special comment '#1'}%
-    % note: \AfterIncludedComment does \input, so #2 goes here!
-    \csarg\def{After#1Comment}{#2\AfterIncludedComment#3}%
-    \csarg\def{#1}{\BeforeIncludedComment\relax
-          \ProcessComment{#1}}%
-    \CommentEndDef{#1}}
-\long\def\processcomment
- #1#2#3#4{\message{Lines-Processing comment '#1'}%
-    \csarg\def{After#1Comment}{#3\AfterIncludedComment#4}%
-    \csarg\def{#1}{\BeforeIncludedComment#2\relax
-          \ProcessComment{#1}}%
-    \CommentEndDef{#1}}
-\def\leveledcomment
- #1#2{\message{Include comment '#1' up to level '#2'}%
-    %\csname #1IsLeveledCommenttrue\endcsname
-    \csarg\let{After#1Comment}\AfterIncludedComment
-    \csarg\def{#1}{\BeforeIncludedComment
-        \ProcessCommentWithArg{#1}}%
-    \CommentEndDef{#1}}
-\else 
-%%%%
-%%%%plain TeX and other formats
-%%%%
-\def\includecomment
- #1{\message{Including comment '#1'}%
-    \csarg\def{#1}{}%
-    \csarg\def{end#1}{}}
-\long\def\specialcomment
- #1#2#3{\message{Special comment '#1'}%
-    \csarg\def{#1}{\def\ThisComment{}\def\AfterComment{#3}#2%
-           \ProcessComment{#1}}%
-    \CommentEndDef{#1}}
-\fi
-
-%%%%
-%%%% general definition of skipped comment
-%%%%
-\def\excludecomment
- #1{\message{Excluding comment '#1'}%
-    \csarg\def{#1}{\let\AfterComment\relax
-           \def\ThisComment####1{}\ProcessComment{#1}}%
-    \csarg\let{After#1Comment}\TossComment
-    \CommentEndDef{#1}}
-
-\if 0%
-\ifx\fmtname\latexename 
-    0%
-\else \ifx\fmtname\latexname 
-          0%
-      \else 
-          1%
-\fi   \fi
-% latex & latex2e:
-\def\EndOfComment#1{\endgroup\end{#1}%
-    \csname After#1Comment\endcsname}
-\def\CommentEndDef#1{{\escapechar=-1\relax
-    \csarg\xdef{End#1Test}{\string\\end\string\{#1\string\}}%
-    }}
-\else
-% plain & other
-\def\EndOfComment#1{\endgroup\AfterComment}
-\def\CommentEndDef#1{{\escapechar=-1\relax
-    \csarg\xdef{End#1Test}{\string\\end#1}%
-    }}
-\fi
-
-\excludecomment{comment}
-
-\endinput
diff --git a/thys/MDP-Algorithms/output/outline/isabelle.sty b/thys/MDP-Algorithms/output/outline/isabelle.sty
deleted file mode 100644
--- a/thys/MDP-Algorithms/output/outline/isabelle.sty
+++ /dev/null
@@ -1,282 +0,0 @@
-%%
-%% macros for Isabelle generated LaTeX output
-%%
-
-%%% Simple document preparation (based on theory token language and symbols)
-
-% isabelle environments
-
-\newcommand{\isabellecontext}{UNKNOWN}
-\newcommand{\setisabellecontext}[1]{\def\isabellecontext{#1}}
-
-\newcommand{\isastyle}{\UNDEF}
-\newcommand{\isastylett}{\UNDEF}
-\newcommand{\isastyleminor}{\UNDEF}
-\newcommand{\isastyleminortt}{\UNDEF}
-\newcommand{\isastylescript}{\UNDEF}
-\newcommand{\isastyletext}{\normalsize\normalfont\rmfamily}
-\newcommand{\isastyletxt}{\normalfont\rmfamily}
-\newcommand{\isastylecmt}{\normalfont\rmfamily}
-
-\newcommand{\isaspacing}{%
-  \sfcode 42 1000 % .
-  \sfcode 63 1000 % ?
-  \sfcode 33 1000 % !
-  \sfcode 58 1000 % :
-  \sfcode 59 1000 % ;
-  \sfcode 44 1000 % ,
-}
-
-%symbol markup -- \emph achieves decent spacing via italic corrections
-\newcommand{\isamath}[1]{\emph{$#1$}}
-\newcommand{\isatext}[1]{\emph{#1}}
-\DeclareRobustCommand{\isascriptstyle}{\def\isamath##1{##1}\def\isatext##1{\mbox{\isaspacing\isastylescript##1}}}
-\newcommand{\isactrlsub}[1]{\emph{\isascriptstyle${}\sb{#1}$}}
-\newcommand{\isactrlsup}[1]{\emph{\isascriptstyle${}\sp{#1}$}}
-\DeclareRobustCommand{\isactrlbsub}{\emph\bgroup\math{}\sb\bgroup\mbox\bgroup\isaspacing\isastylescript}
-\DeclareRobustCommand{\isactrlesub}{\egroup\egroup\endmath\egroup}
-\DeclareRobustCommand{\isactrlbsup}{\emph\bgroup\math{}\sp\bgroup\mbox\bgroup\isaspacing\isastylescript}
-\DeclareRobustCommand{\isactrlesup}{\egroup\egroup\endmath\egroup}
-\newcommand{\isactrlbold}[1]{{\bfseries\upshape\boldmath#1}}
-
-%blackboard-bold (requires font txmia from pxfonts)
-\DeclareSymbolFont{bbbfont}{U}{txmia}{m}{it}
-\SetSymbolFont{bbbfont}{bold}{U}{txmia}{bx}{it}
-\DeclareMathSymbol{\bbbA}{\mathord}{bbbfont}{129}
-\DeclareMathSymbol{\bbbB}{\mathord}{bbbfont}{130}
-\DeclareMathSymbol{\bbbC}{\mathord}{bbbfont}{131}
-\DeclareMathSymbol{\bbbD}{\mathord}{bbbfont}{132}
-\DeclareMathSymbol{\bbbE}{\mathord}{bbbfont}{133}
-\DeclareMathSymbol{\bbbF}{\mathord}{bbbfont}{134}
-\DeclareMathSymbol{\bbbG}{\mathord}{bbbfont}{135}
-\DeclareMathSymbol{\bbbH}{\mathord}{bbbfont}{136}
-\DeclareMathSymbol{\bbbI}{\mathord}{bbbfont}{137}
-\DeclareMathSymbol{\bbbJ}{\mathord}{bbbfont}{138}
-\DeclareMathSymbol{\bbbK}{\mathord}{bbbfont}{139}
-\DeclareMathSymbol{\bbbL}{\mathord}{bbbfont}{140}
-\DeclareMathSymbol{\bbbM}{\mathord}{bbbfont}{141}
-\DeclareMathSymbol{\bbbN}{\mathord}{bbbfont}{142}
-\DeclareMathSymbol{\bbbO}{\mathord}{bbbfont}{143}
-\DeclareMathSymbol{\bbbP}{\mathord}{bbbfont}{144}
-\DeclareMathSymbol{\bbbQ}{\mathord}{bbbfont}{145}
-\DeclareMathSymbol{\bbbR}{\mathord}{bbbfont}{146}
-\DeclareMathSymbol{\bbbS}{\mathord}{bbbfont}{147}
-\DeclareMathSymbol{\bbbT}{\mathord}{bbbfont}{148}
-\DeclareMathSymbol{\bbbU}{\mathord}{bbbfont}{149}
-\DeclareMathSymbol{\bbbV}{\mathord}{bbbfont}{150}
-\DeclareMathSymbol{\bbbW}{\mathord}{bbbfont}{151}
-\DeclareMathSymbol{\bbbX}{\mathord}{bbbfont}{152}
-\DeclareMathSymbol{\bbbY}{\mathord}{bbbfont}{153}
-\DeclareMathSymbol{\bbbZ}{\mathord}{bbbfont}{154}
-
-\newenvironment{isaantiq}{{\isacharat\isacharbraceleft}}{{\isacharbraceright}}
-
-\newdimen\isa@parindent\newdimen\isa@parskip
-
-\newenvironment{isabellebody}{%
-\isamarkuptrue\par%
-\isa@parindent\parindent\parindent0pt%
-\isa@parskip\parskip\parskip0pt%
-\isaspacing\isastyle}{\par}
-
-\newenvironment{isabellebodytt}{%
-\isamarkuptrue\par%
-\isa@parindent\parindent\parindent0pt%
-\isa@parskip\parskip\parskip0pt%
-\isaspacing\isastylett}{\par}
-
-\newenvironment{isabelle}
-{\begin{trivlist}\begin{isabellebody}\item\relax}
-{\end{isabellebody}\end{trivlist}}
-
-\newenvironment{isabellett}
-{\begin{trivlist}\begin{isabellebodytt}\item\relax}
-{\end{isabellebodytt}\end{trivlist}}
-
-\newcommand{\isa}[1]{\emph{\isaspacing\isastyleminor #1}}
-\newcommand{\isatt}[1]{\emph{\isaspacing\isastyleminortt #1}}
-
-\newcommand{\isaindent}[1]{\hphantom{#1}}
-\newcommand{\isanewline}{\mbox{}\par\mbox{}}
-\newcommand{\isasep}{}
-\newcommand{\isadigit}[1]{#1}
-
-\newcommand{\isachardefaults}{%
-\def\isacharbell{\isamath{\bigbox}}%requires stmaryrd
-\chardef\isacharbang=`\!%
-\chardef\isachardoublequote=`\"%
-\chardef\isachardoublequoteopen=`\"%
-\chardef\isachardoublequoteclose=`\"%
-\chardef\isacharhash=`\#%
-\chardef\isachardollar=`\$%
-\chardef\isacharpercent=`\%%
-\chardef\isacharampersand=`\&%
-\chardef\isacharprime=`\'%
-\chardef\isacharparenleft=`\(%
-\chardef\isacharparenright=`\)%
-\chardef\isacharasterisk=`\*%
-\chardef\isacharplus=`\+%
-\chardef\isacharcomma=`\,%
-\chardef\isacharminus=`\-%
-\chardef\isachardot=`\.%
-\chardef\isacharslash=`\/%
-\chardef\isacharcolon=`\:%
-\chardef\isacharsemicolon=`\;%
-\chardef\isacharless=`\<%
-\chardef\isacharequal=`\=%
-\chardef\isachargreater=`\>%
-\chardef\isacharquery=`\?%
-\chardef\isacharat=`\@%
-\chardef\isacharbrackleft=`\[%
-\chardef\isacharbackslash=`\\%
-\chardef\isacharbrackright=`\]%
-\chardef\isacharcircum=`\^%
-\chardef\isacharunderscore=`\_%
-\def\isacharunderscorekeyword{\_}%
-\chardef\isacharbackquote=`\`%
-\chardef\isacharbackquoteopen=`\`%
-\chardef\isacharbackquoteclose=`\`%
-\chardef\isacharbraceleft=`\{%
-\chardef\isacharbar=`\|%
-\chardef\isacharbraceright=`\}%
-\chardef\isachartilde=`\~%
-\def\isacharverbatimopen{\isacharbraceleft\isacharasterisk}%
-\def\isacharverbatimclose{\isacharasterisk\isacharbraceright}%
-\def\isacartoucheopen{\isatext{\guilsinglleft}}%
-\def\isacartoucheclose{\isatext{\guilsinglright}}%
-}
-
-
-% keyword and section markup
-
-\newcommand{\isakeyword}[1]
-{\emph{\normalfont\bfseries\def\isachardot{.}\def\isacharunderscore{\isacharunderscorekeyword}%
-\def\isacharbraceleft{\{}\def\isacharbraceright{\}}#1}}
-\newcommand{\isacommand}[1]{\isakeyword{#1}}
-
-\newcommand{\isakeywordcontrol}[1]
-{\emph{\normalfont\bfseries\itshape\def\isacharunderscore{\isacharunderscorekeyword}#1\,}}
-
-\newcommand{\isamarkupchapter}[1]{\chapter{#1}}
-\newcommand{\isamarkupsection}[1]{\section{#1}}
-\newcommand{\isamarkupsubsection}[1]{\subsection{#1}}
-\newcommand{\isamarkupsubsubsection}[1]{\subsubsection{#1}}
-\newcommand{\isamarkupparagraph}[1]{\paragraph{#1}}
-\newcommand{\isamarkupsubparagraph}[1]{\subparagraph{#1}}
-
-\newif\ifisamarkup
-\newcommand{\isabeginpar}{\par\ifisamarkup\relax\else\medskip\fi}
-\newcommand{\isaendpar}{\par\medskip}
-\newenvironment{isapar}{\parindent\isa@parindent\parskip\isa@parskip\isabeginpar}{\isaendpar}
-\newenvironment{isamarkuptext}{\par\isastyletext\begin{isapar}}{\end{isapar}}
-\newenvironment{isamarkuptxt}{\par\isastyletxt\begin{isapar}}{\end{isapar}}
-\newcommand{\isamarkupcmt}[1]{{\isastylecmt--- #1}}
-
-
-% index entries
-
-\newcommand{\isaindexdef}[1]{\textbf{#1}}
-\newcommand{\isaindexref}[1]{#1}
-
-
-% styles
-
-\def\isabellestyle#1{\csname isabellestyle#1\endcsname}
-
-\newcommand{\isabellestyledefault}{%
-\def\isastyle{\small\normalfont\ttfamily\slshape}%
-\def\isastylett{\small\normalfont\ttfamily}%
-\def\isastyleminor{\small\normalfont\ttfamily\slshape}%
-\def\isastyleminortt{\small\normalfont\ttfamily}%
-\def\isastylescript{\footnotesize\normalfont\ttfamily\slshape}%
-\isachardefaults%
-}
-\isabellestyledefault
-
-\newcommand{\isabellestylett}{%
-\def\isastyle{\small\normalfont\ttfamily}%
-\def\isastylett{\small\normalfont\ttfamily}%
-\def\isastyleminor{\small\normalfont\ttfamily}%
-\def\isastyleminortt{\small\normalfont\ttfamily}%
-\def\isastylescript{\footnotesize\normalfont\ttfamily}%
-\isachardefaults%
-}
-
-\newcommand{\isabellestyleit}{%
-\def\isastyle{\small\normalfont\itshape}%
-\def\isastylett{\small\normalfont\ttfamily}%
-\def\isastyleminor{\normalfont\itshape}%
-\def\isastyleminortt{\normalfont\ttfamily}%
-\def\isastylescript{\footnotesize\normalfont\itshape}%
-\isachardefaults%
-\def\isacharunderscorekeyword{\mbox{-}}%
-\def\isacharbang{\isamath{!}}%
-\def\isachardoublequote{}%
-\def\isachardoublequoteopen{}%
-\def\isachardoublequoteclose{}%
-\def\isacharhash{\isamath{\#}}%
-\def\isachardollar{\isamath{\$}}%
-\def\isacharpercent{\isamath{\%}}%
-\def\isacharampersand{\isamath{\&}}%
-\def\isacharprime{\isamath{\mskip2mu{'}\mskip-2mu}}%
-\def\isacharparenleft{\isamath{(}}%
-\def\isacharparenright{\isamath{)}}%
-\def\isacharasterisk{\isamath{*}}%
-\def\isacharplus{\isamath{+}}%
-\def\isacharcomma{\isamath{\mathord,}}%
-\def\isacharminus{\isamath{-}}%
-\def\isachardot{\isamath{\mathord.}}%
-\def\isacharslash{\isamath{/}}%
-\def\isacharcolon{\isamath{\mathord:}}%
-\def\isacharsemicolon{\isamath{\mathord;}}%
-\def\isacharless{\isamath{<}}%
-\def\isacharequal{\isamath{=}}%
-\def\isachargreater{\isamath{>}}%
-\def\isacharat{\isamath{@}}%
-\def\isacharbrackleft{\isamath{[}}%
-\def\isacharbackslash{\isamath{\backslash}}%
-\def\isacharbrackright{\isamath{]}}%
-\def\isacharunderscore{\mbox{-}}%
-\def\isacharbraceleft{\isamath{\{}}%
-\def\isacharbar{\isamath{\mid}}%
-\def\isacharbraceright{\isamath{\}}}%
-\def\isachartilde{\isamath{{}\sp{\sim}}}%
-\def\isacharbackquoteopen{\isatext{\guilsinglleft}}%
-\def\isacharbackquoteclose{\isatext{\guilsinglright}}%
-}
-
-\newcommand{\isabellestyleliteral}{%
-\isabellestyleit%
-\def\isacharunderscore{\_}%
-\def\isacharunderscorekeyword{\_}%
-\chardef\isacharbackquoteopen=`\`%
-\chardef\isacharbackquoteclose=`\`%
-}
-
-\newcommand{\isabellestyleliteralunderscore}{%
-\isabellestyleliteral%
-\def\isacharunderscore{\textunderscore}%
-\def\isacharunderscorekeyword{\textunderscore}%
-}
-
-\newcommand{\isabellestylesl}{%
-\isabellestyleit%
-\def\isastyle{\small\normalfont\slshape}%
-\def\isastylett{\small\normalfont\ttfamily}%
-\def\isastyleminor{\normalfont\slshape}%
-\def\isastyleminortt{\normalfont\ttfamily}%
-\def\isastylescript{\footnotesize\normalfont\slshape}%
-}
-
-
-% cancel text
-
-\usepackage[normalem]{ulem}
-\newcommand{\isamarkupcancel}[1]{\isa{\xout{#1}}}
-
-
-% tags
-
-\newcommand{\isafold}[1]{\emph{$\langle\mathord{\mathit{#1}}\rangle$}}
-
-\IfFileExists{isabelletags.sty}{\usepackage{isabelletags}}{}
diff --git a/thys/MDP-Algorithms/output/outline/isabellesym.sty b/thys/MDP-Algorithms/output/outline/isabellesym.sty
deleted file mode 100644
--- a/thys/MDP-Algorithms/output/outline/isabellesym.sty
+++ /dev/null
@@ -1,496 +0,0 @@
-%%
-%% definitions of standard Isabelle symbols
-%%
-
-\newcommand{\isasymzero}{\isamath{\mathbf{0}}}  %requires amssymb
-\newcommand{\isasymone}{\isamath{\mathbf{1}}}  %requires amssymb
-\newcommand{\isasymtwo}{\isamath{\mathbf{2}}}  %requires amssymb
-\newcommand{\isasymthree}{\isamath{\mathbf{3}}}  %requires amssymb
-\newcommand{\isasymfour}{\isamath{\mathbf{4}}}  %requires amssymb
-\newcommand{\isasymfive}{\isamath{\mathbf{5}}}  %requires amssymb
-\newcommand{\isasymsix}{\isamath{\mathbf{6}}}  %requires amssymb
-\newcommand{\isasymseven}{\isamath{\mathbf{7}}}  %requires amssymb
-\newcommand{\isasymeight}{\isamath{\mathbf{8}}}  %requires amssymb
-\newcommand{\isasymnine}{\isamath{\mathbf{9}}}  %requires amssymb
-\newcommand{\isasymA}{\isamath{\mathcal{A}}}
-\newcommand{\isasymB}{\isamath{\mathcal{B}}}
-\newcommand{\isasymC}{\isamath{\mathcal{C}}}
-\newcommand{\isasymD}{\isamath{\mathcal{D}}}
-\newcommand{\isasymE}{\isamath{\mathcal{E}}}
-\newcommand{\isasymF}{\isamath{\mathcal{F}}}
-\newcommand{\isasymG}{\isamath{\mathcal{G}}}
-\newcommand{\isasymH}{\isamath{\mathcal{H}}}
-\newcommand{\isasymI}{\isamath{\mathcal{I}}}
-\newcommand{\isasymJ}{\isamath{\mathcal{J}}}
-\newcommand{\isasymK}{\isamath{\mathcal{K}}}
-\newcommand{\isasymL}{\isamath{\mathcal{L}}}
-\newcommand{\isasymM}{\isamath{\mathcal{M}}}
-\newcommand{\isasymN}{\isamath{\mathcal{N}}}
-\newcommand{\isasymO}{\isamath{\mathcal{O}}}
-\newcommand{\isasymP}{\isamath{\mathcal{P}}}
-\newcommand{\isasymQ}{\isamath{\mathcal{Q}}}
-\newcommand{\isasymR}{\isamath{\mathcal{R}}}
-\newcommand{\isasymS}{\isamath{\mathcal{S}}}
-\newcommand{\isasymT}{\isamath{\mathcal{T}}}
-\newcommand{\isasymU}{\isamath{\mathcal{U}}}
-\newcommand{\isasymV}{\isamath{\mathcal{V}}}
-\newcommand{\isasymW}{\isamath{\mathcal{W}}}
-\newcommand{\isasymX}{\isamath{\mathcal{X}}}
-\newcommand{\isasymY}{\isamath{\mathcal{Y}}}
-\newcommand{\isasymZ}{\isamath{\mathcal{Z}}}
-\newcommand{\isasyma}{\isamath{\mathrm{a}}}
-\newcommand{\isasymb}{\isamath{\mathrm{b}}}
-\newcommand{\isasymc}{\isamath{\mathrm{c}}}
-\newcommand{\isasymd}{\isamath{\mathrm{d}}}
-\newcommand{\isasyme}{\isamath{\mathrm{e}}}
-\newcommand{\isasymf}{\isamath{\mathrm{f}}}
-\newcommand{\isasymg}{\isamath{\mathrm{g}}}
-\newcommand{\isasymh}{\isamath{\mathrm{h}}}
-\newcommand{\isasymi}{\isamath{\mathrm{i}}}
-\newcommand{\isasymj}{\isamath{\mathrm{j}}}
-\newcommand{\isasymk}{\isamath{\mathrm{k}}}
-\newcommand{\isasyml}{\isamath{\mathrm{l}}}
-\newcommand{\isasymm}{\isamath{\mathrm{m}}}
-\newcommand{\isasymn}{\isamath{\mathrm{n}}}
-\newcommand{\isasymo}{\isamath{\mathrm{o}}}
-\newcommand{\isasymp}{\isamath{\mathrm{p}}}
-\newcommand{\isasymq}{\isamath{\mathrm{q}}}
-\newcommand{\isasymr}{\isamath{\mathrm{r}}}
-\newcommand{\isasyms}{\isamath{\mathrm{s}}}
-\newcommand{\isasymt}{\isamath{\mathrm{t}}}
-\newcommand{\isasymu}{\isamath{\mathrm{u}}}
-\newcommand{\isasymv}{\isamath{\mathrm{v}}}
-\newcommand{\isasymw}{\isamath{\mathrm{w}}}
-\newcommand{\isasymx}{\isamath{\mathrm{x}}}
-\newcommand{\isasymy}{\isamath{\mathrm{y}}}
-\newcommand{\isasymz}{\isamath{\mathrm{z}}}
-\newcommand{\isasymAA}{\isamath{\mathfrak{A}}}  %requires eufrak
-\newcommand{\isasymBB}{\isamath{\mathfrak{B}}}  %requires eufrak
-\newcommand{\isasymCC}{\isamath{\mathfrak{C}}}  %requires eufrak
-\newcommand{\isasymDD}{\isamath{\mathfrak{D}}}  %requires eufrak
-\newcommand{\isasymEE}{\isamath{\mathfrak{E}}}  %requires eufrak
-\newcommand{\isasymFF}{\isamath{\mathfrak{F}}}  %requires eufrak
-\newcommand{\isasymGG}{\isamath{\mathfrak{G}}}  %requires eufrak
-\newcommand{\isasymHH}{\isamath{\mathfrak{H}}}  %requires eufrak
-\newcommand{\isasymII}{\isamath{\mathfrak{I}}}  %requires eufrak
-\newcommand{\isasymJJ}{\isamath{\mathfrak{J}}}  %requires eufrak
-\newcommand{\isasymKK}{\isamath{\mathfrak{K}}}  %requires eufrak
-\newcommand{\isasymLL}{\isamath{\mathfrak{L}}}  %requires eufrak
-\newcommand{\isasymMM}{\isamath{\mathfrak{M}}}  %requires eufrak
-\newcommand{\isasymNN}{\isamath{\mathfrak{N}}}  %requires eufrak
-\newcommand{\isasymOO}{\isamath{\mathfrak{O}}}  %requires eufrak
-\newcommand{\isasymPP}{\isamath{\mathfrak{P}}}  %requires eufrak
-\newcommand{\isasymQQ}{\isamath{\mathfrak{Q}}}  %requires eufrak
-\newcommand{\isasymRR}{\isamath{\mathfrak{R}}}  %requires eufrak
-\newcommand{\isasymSS}{\isamath{\mathfrak{S}}}  %requires eufrak
-\newcommand{\isasymTT}{\isamath{\mathfrak{T}}}  %requires eufrak
-\newcommand{\isasymUU}{\isamath{\mathfrak{U}}}  %requires eufrak
-\newcommand{\isasymVV}{\isamath{\mathfrak{V}}}  %requires eufrak
-\newcommand{\isasymWW}{\isamath{\mathfrak{W}}}  %requires eufrak
-\newcommand{\isasymXX}{\isamath{\mathfrak{X}}}  %requires eufrak
-\newcommand{\isasymYY}{\isamath{\mathfrak{Y}}}  %requires eufrak
-\newcommand{\isasymZZ}{\isamath{\mathfrak{Z}}}  %requires eufrak
-\newcommand{\isasymaa}{\isamath{\mathfrak{a}}}  %requires eufrak
-\newcommand{\isasymbb}{\isamath{\mathfrak{b}}}  %requires eufrak
-\newcommand{\isasymcc}{\isamath{\mathfrak{c}}}  %requires eufrak
-\newcommand{\isasymdd}{\isamath{\mathfrak{d}}}  %requires eufrak
-\newcommand{\isasymee}{\isamath{\mathfrak{e}}}  %requires eufrak
-\newcommand{\isasymff}{\isamath{\mathfrak{f}}}  %requires eufrak
-\newcommand{\isasymgg}{\isamath{\mathfrak{g}}}  %requires eufrak
-\newcommand{\isasymhh}{\isamath{\mathfrak{h}}}  %requires eufrak
-\newcommand{\isasymii}{\isamath{\mathfrak{i}}}  %requires eufrak
-\newcommand{\isasymjj}{\isamath{\mathfrak{j}}}  %requires eufrak
-\newcommand{\isasymkk}{\isamath{\mathfrak{k}}}  %requires eufrak
-\newcommand{\isasymll}{\isamath{\mathfrak{l}}}  %requires eufrak
-\newcommand{\isasymmm}{\isamath{\mathfrak{m}}}  %requires eufrak
-\newcommand{\isasymnn}{\isamath{\mathfrak{n}}}  %requires eufrak
-\newcommand{\isasymoo}{\isamath{\mathfrak{o}}}  %requires eufrak
-\newcommand{\isasympp}{\isamath{\mathfrak{p}}}  %requires eufrak
-\newcommand{\isasymqq}{\isamath{\mathfrak{q}}}  %requires eufrak
-\newcommand{\isasymrr}{\isamath{\mathfrak{r}}}  %requires eufrak
-\newcommand{\isasymss}{\isamath{\mathfrak{s}}}  %requires eufrak
-\newcommand{\isasymtt}{\isamath{\mathfrak{t}}}  %requires eufrak
-\newcommand{\isasymuu}{\isamath{\mathfrak{u}}}  %requires eufrak
-\newcommand{\isasymvv}{\isamath{\mathfrak{v}}}  %requires eufrak
-\newcommand{\isasymww}{\isamath{\mathfrak{w}}}  %requires eufrak
-\newcommand{\isasymxx}{\isamath{\mathfrak{x}}}  %requires eufrak
-\newcommand{\isasymyy}{\isamath{\mathfrak{y}}}  %requires eufrak
-\newcommand{\isasymzz}{\isamath{\mathfrak{z}}}  %requires eufrak
-\newcommand{\isasymalpha}{\isamath{\alpha}}
-\newcommand{\isasymbeta}{\isamath{\beta}}
-\newcommand{\isasymgamma}{\isamath{\gamma}}
-\newcommand{\isasymdelta}{\isamath{\delta}}
-\newcommand{\isasymepsilon}{\isamath{\varepsilon}}
-\newcommand{\isasymzeta}{\isamath{\zeta}}
-\newcommand{\isasymeta}{\isamath{\eta}}
-\newcommand{\isasymtheta}{\isamath{\vartheta}}
-\newcommand{\isasymiota}{\isamath{\iota}}
-\newcommand{\isasymkappa}{\isamath{\kappa}}
-\newcommand{\isasymlambda}{\isamath{\lambda}}
-\newcommand{\isasymmu}{\isamath{\mu}}
-\newcommand{\isasymnu}{\isamath{\nu}}
-\newcommand{\isasymxi}{\isamath{\xi}}
-\newcommand{\isasympi}{\isamath{\pi}}
-\newcommand{\isasymrho}{\isamath{\varrho}}
-\newcommand{\isasymsigma}{\isamath{\sigma}}
-\newcommand{\isasymtau}{\isamath{\tau}}
-\newcommand{\isasymupsilon}{\isamath{\upsilon}}
-\newcommand{\isasymphi}{\isamath{\varphi}}
-\newcommand{\isasymchi}{\isamath{\chi}}
-\newcommand{\isasympsi}{\isamath{\psi}}
-\newcommand{\isasymomega}{\isamath{\omega}}
-\newcommand{\isasymGamma}{\isamath{\Gamma}}
-\newcommand{\isasymDelta}{\isamath{\Delta}}
-\newcommand{\isasymTheta}{\isamath{\Theta}}
-\newcommand{\isasymLambda}{\isamath{\Lambda}}
-\newcommand{\isasymXi}{\isamath{\Xi}}
-\newcommand{\isasymPi}{\isamath{\Pi}}
-\newcommand{\isasymSigma}{\isamath{\Sigma}}
-\newcommand{\isasymUpsilon}{\isamath{\Upsilon}}
-\newcommand{\isasymPhi}{\isamath{\Phi}}
-\newcommand{\isasymPsi}{\isamath{\Psi}}
-\newcommand{\isasymOmega}{\isamath{\Omega}}
-\newcommand{\isasymbbbA}{\isamath{\bbbA}}  %requires font txmia from txfonts
-\newcommand{\isasymbool}{\isamath{\bbbB}}  %requires font txmia from txfonts
-\newcommand{\isasymcomplex}{\isamath{\bbbC}}  %requires font txmia from txfonts
-\newcommand{\isasymbbbD}{\isamath{\bbbD}}  %requires font txmia from txfonts
-\newcommand{\isasymbbbE}{\isamath{\bbbE}}  %requires font txmia from txfonts
-\newcommand{\isasymbbbF}{\isamath{\bbbF}}  %requires font txmia from txfonts
-\newcommand{\isasymbbbG}{\isamath{\bbbG}}  %requires font txmia from txfonts
-\newcommand{\isasymbbbH}{\isamath{\bbbH}}  %requires font txmia from txfonts
-\newcommand{\isasymbbbI}{\isamath{\bbbI}}  %requires font txmia from txfonts
-\newcommand{\isasymbbbJ}{\isamath{\bbbJ}}  %requires font txmia from txfonts
-\newcommand{\isasymbbbK}{\isamath{\bbbK}}  %requires font txmia from txfonts
-\newcommand{\isasymbbbL}{\isamath{\bbbL}}  %requires font txmia from txfonts
-\newcommand{\isasymbbbM}{\isamath{\bbbM}}  %requires font txmia from txfonts
-\newcommand{\isasymnat}{\isamath{\bbbN}}  %requires font txmia from txfonts
-\newcommand{\isasymbbbO}{\isamath{\bbbO}}  %requires font txmia from txfonts
-\newcommand{\isasymbbbP}{\isamath{\bbbP}}  %requires font txmia from txfonts
-\newcommand{\isasymrat}{\isamath{\bbbQ}}  %requires font txmia from txfonts
-\newcommand{\isasymreal}{\isamath{\bbbR}}  %requires font txmia from txfonts
-\newcommand{\isasymbbbS}{\isamath{\bbbS}}  %requires font txmia from txfonts
-\newcommand{\isasymbbbT}{\isamath{\bbbT}}  %requires font txmia from txfonts
-\newcommand{\isasymbbbU}{\isamath{\bbbU}}  %requires font txmia from txfonts
-\newcommand{\isasymbbbV}{\isamath{\bbbV}}  %requires font txmia from txfonts
-\newcommand{\isasymbbbW}{\isamath{\bbbW}}  %requires font txmia from txfonts
-\newcommand{\isasymbbbX}{\isamath{\bbbX}}  %requires font txmia from txfonts
-\newcommand{\isasymbbbY}{\isamath{\bbbY}}  %requires font txmia from txfonts
-\newcommand{\isasymint}{\isamath{\bbbZ}}  %requires font txmia from txfonts
-\newcommand{\isasymleftarrow}{\isamath{\leftarrow}}
-\newcommand{\isasymrightarrow}{\isamath{\rightarrow}}
-\newcommand{\isasymlongleftarrow}{\isamath{\longleftarrow}}
-\newcommand{\isasymlongrightarrow}{\isamath{\longrightarrow}}
-\newcommand{\isasymlonglongleftarrow}{\isamath{\xleftarrow{\hphantom{AAA}}}}  %requires amsmath
-\newcommand{\isasymlonglongrightarrow}{\isamath{\xrightarrow{\hphantom{AAA}}}}  %requires amsmath
-\newcommand{\isasymlonglonglongleftarrow}{\isamath{\xleftarrow{\hphantom{AAAA}}}}  %requires amsmath
-\newcommand{\isasymlonglonglongrightarrow}{\isamath{\xrightarrow{\hphantom{AAAA}}}}  %requires amsmath
-\newcommand{\isasymLeftarrow}{\isamath{\Leftarrow}}
-\newcommand{\isasymRightarrow}{\isamath{\Rightarrow}}
-\newcommand{\isasymLongleftarrow}{\isamath{\Longleftarrow}}
-\newcommand{\isasymLongrightarrow}{\isamath{\Longrightarrow}}
-\newcommand{\isasymLleftarrow}{\isamath{\Lleftarrow}}  %requires amssymb
-\newcommand{\isasymRrightarrow}{\isamath{\Rrightarrow}}  %requires amssymb
-\newcommand{\isasymleftrightarrow}{\isamath{\leftrightarrow}}
-\newcommand{\isasymLeftrightarrow}{\isamath{\Leftrightarrow}}
-\newcommand{\isasymlongleftrightarrow}{\isamath{\longleftrightarrow}}
-\newcommand{\isasymLongleftrightarrow}{\isamath{\Longleftrightarrow}}
-\newcommand{\isasymmapsto}{\isamath{\mapsto}}
-\newcommand{\isasymlongmapsto}{\isamath{\longmapsto}}
-\newcommand{\isasymmidarrow}{\isamath{\relbar}}
-\newcommand{\isasymMidarrow}{\isamath{\Relbar}}
-\newcommand{\isasymhookleftarrow}{\isamath{\hookleftarrow}}
-\newcommand{\isasymhookrightarrow}{\isamath{\hookrightarrow}}
-\newcommand{\isasymleftharpoondown}{\isamath{\leftharpoondown}}
-\newcommand{\isasymrightharpoondown}{\isamath{\rightharpoondown}}
-\newcommand{\isasymleftharpoonup}{\isamath{\leftharpoonup}}
-\newcommand{\isasymrightharpoonup}{\isamath{\rightharpoonup}}
-\newcommand{\isasymrightleftharpoons}{\isamath{\rightleftharpoons}}
-\newcommand{\isasymleadsto}{\isamath{\leadsto}}  %requires amssymb
-\newcommand{\isasymdownharpoonleft}{\isamath{\downharpoonleft}}  %requires amssymb
-\newcommand{\isasymdownharpoonright}{\isamath{\downharpoonright}}  %requires amssymb
-\newcommand{\isasymupharpoonleft}{\isamath{\upharpoonleft}}  %requires amssymb
-\newcommand{\isasymupharpoonright}{\isamath{\upharpoonright}}  %requires amssymb
-\newcommand{\isasymrestriction}{\isamath{\restriction}}  %requires amssymb
-\newcommand{\isasymColon}{\isamath{\mathrel{::}}}
-\newcommand{\isasymup}{\isamath{\uparrow}}
-\newcommand{\isasymUp}{\isamath{\Uparrow}}
-\newcommand{\isasymdown}{\isamath{\downarrow}}
-\newcommand{\isasymDown}{\isamath{\Downarrow}}
-\newcommand{\isasymupdown}{\isamath{\updownarrow}}
-\newcommand{\isasymUpdown}{\isamath{\Updownarrow}}
-\newcommand{\isasymlangle}{\isamath{\langle}}
-\newcommand{\isasymrangle}{\isamath{\rangle}}
-\newcommand{\isasymllangle}{\isamath{\langle\mskip-5mu\langle}}
-\newcommand{\isasymrrangle}{\isamath{\rangle\mskip-5mu\rangle}}
-\newcommand{\isasymlceil}{\isamath{\lceil}}
-\newcommand{\isasymrceil}{\isamath{\rceil}}
-\newcommand{\isasymlfloor}{\isamath{\lfloor}}
-\newcommand{\isasymrfloor}{\isamath{\rfloor}}
-\newcommand{\isasymlparr}{\isamath{\mathopen{(\mkern-3.3mu\mid}}}
-\newcommand{\isasymrparr}{\isamath{\mathclose{\mid\mkern-3.3mu)}}}
-\newcommand{\isasymlbrakk}{\isamath{\mathopen{\lbrack\mkern-3mu\lbrack}}}
-\newcommand{\isasymrbrakk}{\isamath{\mathclose{\rbrack\mkern-3mu\rbrack}}}
-\newcommand{\isasymlbrace}{\isamath{\mathopen{\lbrace\mkern-4.3mu\mid}}}
-\newcommand{\isasymrbrace}{\isamath{\mathclose{\mid\mkern-4.3mu\rbrace}}}
-\newcommand{\isasymlblot}{\isamath{{\langle}\mkern -3.5mu{|}}}
-\newcommand{\isasymrblot}{\isamath{{|}\mkern -3.5mu{\rangle}}}
-\newcommand{\isasymguillemotleft}{\isatext{\guillemotleft}}
-\newcommand{\isasymguillemotright}{\isatext{\guillemotright}}
-\newcommand{\isasymbottom}{\isamath{\bot}}
-\newcommand{\isasymtop}{\isamath{\top}}
-\newcommand{\isasymand}{\isamath{\wedge}}
-\newcommand{\isasymAnd}{\isamath{\bigwedge}}
-\newcommand{\isasymor}{\isamath{\vee}}
-\newcommand{\isasymOr}{\isamath{\bigvee}}
-\newcommand{\isasymforall}{\isamath{\forall\,}}
-\newcommand{\isasymexists}{\isamath{\exists\,}}
-\newcommand{\isasymnot}{\isamath{\neg}}
-\newcommand{\isasymnexists}{\isamath{\nexists\,}}  %requires amssymb
-\newcommand{\isasymcircle}{\isamath{\ocircle}}  %requires wasysym
-\newcommand{\isasymbox}{\isamath{\Box}}  %requires amssymb
-\newcommand{\isasymdiamond}{\isamath{\Diamond}}  %requires amssymb
-\newcommand{\isasymdiamondop}{\isamath{\diamond}}
-\newcommand{\isasymsurd}{\isamath{\surd}}
-\newcommand{\isasymturnstile}{\isamath{\vdash}}
-\newcommand{\isasymTurnstile}{\isamath{\models}}
-\newcommand{\isasymtturnstile}{\isamath{\vdash\!\!\!\vdash}}
-\newcommand{\isasymTTurnstile}{\isamath{\mid\!\models}}
-\newcommand{\isasymstileturn}{\isamath{\dashv}}
-\newcommand{\isasymle}{\isamath{\le}}
-\newcommand{\isasymge}{\isamath{\ge}}
-\newcommand{\isasymlless}{\isamath{\ll}}
-\newcommand{\isasymggreater}{\isamath{\gg}}
-\newcommand{\isasymlesssim}{\isamath{\lesssim}}  %requires amssymb
-\newcommand{\isasymgreatersim}{\isamath{\gtrsim}}  %requires amssymb
-\newcommand{\isasymlessapprox}{\isamath{\lessapprox}}  %requires amssymb
-\newcommand{\isasymgreaterapprox}{\isamath{\gtrapprox}}  %requires amssymb
-\newcommand{\isasymin}{\isamath{\in}}
-\newcommand{\isasymnotin}{\isamath{\notin}}
-\newcommand{\isasymsubset}{\isamath{\subset}}
-\newcommand{\isasymsupset}{\isamath{\supset}}
-\newcommand{\isasymsubseteq}{\isamath{\subseteq}}
-\newcommand{\isasymsupseteq}{\isamath{\supseteq}}
-\newcommand{\isasymsqsubset}{\isamath{\sqsubset}}  %requires amssymb
-\newcommand{\isasymsqsupset}{\isamath{\sqsupset}}  %requires amssymb
-\newcommand{\isasymsqsubseteq}{\isamath{\sqsubseteq}}
-\newcommand{\isasymsqsupseteq}{\isamath{\sqsupseteq}}
-\newcommand{\isasyminter}{\isamath{\cap}}
-\newcommand{\isasymInter}{\isamath{\bigcap\,}}
-\newcommand{\isasymunion}{\isamath{\cup}}
-\newcommand{\isasymUnion}{\isamath{\bigcup\,}}
-\newcommand{\isasymsqunion}{\isamath{\sqcup}}
-\newcommand{\isasymSqunion}{\isamath{\bigsqcup\,}}
-\newcommand{\isasymsqinter}{\isamath{\sqcap}}
-\newcommand{\isasymSqinter}{\isamath{\bigsqcap\,}}  %requires stmaryrd
-\newcommand{\isasymsetminus}{\isamath{\setminus}}
-\newcommand{\isasympropto}{\isamath{\propto}}
-\newcommand{\isasymuplus}{\isamath{\uplus}}
-\newcommand{\isasymUplus}{\isamath{\biguplus\,}}
-\newcommand{\isasymnoteq}{\isamath{\not=}}
-\newcommand{\isasymsim}{\isamath{\sim}}
-\newcommand{\isasymdoteq}{\isamath{\doteq}}
-\newcommand{\isasymsimeq}{\isamath{\simeq}}
-\newcommand{\isasymapprox}{\isamath{\approx}}
-\newcommand{\isasymasymp}{\isamath{\asymp}}
-\newcommand{\isasymcong}{\isamath{\cong}}
-\newcommand{\isasymsmile}{\isamath{\smile}}
-\newcommand{\isasymequiv}{\isamath{\equiv}}
-\newcommand{\isasymfrown}{\isamath{\frown}}
-\newcommand{\isasymJoin}{\isamath{\Join}}  %requires amssymb
-\newcommand{\isasymbowtie}{\isamath{\bowtie}}
-\newcommand{\isasymprec}{\isamath{\prec}}
-\newcommand{\isasymsucc}{\isamath{\succ}}
-\newcommand{\isasympreceq}{\isamath{\preceq}}
-\newcommand{\isasymsucceq}{\isamath{\succeq}}
-\newcommand{\isasymparallel}{\isamath{\parallel}}
-\newcommand{\isasymParallel}{\isamath{\bigparallel}}  %requires stmaryrd
-\newcommand{\isasyminterleace}{\isamath{\interleave}}  %requires stmaryrd
-\newcommand{\isasymsslash}{\isamath{\sslash}}  %requires stmaryrd
-\newcommand{\isasymbar}{\isamath{\mid}}
-\newcommand{\isasymbbar}{\isamath{[\mskip-1.5mu]}}
-\newcommand{\isasymplusminus}{\isamath{\pm}}
-\newcommand{\isasymminusplus}{\isamath{\mp}}
-\newcommand{\isasymtimes}{\isamath{\times}}
-\newcommand{\isasymdiv}{\isamath{\div}}
-\newcommand{\isasymcdot}{\isamath{\cdot}}
-\newcommand{\isasymsqdot}{\isamath{\sbox\z@{$\centerdot$}\ht\z@=.33333\ht\z@\vcenter{\box\z@}}}  %requires amssymb
-\newcommand{\isasymstar}{\isamath{\star}}
-\newcommand{\isasymbullet}{\boldmath\isamath{\mathchoice{\displaystyle{\cdot}}{\textstyle{\cdot}}{\scriptstyle{\bullet}}{\scriptscriptstyle{\bullet}}}}
-\newcommand{\isasymcirc}{\isamath{\circ}}
-\newcommand{\isasymdagger}{\isamath{\dagger}}
-\newcommand{\isasymddagger}{\isamath{\ddagger}}
-\newcommand{\isasymlhd}{\isamath{\lhd}}  %requires amssymb
-\newcommand{\isasymrhd}{\isamath{\rhd}}  %requires amssymb
-\newcommand{\isasymunlhd}{\isamath{\unlhd}}  %requires amssymb
-\newcommand{\isasymunrhd}{\isamath{\unrhd}}  %requires amssymb
-\newcommand{\isasymtriangleleft}{\isamath{\triangleleft}}
-\newcommand{\isasymtriangleright}{\isamath{\triangleright}}
-\newcommand{\isasymtriangle}{\isamath{\triangle}}
-\newcommand{\isasymtriangleq}{\isamath{\triangleq}}  %requires amssymb
-\newcommand{\isasymoplus}{\isamath{\oplus}}
-\newcommand{\isasymOplus}{\isamath{\bigoplus\,}}
-\newcommand{\isasymotimes}{\isamath{\otimes}}
-\newcommand{\isasymOtimes}{\isamath{\bigotimes\,}}
-\newcommand{\isasymodot}{\isamath{\odot}}
-\newcommand{\isasymOdot}{\isamath{\bigodot\,}}
-\newcommand{\isasymominus}{\isamath{\ominus}}
-\newcommand{\isasymoslash}{\isamath{\oslash}}
-\newcommand{\isasymdots}{\isamath{\dots}}
-\newcommand{\isasymcdots}{\isamath{\cdots}}
-\newcommand{\isasymSum}{\isamath{\sum\,}}
-\newcommand{\isasymProd}{\isamath{\prod\,}}
-\newcommand{\isasymCoprod}{\isamath{\coprod\,}}
-\newcommand{\isasyminfinity}{\isamath{\infty}}
-\newcommand{\isasymintegral}{\isamath{\int\,}}
-\newcommand{\isasymointegral}{\isamath{\oint\,}}
-\newcommand{\isasymclubsuit}{\isamath{\clubsuit}}
-\newcommand{\isasymdiamondsuit}{\isamath{\diamondsuit}}
-\newcommand{\isasymheartsuit}{\isamath{\heartsuit}}
-\newcommand{\isasymspadesuit}{\isamath{\spadesuit}}
-\newcommand{\isasymaleph}{\isamath{\aleph}}
-\newcommand{\isasymemptyset}{\isamath{\emptyset}}
-\newcommand{\isasymnabla}{\isamath{\nabla}}
-\newcommand{\isasympartial}{\isamath{\partial}}
-\newcommand{\isasymRe}{\isamath{\Re}}
-\newcommand{\isasymIm}{\isamath{\Im}}
-\newcommand{\isasymflat}{\isamath{\flat}}
-\newcommand{\isasymnatural}{\isamath{\natural}}
-\newcommand{\isasymsharp}{\isamath{\sharp}}
-\newcommand{\isasymangle}{\isamath{\angle}}
-\newcommand{\isasymcopyright}{\isatext{\normalfont\rmfamily\copyright}}
-\newcommand{\isasymregistered}{\isatext{\normalfont\rmfamily\textregistered}}
-\newcommand{\isasyminverse}{\isamath{{}^{-1}}}
-\newcommand{\isasymonequarter}{\isatext{\normalfont\rmfamily\textonequarter}}  %requires textcomp
-\newcommand{\isasymonehalf}{\isatext{\normalfont\rmfamily\textonehalf}}  %requires textcomp
-\newcommand{\isasymthreequarters}{\isatext{\normalfont\rmfamily\textthreequarters}}  %requires textcomp
-\newcommand{\isasymordfeminine}{\isatext{\normalfont\rmfamily\textordfeminine}}
-\newcommand{\isasymordmasculine}{\isatext{\normalfont\rmfamily\textordmasculine}}
-\newcommand{\isasymsection}{\isatext{\normalfont\rmfamily\S}}
-\newcommand{\isasymparagraph}{\isatext{\normalfont\rmfamily\P}}
-\newcommand{\isasymexclamdown}{\isatext{\normalfont\rmfamily\textexclamdown}}
-\newcommand{\isasymquestiondown}{\isatext{\normalfont\rmfamily\textquestiondown}}
-\newcommand{\isasymeuro}{\isatext{\euro}}  %requires eurosym
-\newcommand{\isasympounds}{\isamath{\pounds}}
-\newcommand{\isasymyen}{\isatext{\yen}}  %requires amssymb
-\newcommand{\isasymcent}{\isatext{\textcent}}  %requires textcomp
-\newcommand{\isasymcurrency}{\isatext{\textcurrency}} %requires textcomp
-\newcommand{\isasymdegree}{\isatext{\normalfont\rmfamily\textdegree}}  %requires textcomp
-\newcommand{\isasymhyphen}{\isatext{\normalfont\rmfamily-}}
-\newcommand{\isasymamalg}{\isamath{\amalg}}
-\newcommand{\isasymmho}{\isamath{\mho}}  %requires amssymb
-\newcommand{\isasymlozenge}{\isamath{\lozenge}}  %requires amssymb
-\newcommand{\isasymwp}{\isamath{\wp}}
-\newcommand{\isasymwrong}{\isamath{\wr}}
-\newcommand{\isasymacute}{\isatext{\'\relax}}
-\newcommand{\isasymindex}{\isatext{\i}}
-\newcommand{\isasymdieresis}{\isatext{\"\relax}}
-\newcommand{\isasymcedilla}{\isatext{\c\relax}}
-\newcommand{\isasymhungarumlaut}{\isatext{\H\relax}}
-\newcommand{\isasymsome}{\isamath{\epsilon\,}}
-\newcommand{\isasymbind}{\isamath{\mathbin{>\!\!\!>\mkern-6.7mu=}}}
-\newcommand{\isasymthen}{\isamath{\mathbin{>\!\!\!>}}}
-
-%Z notation
-\newcommand{\isaZhbar}[1]{\rlap{\raise.0001ex\hbox{\isamath{-}}}#1}
-\newcommand{\isaZpvbar}[1]{\ooalign{\hfil\isamath{\mapstochar\mkern 5mu}\hfil\cr#1}}
-\newcommand{\isaZfvbar}[1]{\ooalign{\hfil\isamath{\mapstochar\mkern 3mu\mapstochar\mkern 5mu}\hfil\cr#1}}
-\newcommand{\isaZdarrow}[3]{\ooalign{\isamath{#1}\hfil\cr\isamath{\mkern#3mu\isamath{#2}}}}
-\newcommand{\isasymZcomp}{\isamath{\fatsemi}}  %requires stmaryrd
-\newcommand{\isasymZinj}{\isamath{\rightarrowtail}}  %requires amssymb
-\newcommand{\isasymZpinj}{\isaZpvbar{\isamath{\rightarrowtail}}}  %requires amssymb
-\newcommand{\isasymZfinj}{\isaZfvbar{\isasymZinj}}  %requires amssymb
-\newcommand{\isasymZsurj}{\isaZdarrow{\rightarrow}{\rightarrow}{4}}  %requires amssymb
-\newcommand{\isasymZpsurj}{\isaZpvbar{\isasymZsurj}}  %requires amssymb
-\newcommand{\isasymZbij}{\isaZdarrow{\rightarrowtail}{\rightarrow}{5}}  %requires amssymb
-\newcommand{\isasymZpfun}{\isaZpvbar{\isamath{\rightarrow}}}
-\newcommand{\isasymZffun}{\isaZfvbar{\isamath{\rightarrow}}}
-\newcommand{\isasymZdres}{\isamath{\lhd}}  %requires amssymb
-\newcommand{\isasymZndres}{\isaZhbar{\isamath{\lhd}}}  %requires amssymb
-\newcommand{\isasymZrres}{\isamath{\rhd}}  %requires amssymb
-\newcommand{\isasymZnrres}{\isaZhbar{\isamath{\rhd}}}  %requires amssymb
-\newcommand{\isasymZspot}{\isamath{\bullet}}
-\newcommand{\isasymZproject}{\isamath{\upharpoonright}}  %requires amssymb
-\newcommand{\isasymZsemi}{\isatext{\raise 0.66ex\hbox{\oalign{\hfil\isamath{\scriptscriptstyle\mathrm{o}}\hfil\cr\hfil\isamath{\scriptscriptstyle\mathrm{9}}\hfil}}}}
-\newcommand{\isasymZtypecolon}{\isatext{\raise 0.6ex\hbox{\oalign{\hfil\isamath{\scriptscriptstyle\mathrm{o}}\hfil\cr\hfil\isamath{\scriptscriptstyle\mathrm{o}}\hfil}}}}
-\newcommand{\isasymZhide}{\isamath{\backslash}}
-\newcommand{\isasymZcat}{\isatext{\raise 0.8ex\hbox{\isamath{\mathchar\frown}}}}
-\newcommand{\isasymZinbag}{\isatext{\ooalign{\isamath{\sqsubset\mkern-1mu}\cr\isamath{-\mkern-1mu}\cr}}}
-
-\newcommand{\isasymhole}{\isatext{\normalfont\rmfamily\wasylozenge}}  %requires wasysym
-\newcommand{\isasymnewline}{\isatext{\fbox{$\hookleftarrow$}}}
-\newcommand{\isasymcomment}{\isatext{\isastylecmt---}}
-\newcommand{\isasymproof}{\isamath{\,\langle\mathit{proof}\rangle}}
-\newcommand{\isasymopen}{\isatext{\guilsinglleft}}
-\newcommand{\isasymclose}{\isatext{\guilsinglright}}
-\newcommand{\isasymcheckmark}{\isatext{\ding{51}}}  %requires pifont
-\newcommand{\isasymcrossmark}{\isatext{\ding{55}}}  %requires pifont
-\newcommand{\isactrlmarker}{\isatext{\ding{48}}}  %requires pifont
-\newcommand{\isactrltry}{\isakeywordcontrol{try}}
-\newcommand{\isactrlcan}{\isakeywordcontrol{can}}
-\newcommand{\isactrlassert}{\isakeywordcontrol{assert}}
-\newcommand{\isactrlcancel}{\isakeywordcontrol{cancel}}
-\newcommand{\isactrlbinding}{\isakeywordcontrol{binding}}
-\newcommand{\isactrlclass}{\isakeywordcontrol{class}}
-\newcommand{\isactrlclassUNDERSCOREsyntax}{\isakeywordcontrol{class{\isacharunderscore}syntax}}
-\newcommand{\isactrlcommandUNDERSCOREkeyword}{\isakeywordcontrol{command{\isacharunderscore}keyword}}
-\newcommand{\isactrlconst}{\isakeywordcontrol{const}}
-\newcommand{\isactrlconstUNDERSCOREabbrev}{\isakeywordcontrol{const{\isacharunderscore}abbrev}}
-\newcommand{\isactrlconstUNDERSCOREname}{\isakeywordcontrol{const{\isacharunderscore}name}}
-\newcommand{\isactrlconstUNDERSCOREsyntax}{\isakeywordcontrol{const{\isacharunderscore}syntax}}
-\newcommand{\isactrlcontext}{\isakeywordcontrol{context}}
-\newcommand{\isactrlcprop}{\isakeywordcontrol{cprop}}
-\newcommand{\isactrlcterm}{\isakeywordcontrol{cterm}}
-\newcommand{\isactrlctyp}{\isakeywordcontrol{ctyp}}
-\newcommand{\isactrldir}{\isakeywordcontrol{dir}}
-\newcommand{\isactrlfile}{\isakeywordcontrol{file}}
-\newcommand{\isactrlhere}{\isakeywordcontrol{here}}
-\newcommand{\isactrlinstantiate}{\isakeywordcontrol{instantiate}}
-\newcommand{\isactrlkeyword}{\isakeywordcontrol{keyword}}
-\newcommand{\isactrllatex}{\isakeywordcontrol{latex}}
-\newcommand{\isactrllocale}{\isakeywordcontrol{locale}}
-\newcommand{\isactrlmakeUNDERSCOREjudgment}{\isakeywordcontrol{make{\isacharunderscore}judgment}}
-\newcommand{\isactrldestUNDERSCOREjudgment}{\isakeywordcontrol{dest{\isacharunderscore}judgment}}
-\newcommand{\isactrlmakeUNDERSCOREstring}{\isakeywordcontrol{make{\isacharunderscore}string}}
-\newcommand{\isactrlmasterUNDERSCOREdir}{\isakeywordcontrol{master{\isacharunderscore}dir}}
-\newcommand{\isactrlmethod}{\isakeywordcontrol{method}}
-\newcommand{\isactrlnamedUNDERSCOREtheorems}{\isakeywordcontrol{named{\isacharunderscore}theorems}}
-\newcommand{\isactrlnonterminal}{\isakeywordcontrol{nonterminal}}
-\newcommand{\isactrloracleUNDERSCOREname}{\isakeywordcontrol{oracle{\isacharunderscore}name}}
-\newcommand{\isactrlpath}{\isakeywordcontrol{path}}
-\newcommand{\isactrlpathUNDERSCOREbinding}{\isakeywordcontrol{path{\isacharunderscore}binding}}
-\newcommand{\isactrlplugin}{\isakeywordcontrol{plugin}}
-\newcommand{\isactrlprint}{\isakeywordcontrol{print}}
-\newcommand{\isactrlprop}{\isakeywordcontrol{prop}}
-\newcommand{\isactrlscala}{\isakeywordcontrol{scala}}
-\newcommand{\isactrlscalaUNDERSCOREfunction}{\isakeywordcontrol{scala{\isacharunderscore}function}}
-\newcommand{\isactrlscalaUNDERSCOREmethod}{\isakeywordcontrol{scala{\isacharunderscore}method}}
-\newcommand{\isactrlscalaUNDERSCOREobject}{\isakeywordcontrol{scala{\isacharunderscore}object}}
-\newcommand{\isactrlscalaUNDERSCOREtype}{\isakeywordcontrol{scala{\isacharunderscore}type}}
-\newcommand{\isactrlsimproc}{\isakeywordcontrol{simproc}}
-\newcommand{\isactrlsort}{\isakeywordcontrol{sort}}
-\newcommand{\isactrlsyntaxUNDERSCOREconst}{\isakeywordcontrol{syntax{\isacharunderscore}const}}
-\newcommand{\isactrlsystemUNDERSCOREoption}{\isakeywordcontrol{system{\isacharunderscore}option}}
-\newcommand{\isactrlterm}{\isakeywordcontrol{term}}
-\newcommand{\isactrltheory}{\isakeywordcontrol{theory}}
-\newcommand{\isactrltheoryUNDERSCOREcontext}{\isakeywordcontrol{theory{\isacharunderscore}context}}
-\newcommand{\isactrltyp}{\isakeywordcontrol{typ}}
-\newcommand{\isactrltypeUNDERSCOREabbrev}{\isakeywordcontrol{type{\isacharunderscore}abbrev}}
-\newcommand{\isactrltypeUNDERSCOREname}{\isakeywordcontrol{type{\isacharunderscore}name}}
-\newcommand{\isactrltypeUNDERSCOREsyntax}{\isakeywordcontrol{type{\isacharunderscore}syntax}}
-\newcommand{\isactrlundefined}{\isakeywordcontrol{undefined}}
-\newcommand{\isactrltvar}{\isakeywordcontrol{tvar}}
-\newcommand{\isactrlvar}{\isakeywordcontrol{var}}
-\newcommand{\isactrlConst}{\isakeywordcontrol{Const}}
-\newcommand{\isactrlConstUNDERSCORE}{\isakeywordcontrol{Const{\isacharunderscore}}}
-\newcommand{\isactrlConstUNDERSCOREfn}{\isakeywordcontrol{Const{\isacharunderscore}fn}}
-\newcommand{\isactrlType}{\isakeywordcontrol{Type}}
-\newcommand{\isactrlTypeUNDERSCOREfn}{\isakeywordcontrol{Type{\isacharunderscore}fn}}
-
-\newcommand{\isactrlcode}{\isakeywordcontrol{code}}
-\newcommand{\isactrlcomputation}{\isakeywordcontrol{computation}}
-\newcommand{\isactrlcomputationUNDERSCOREconv}{\isakeywordcontrol{computation{\isacharunderscore}conv}}
-\newcommand{\isactrlcomputationUNDERSCOREcheck}{\isakeywordcontrol{computation{\isacharunderscore}check}}
-\newcommand{\isactrlifUNDERSCORElinux}{\isakeywordcontrol{if{\isacharunderscore}linux}}
-\newcommand{\isactrlifUNDERSCOREmacos}{\isakeywordcontrol{if{\isacharunderscore}macos}}
-\newcommand{\isactrlifUNDERSCOREwindows}{\isakeywordcontrol{if{\isacharunderscore}windows}}
-\newcommand{\isactrlifUNDERSCOREunix}{\isakeywordcontrol{if{\isacharunderscore}unix}}
diff --git a/thys/MDP-Algorithms/output/outline/isabelletags.sty b/thys/MDP-Algorithms/output/outline/isabelletags.sty
deleted file mode 100644
--- a/thys/MDP-Algorithms/output/outline/isabelletags.sty
+++ /dev/null
@@ -1,20 +0,0 @@
-%plain TeX version of comment package -- much faster!
-\let\isafmtname\fmtname\def\fmtname{plain}
-\usepackage{comment}
-\let\fmtname\isafmtname
-
-\newcommand{\isakeeptag}[1]%
-{\includecomment{isadelim#1}\includecomment{isatag#1}\csarg\def{isafold#1}{}}
-\newcommand{\isadroptag}[1]%
-{\excludecomment{isadelim#1}\excludecomment{isatag#1}\csarg\def{isafold#1}{}}
-\newcommand{\isafoldtag}[1]%
-{\includecomment{isadelim#1}\excludecomment{isatag#1}\csarg\def{isafold#1}{\isafold{#1}}}
-
-\isafoldtag{ML}
-\isakeeptag{document}
-\isakeeptag{important}
-\isadroptag{invisible}
-\isafoldtag{proof}
-\isakeeptag{theory}
-\isakeeptag{unimportant}
-\isakeeptag{visible}
diff --git a/thys/MDP-Algorithms/output/outline/pdfsetup.sty b/thys/MDP-Algorithms/output/outline/pdfsetup.sty
deleted file mode 100644
--- a/thys/MDP-Algorithms/output/outline/pdfsetup.sty
+++ /dev/null
@@ -1,7 +0,0 @@
-%%
-%% default hyperref setup (both for pdf and dvi output)
-%%
-
-\usepackage{color}
-\definecolor{linkcolor}{rgb}{0,0,0.5}
-\usepackage[colorlinks=true,linkcolor=linkcolor,citecolor=linkcolor,filecolor=linkcolor,urlcolor=linkcolor,pdfpagelabels]{hyperref}
diff --git a/thys/MDP-Algorithms/output/outline/railsetup.sty b/thys/MDP-Algorithms/output/outline/railsetup.sty
deleted file mode 100644
--- a/thys/MDP-Algorithms/output/outline/railsetup.sty
+++ /dev/null
@@ -1,1202 +0,0 @@
-% rail.sty - style file to support railroad diagrams
-%
-% 09-Jul-90 L. Rooijakkers
-% 08-Oct-90 L. Rooijakkers	fixed centering bug when \rail@tmpc<0.
-% 07-Feb-91 L. Rooijakkers	added \railoptions command, indexing
-% 08-Feb-91 L. Rooijakkers	minor fixes
-% 28-Jun-94 K. Barthelmann	turned into LaTeX2e package
-% 08-Dec-96 K. Barthelmann	replaced \@writefile
-% 13-Dec-96 K. Barthelmann	cleanup
-% 22-Feb-98 K. Barthelmann	fixed catcodes of special characters
-% 18-Apr-98 K. Barthelmann	fixed \par handling
-% 19-May-98 J. Olsson		Added new macros to support arrow heads.
-% 26-Jul-98 K. Barthelmann	changed \par to output newlines
-% 02-May-11 M. Wenzel           default setup for Isabelle
-%
-% This style file needs to be used in conjunction with the 'rail'
-% program. Running LaTeX as 'latex file' produces file.rai, which should be
-% processed by Rail with 'rail file'. This produces file.rao, which will
-% be picked up by LaTeX on the next 'latex file' run.
-%
-% LaTeX will warn if there is no file.rao or it's out of date.
-%
-% The macros in this file thus consist of two parts: those that read and
-% write the .rai and .rao files, and those that do the actual formatting
-% of the railroad diagrams.
-
-\NeedsTeXFormat{LaTeX2e}
-\ProvidesPackage{rail}[1998/05/19]
-
-% railroad diagram formatting parameters (user level)
-% all of these are copied into their internal versions by \railinit
-%
-% \railunit : \unitlength within railroad diagrams
-% \railextra : extra length at outside of diagram
-% \railboxheight : height of ovals and frames
-% \railboxskip : vertical space between lines
-% \railboxleft : space to the left of a box
-% \railboxright : space to the right of a box
-% \railovalspace : extra space around contents of oval
-% \railframespace : extra space around contents of frame
-% \railtextleft : space to the left of text
-% \railtextright : space to the right of text
-% \railtextup : space to lift text up
-% \railjoinsize : circle size of join/split arcs
-% \railjoinadjust : space to adjust join
-%
-% \railnamesep : separator between name and rule body
-
-\newlength\railunit
-\newlength\railextra
-\newlength\railboxheight
-\newlength\railboxskip
-\newlength\railboxleft
-\newlength\railboxright
-\newlength\railovalspace
-\newlength\railframespace
-\newlength\railtextleft
-\newlength\railtextright
-\newlength\railtextup
-\newlength\railjoinsize
-\newlength\railjoinadjust
-\newlength\railnamesep
-
-% initialize the parameters
-
-\setlength\railunit{1sp}
-\setlength\railextra{4ex}
-\setlength\railboxleft{1ex}
-\setlength\railboxright{1ex}
-\setlength\railovalspace{2ex}
-\setlength\railframespace{2ex}
-\setlength\railtextleft{1ex}
-\setlength\railtextright{1ex}
-\setlength\railjoinadjust{0pt}
-\setlength\railnamesep{1ex}
-
-\DeclareOption{10pt}{
-  \setlength\railboxheight{16pt}
-  \setlength\railboxskip{24pt}
-  \setlength\railtextup{5pt}
-  \setlength\railjoinsize{16pt}
-}
-\DeclareOption{11pt}{
-  \setlength\railboxheight{16pt}
-  \setlength\railboxskip{24pt}
-  \setlength\railtextup{5pt}
-  \setlength\railjoinsize{16pt}
-}
-\DeclareOption{12pt}{
-  \setlength\railboxheight{20pt}
-  \setlength\railboxskip{28pt}
-  \setlength\railtextup{6pt}
-  \setlength\railjoinsize{20pt}
-}
-
-\ExecuteOptions{10pt}
-\ProcessOptions
-
-% internal versions of the formatting parameters
-%
-% \rail@extra   : \railextra
-% \rail@boxht   : \railboxheight
-% \rail@boxsp   : \railboxskip
-% \rail@boxlf   : \railboxleft
-% \rail@boxrt   : \railboxright
-% \rail@boxhht  : \railboxheight / 2
-% \rail@ovalsp  : \railovalspace
-% \rail@framesp : \railframespace
-% \rail@textlf  : \railtextleft
-% \rail@textrt  : \railtextright
-% \rail@textup  : \railtextup
-% \rail@joinsz  : \railjoinsize
-% \rail@joinhsz : \railjoinsize / 2
-% \rail@joinadj : \railjoinadjust
-%
-% \railinit : internalize all of the parameters.
-
-\newcount\rail@extra
-\newcount\rail@boxht
-\newcount\rail@boxsp
-\newcount\rail@boxlf
-\newcount\rail@boxrt
-\newcount\rail@boxhht
-\newcount\rail@ovalsp
-\newcount\rail@framesp
-\newcount\rail@textlf
-\newcount\rail@textrt
-\newcount\rail@textup
-\newcount\rail@joinsz
-\newcount\rail@joinhsz
-\newcount\rail@joinadj
-
-\newcommand\railinit{
-\rail@extra=\railextra
-\divide\rail@extra by \railunit
-\rail@boxht=\railboxheight
-\divide\rail@boxht by \railunit
-\rail@boxsp=\railboxskip
-\divide\rail@boxsp by \railunit
-\rail@boxlf=\railboxleft
-\divide\rail@boxlf by \railunit
-\rail@boxrt=\railboxright
-\divide\rail@boxrt by \railunit
-\rail@boxhht=\railboxheight
-\divide\rail@boxhht by \railunit
-\divide\rail@boxhht by 2
-\rail@ovalsp=\railovalspace
-\divide\rail@ovalsp by \railunit
-\rail@framesp=\railframespace
-\divide\rail@framesp by \railunit
-\rail@textlf=\railtextleft
-\divide\rail@textlf by \railunit
-\rail@textrt=\railtextright
-\divide\rail@textrt by \railunit
-\rail@textup=\railtextup
-\divide\rail@textup by \railunit
-\rail@joinsz=\railjoinsize
-\divide\rail@joinsz by \railunit
-\rail@joinhsz=\railjoinsize
-\divide\rail@joinhsz by \railunit
-\divide\rail@joinhsz by 2
-\rail@joinadj=\railjoinadjust
-\divide\rail@joinadj by \railunit
-}
-
-\AtBeginDocument{\railinit}
-
-% \rail@param : declarations for list environment
-%
-% \railparam{TEXT} : sets \rail@param to TEXT
-%
-% \rail@reserved : characters reserved for grammar
-
-\newcommand\railparam[1]{
-\def\rail@param{
-  \setlength\leftmargin{0pt}\setlength\rightmargin{0pt}
-  \setlength\labelwidth{0pt}\setlength\labelsep{0pt}
-  \setlength\itemindent{0pt}\setlength\listparindent{0pt}
-  #1
-}
-}
-\railparam{}
-
-\newtoks\rail@reserved
-\rail@reserved={:;|*+?[]()'"}
-
-% \rail@termfont : format setup for terminals
-%
-% \rail@nontfont : format setup for nonterminals
-%
-% \rail@annofont : format setup for annotations
-%
-% \rail@rulefont : format setup for rule names
-%
-% \rail@indexfont : format setup for index entry
-%
-% \railtermfont{TEXT} : set terminal format setup to TEXT
-%
-% \railnontermfont{TEXT} : set nonterminal format setup to TEXT
-%
-% \railannotatefont{TEXT} : set annotation format setup to TEXT
-%
-% \railnamefont{TEXT} : set rule name format setup to TEXT
-%
-% \railindexfont{TEXT} : set index entry format setup to TEXT
-
-\def\rail@termfont{\ttfamily\upshape}
-\def\rail@nontfont{\rmfamily\upshape}
-\def\rail@annofont{\rmfamily\itshape}
-\def\rail@namefont{\rmfamily\itshape}
-\def\rail@indexfont{\rmfamily\itshape}
-
-\newcommand\railtermfont[1]{
-\def\rail@termfont{#1}
-}
-
-\newcommand\railnontermfont[1]{
-\def\rail@nontfont{#1}
-}
-
-\newcommand\railannotatefont[1]{
-\def\rail@annofont{#1}
-}
-
-\newcommand\railnamefont[1]{
-\def\rail@namefont{#1}
-}
-
-\newcommand\railindexfont[1]{
-\def\rail@indexfont{#1}
-}
-
-% railroad read/write macros
-%
-% \begin{rail} TEXT \end{rail} : TEXT is written out to the .rai file,
-%                                as \rail@i{NR}{TEXT}. Then the matching
-%                                \rail@o{NR}{FMT} from the .rao file is
-%                                executed (if defined).
-%
-% \railoptions{OPTIONS} : OPTIONS are written out to the .rai file,
-%                         as \rail@p{OPTIONS}.
-%
-% \railterm{IDENT,IDENT,...} : format IDENT as terminals. writes out
-%                              \rail@t{IDENT} to the .rai file
-%
-% \railalias{IDENT}{TEXT} : format IDENT as TEXT. defines \rail@t@IDENT as
-%                           TEXT.
-%
-% \railtoken{IDENT}{TEXT} : abbreviates \railalias{IDENT}{TEXT}\railterm{IDENT}
-%                           (for backward compatibility)
-%
-% \rail@setcodes : guards special characters
-%
-% \rail@makeother{CHARACTER} : sets \catcode of CHARACTER to "other"
-%                              used inside a loop for \rail@setcodes
-%
-% \rail@nr : railroad diagram counter
-%
-% \ifrail@match : current \rail@i{NR}{TEXT} matches
-%
-% \rail@first : actions to be done first. read in .rao file,
-%               open .rai file if \@filesw true, undefine \rail@first.
-%               executed from \begin{rail}, \railoptions and \railterm.
-%
-% \rail@i{NR}{TEXT} : defines \rail@i@NR as TEXT. written to the .rai
-%                     file by \rail, read from the .rao file by
-%                     \rail@first
-%
-% \rail@t{IDENT} : tells Rail that IDENT is to be custom formatted,
-%                  written to the .rai file by \railterm.
-%
-% \rail@o{NR}{TEXT} : defines \rail@o@NR as TEXT, read from the .rao
-%                     file by \rail@first.
-%
-% \rail@p{OPTIONS} : pass options to rail, written to the .rai file by
-%                    \railoptions
-%
-% \rail@write{TEXT} : write TEXT to the .rai file
-%
-% \rail@warn : warn user for mismatching diagrams
-%
-% \rail@endwarn : either \relax or \rail@warn
-%
-% \ifrail@all : checked at the end of the document
-
-\def\rail@makeother#1{
-  \expandafter\catcode\expandafter`\csname\string #1\endcsname=12
-}
-
-\def\rail@setcodes{
-\let\par=\relax
-\let\\=\relax
-\expandafter\@tfor\expandafter\rail@symbol\expandafter:\expandafter=%
-  \the\rail@reserved
-\do{\expandafter\rail@makeother\rail@symbol}
-}
-
-\newcount\rail@nr
-
-\newif\ifrail@all
-\rail@alltrue
-
-\newif\ifrail@match
-
-\def\rail@first{
-\begingroup
-\makeatletter
-\rail@setcodes
-\InputIfFileExists{\jobname.rao}{}{\PackageInfo{rail}{No file \jobname.rao}}
-\makeatother
-\endgroup
-\if@filesw
-\newwrite\tf@rai
-\immediate\openout\tf@rai=\jobname.rai
-\fi
-\global\let\rail@first=\relax
-}
-
-\long\def\rail@body#1\end{
-{
-\newlinechar=`^^J
-\def\par{\string\par^^J}
-\rail@write{\string\rail@i{\number\rail@nr}{#1}}
-}
-\xdef\rail@i@{#1}
-\end
-}
-
-\newenvironment{rail}{
-\global\advance\rail@nr by 1
-\rail@first
-\begingroup
-\rail@setcodes
-\rail@body
-}{
-\endgroup
-\rail@matchtrue
-\@ifundefined{rail@o@\number\rail@nr}{\rail@matchfalse}{}
-\expandafter\ifx\csname rail@i@\number\rail@nr\endcsname\rail@i@
-\else
-\rail@matchfalse
-\fi
-\ifrail@match
-\csname rail@o@\number\rail@nr\endcsname
-\else
-\PackageWarning{rail}{Railroad diagram {\number\rail@nr} doesn't match}
-\global\let\rail@endwarn=\rail@warn
-\begin{list}{}{\rail@param}
-\rail@begin{1}{}
-\rail@setbox{\bfseries ???}
-\rail@oval
-\rail@end
-\end{list}
-\fi
-}
-
-\newcommand\railoptions[1]{
-\rail@first
-\rail@write{\string\rail@p{#1}}
-}
-
-\newcommand\railterm[1]{
-\rail@first
-\@for\rail@@:=#1\do{
-\rail@write{\string\rail@t{\rail@@}}
-}
-}
-
-\newcommand\railalias[2]{
-\expandafter\def\csname rail@t@#1\endcsname{#2}
-}
-
-\newcommand\railtoken[2]{\railalias{#1}{#2}\railterm{#1}}
-
-\long\def\rail@i#1#2{
-\expandafter\gdef\csname rail@i@#1\endcsname{#2}
-}
-
-\def\rail@o#1#2{
-\expandafter\gdef\csname rail@o@#1\endcsname{
-\begin{list}{}{\rail@param}
-#2
-\end{list}
-}
-}
-
-\def\rail@t#1{}
-
-\def\rail@p#1{}
-
-\long\def\rail@write#1{\@ifundefined{tf@rai}{}{\immediate\write\tf@rai{#1}}}
-
-\def\rail@warn{
-\PackageWarningNoLine{rail}{Railroad diagram(s) may have changed.
-                            Use 'rail' and rerun}
-}
-
-\let\rail@endwarn=\relax
-
-\AtEndDocument{\rail@endwarn}
-
-% index entry macro
-%
-% \rail@index{IDENT} : add index entry for IDENT
-
-\def\rail@index#1{
-\index{\rail@indexfont#1}
-}
-
-% railroad formatting primitives
-%
-% \rail@x : current x
-% \rail@y : current y
-% \rail@ex : current end x
-% \rail@sx : starting x for \rail@cr
-% \rail@rx : rightmost previous x for \rail@cr
-%
-% \rail@tmpa : temporary count
-% \rail@tmpb : temporary count
-% \rail@tmpc : temporary count
-%
-% \rail@put : put at (\rail@x,\rail@y)
-% \rail@vput : put vector at (\rail@x,\rail@y)
-%
-% \rail@eline : end line by drawing from \rail@ex to \rail@x
-%
-% \rail@vreline : end line by drawing a vector from \rail@x to \rail@ex
-%
-% \rail@vleline : end line by drawing a vector from \rail@ex to \rail@x
-%
-% \rail@sety{LEVEL} : set \rail@y to level LEVEL
-
-\newcount\rail@x
-\newcount\rail@y
-\newcount\rail@ex
-\newcount\rail@sx
-\newcount\rail@rx
-
-\newcount\rail@tmpa
-\newcount\rail@tmpb
-\newcount\rail@tmpc
-
-\def\rail@put{\put(\number\rail@x,\number\rail@y)}
-
-\def\rail@vput{\put(\number\rail@ex,\number\rail@y)}
-
-\def\rail@eline{
-\rail@tmpb=\rail@x
-\advance\rail@tmpb by -\rail@ex
-\rail@put{\line(-1,0){\number\rail@tmpb}}
-}
-
-\def\rail@vreline{
-\rail@tmpb=\rail@x
-\advance\rail@tmpb by -\rail@ex
-\rail@vput{\vector(1,0){\number\rail@tmpb}}
-}
-
-\def\rail@vleline{
-\rail@tmpb=\rail@x
-\advance\rail@tmpb by -\rail@ex
-\rail@put{\vector(-1,0){\number\rail@tmpb}}
-}
-
-\def\rail@sety#1{
-\rail@y=#1
-\multiply\rail@y by -\rail@boxsp
-\advance\rail@y by -\rail@boxht
-}
-
-% \rail@begin{HEIGHT}{NAME} : begin a railroad diagram of height HEIGHT
-%
-% \rail@end : end a railroad diagram
-%
-% \rail@expand{IDENT} : expand IDENT
-
-\def\rail@begin#1#2{
-\item
-\begin{minipage}[t]{\linewidth}
-\ifx\@empty#2\else
-{\rail@namefont \rail@expand{#2}}\\*[\railnamesep]
-\fi
-\unitlength=\railunit
-\rail@tmpa=#1
-\multiply\rail@tmpa by \rail@boxsp
-\begin{picture}(0,\number\rail@tmpa)(0,-\number\rail@tmpa)
-\rail@ex=0
-\rail@rx=0
-\rail@x=\rail@extra
-\rail@sx=\rail@x
-\rail@sety{0}
-}
-
-\def\rail@end{
-\advance\rail@x by \rail@extra
-\rail@eline
-\end{picture}
-\end{minipage}
-}
-
-\def\rail@vend{
-\advance\rail@x by \rail@extra
-\rail@vreline
-\end{picture}
-\end{minipage}
-}
-
-\def\rail@expand#1{\@ifundefined{rail@t@#1}{#1}{\csname rail@t@#1\endcsname}}
-
-% \rail@token{TEXT}[ANNOT] : format token TEXT with annotation
-% \rail@ltoken{TEXT}[ANNOT] : format token TEXT with annotation, arrow left
-% \rail@rtoken{TEXT}[ANNOT] : format token TEXT with annotation, arrow right
-%
-% \rail@ctoken{TEXT}[ANNOT] : format token TEXT centered with annotation
-% \rail@lctoken{TEXT}[ANNOT] : format token TEXT centered with annotation, arrow left
-% \rail@rctoken{TEXT}[ANNOT] : format token TEXT centered with annotation, arrow right
-%
-% \rail@nont{TEXT}[ANNOT] : format nonterminal TEXT with annotation
-% \rail@lnont{TEXT}[ANNOT] : format nonterminal TEXT with annotation, arrow left
-% \rail@rnont{TEXT}[ANNOT] : format nonterminal TEXT with annotation. arrow right
-%
-% \rail@cnont{TEXT}[ANNOT] : format nonterminal TEXT centered with annotation
-% \rail@lcnont{TEXT}[ANNOT] : format nonterminal TEXT centered with annotation,
-%                             arrow left
-% \rail@rcnont{TEXT}[ANNOT] : format nonterminal TEXT centered with annotation,
-%                             arrow right
-%
-% \rail@term{TEXT}[ANNOT] : format terminal TEXT with annotation
-% \rail@lterm{TEXT}[ANNOT] : format terminal TEXT with annotation, arrow left
-% \rail@rterm{TEXT}[ANNOT] : format terminal TEXT with annotation, arrow right
-%
-% \rail@cterm{TEXT}[ANNOT] : format terminal TEXT centered with annotation
-% \rail@lcterm{TEXT}[ANNOT] : format terminal TEXT centered with annotation, arrow left
-% \rail@rcterm{TEXT}[ANNOT] : format terminal TEXT centered with annotation,
-%                             arrow right
-%
-% \rail@annote[TEXT] : format TEXT as annotation
-
-\def\rail@token#1[#2]{
-\rail@setbox{%
-{\rail@termfont \rail@expand{#1}}\ifx\@empty#2\else\ {\rail@annofont #2}\fi
-}
-\rail@oval
-}
-
-\def\rail@ltoken#1[#2]{
-\rail@setbox{%
-{\rail@termfont \rail@expand{#1}}\ifx\@empty#2\else\ {\rail@annofont #2}\fi
-}
-\rail@vloval
-}
-
-\def\rail@rtoken#1[#2]{
-\rail@setbox{%
-{\rail@termfont \rail@expand{#1}}\ifx\@empty#2\else\ {\rail@annofont #2}\fi
-}
-\rail@vroval
-}
-
-\def\rail@ctoken#1[#2]{
-\rail@setbox{%
-{\rail@termfont \rail@expand{#1}}\ifx\@empty#2\else\ {\rail@annofont #2}\fi
-}
-\rail@coval
-}
-
-\def\rail@lctoken#1[#2]{
-\rail@setbox{%
-{\rail@termfont \rail@expand{#1}}\ifx\@empty#2\else\ {\rail@annofont #2}\fi
-}
-\rail@vlcoval
-}
-
-\def\rail@rctoken#1[#2]{
-\rail@setbox{%
-{\rail@termfont \rail@expand{#1}}\ifx\@empty#2\else\ {\rail@annofont #2}\fi
-}
-\rail@vrcoval
-}
-
-\def\rail@nont#1[#2]{
-\rail@setbox{%
-{\rail@nontfont \rail@expand{#1}}\ifx\@empty#2\else\ {\rail@annofont #2}\fi
-}
-\rail@frame
-}
-
-\def\rail@lnont#1[#2]{
-\rail@setbox{%
-{\rail@nontfont \rail@expand{#1}}\ifx\@empty#2\else\ {\rail@annofont #2}\fi
-}
-\rail@vlframe
-}
-
-\def\rail@rnont#1[#2]{
-\rail@setbox{%
-{\rail@nontfont \rail@expand{#1}}\ifx\@empty#2\else\ {\rail@annofont #2}\fi
-}
-\rail@vrframe
-}
-
-\def\rail@cnont#1[#2]{
-\rail@setbox{%
-{\rail@nontfont \rail@expand{#1}}\ifx\@empty#2\else\ {\rail@annofont #2}\fi
-}
-\rail@cframe
-}
-
-\def\rail@lcnont#1[#2]{
-\rail@setbox{%
-{\rail@nontfont \rail@expand{#1}}\ifx\@empty#2\else\ {\rail@annofont #2}\fi
-}
-\rail@vlcframe
-}
-
-\def\rail@rcnont#1[#2]{
-\rail@setbox{%
-{\rail@nontfont \rail@expand{#1}}\ifx\@empty#2\else\ {\rail@annofont #2}\fi
-}
-\rail@vrcframe
-}
-
-\def\rail@term#1[#2]{
-\rail@setbox{%
-{\rail@termfont #1}\ifx\@empty#2\else\ {\rail@annofont #2}\fi
-}
-\rail@oval
-}
-
-\def\rail@lterm#1[#2]{
-\rail@setbox{%
-{\rail@termfont #1}\ifx\@empty#2\else\ {\rail@annofont #2}\fi
-}
-\rail@vloval
-}
-
-\def\rail@rterm#1[#2]{
-\rail@setbox{%
-{\rail@termfont #1}\ifx\@empty#2\else\ {\rail@annofont #2}\fi
-}
-\rail@vroval
-}
-
-\def\rail@cterm#1[#2]{
-\rail@setbox{%
-{\rail@termfont #1}\ifx\@empty#2\else\ {\rail@annofont #2}\fi
-}
-\rail@coval
-}
-
-\def\rail@lcterm#1[#2]{
-\rail@setbox{%
-{\rail@termfont #1}\ifx\@empty#2\else\ {\rail@annofont #2}\fi
-}
-\rail@vlcoval
-}
-
-\def\rail@rcterm#1[#2]{
-\rail@setbox{%
-{\rail@termfont #1}\ifx\@empty#2\else\ {\rail@annofont #2}\fi
-}
-\rail@vrcoval
-}
-
-\def\rail@annote[#1]{
-\rail@setbox{\rail@annofont #1}
-\rail@text
-}
-
-% \rail@box : temporary box for \rail@oval and \rail@frame
-%
-% \rail@setbox{TEXT} : set \rail@box to TEXT, set \rail@tmpa to width
-%
-% \rail@oval : format \rail@box of width \rail@tmpa inside an oval
-% \rail@vloval : format \rail@box of width \rail@tmpa inside an oval, vector left
-% \rail@vroval : format \rail@box of width \rail@tmpa inside an oval, vector right
-%
-% \rail@coval : same as \rail@oval, but centered between \rail@x and
-%               \rail@mx
-% \rail@vlcoval : same as \rail@oval, but centered between \rail@x and
-%                 \rail@mx, vector left
-% \rail@vrcoval : same as \rail@oval, but centered between \rail@x and
-%                 \rail@mx, vector right
-%
-% \rail@frame : format \rail@box of width \rail@tmpa inside a frame
-% \rail@vlframe : format \rail@box of width \rail@tmpa inside a frame, vector left
-% \rail@vrframe : format \rail@box of width \rail@tmpa inside a frame, vector right
-%
-% \rail@cframe : same as \rail@frame, but centered between \rail@x and
-%                \rail@mx
-% \rail@vlcframe : same as \rail@frame, but centered between \rail@x and
-%                  \rail@mx, vector left
-% \rail@vrcframe : same as \rail@frame, but centered between \rail@x and
-%                  \rail@mx, vector right
-%
-% \rail@text : format \rail@box of width \rail@tmpa above the line
-
-\newbox\rail@box
-
-\def\rail@setbox#1{
-\setbox\rail@box\hbox{\strut#1}
-\rail@tmpa=\wd\rail@box
-\divide\rail@tmpa by \railunit
-}
-
-\def\rail@oval{
-\advance\rail@x by \rail@boxlf
-\rail@eline
-\advance\rail@tmpa by \rail@ovalsp
-\ifnum\rail@tmpa<\rail@boxht\rail@tmpa=\rail@boxht\fi
-\rail@tmpb=\rail@tmpa
-\divide\rail@tmpb by 2
-\advance\rail@y by -\rail@boxhht
-\rail@put{\makebox(\number\rail@tmpa,\number\rail@boxht){\box\rail@box}}
-\advance\rail@y by \rail@boxhht
-\advance\rail@x by \rail@tmpb
-\rail@put{\oval(\number\rail@tmpa,\number\rail@boxht)}
-\advance\rail@x by \rail@tmpb
-\rail@ex=\rail@x
-\advance\rail@x by \rail@boxrt
-}
-
-\def\rail@vloval{
-\advance\rail@x by \rail@boxlf
-\rail@eline
-\advance\rail@tmpa by \rail@ovalsp
-\ifnum\rail@tmpa<\rail@boxht\rail@tmpa=\rail@boxht\fi
-\rail@tmpb=\rail@tmpa
-\divide\rail@tmpb by 2
-\advance\rail@y by -\rail@boxhht
-\rail@put{\makebox(\number\rail@tmpa,\number\rail@boxht){\box\rail@box}}
-\advance\rail@y by \rail@boxhht
-\advance\rail@x by \rail@tmpb
-\rail@put{\oval(\number\rail@tmpa,\number\rail@boxht)}
-\advance\rail@x by \rail@tmpb
-\rail@ex=\rail@x
-\advance\rail@x by \rail@boxrt
-\rail@vleline
-}
-
-\def\rail@vroval{
-\advance\rail@x by \rail@boxlf
-\rail@vreline
-\advance\rail@tmpa by \rail@ovalsp
-\ifnum\rail@tmpa<\rail@boxht\rail@tmpa=\rail@boxht\fi
-\rail@tmpb=\rail@tmpa
-\divide\rail@tmpb by 2
-\advance\rail@y by -\rail@boxhht
-\rail@put{\makebox(\number\rail@tmpa,\number\rail@boxht){\box\rail@box}}
-\advance\rail@y by \rail@boxhht
-\advance\rail@x by \rail@tmpb
-\rail@put{\oval(\number\rail@tmpa,\number\rail@boxht)}
-\advance\rail@x by \rail@tmpb
-\rail@ex=\rail@x
-\advance\rail@x by \rail@boxrt
-}
-
-\def\rail@coval{
-\rail@tmpb=\rail@tmpa
-\advance\rail@tmpb by \rail@ovalsp
-\ifnum\rail@tmpb<\rail@boxht\rail@tmpb=\rail@boxht\fi
-\advance\rail@tmpb by \rail@boxlf
-\advance\rail@tmpb by \rail@boxrt
-\rail@tmpc=\rail@mx
-\advance\rail@tmpc by -\rail@x
-\advance\rail@tmpc by -\rail@tmpb
-\divide\rail@tmpc by 2
-\ifnum\rail@tmpc>0
-\advance\rail@x by \rail@tmpc
-\fi
-\rail@oval
-}
-
-\def\rail@vlcoval{
-\rail@tmpb=\rail@tmpa
-\advance\rail@tmpb by \rail@ovalsp
-\ifnum\rail@tmpb<\rail@boxht\rail@tmpb=\rail@boxht\fi
-\advance\rail@tmpb by \rail@boxlf
-\advance\rail@tmpb by \rail@boxrt
-\rail@tmpc=\rail@mx
-\advance\rail@tmpc by -\rail@x
-\advance\rail@tmpc by -\rail@tmpb
-\divide\rail@tmpc by 2
-\ifnum\rail@tmpc>0
-\advance\rail@x by \rail@tmpc
-\fi
-\rail@vloval
-}
-
-\def\rail@vrcoval{
-\rail@tmpb=\rail@tmpa
-\advance\rail@tmpb by \rail@ovalsp
-\ifnum\rail@tmpb<\rail@boxht\rail@tmpb=\rail@boxht\fi
-\advance\rail@tmpb by \rail@boxlf
-\advance\rail@tmpb by \rail@boxrt
-\rail@tmpc=\rail@mx
-\advance\rail@tmpc by -\rail@x
-\advance\rail@tmpc by -\rail@tmpb
-\divide\rail@tmpc by 2
-\ifnum\rail@tmpc>0
-\advance\rail@x by \rail@tmpc
-\fi
-\rail@vroval
-}
-
-\def\rail@frame{
-\advance\rail@x by \rail@boxlf
-\rail@eline
-\advance\rail@tmpa by \rail@framesp
-\ifnum\rail@tmpa<\rail@boxht\rail@tmpa=\rail@boxht\fi
-\advance\rail@y by -\rail@boxhht
-\rail@put{\framebox(\number\rail@tmpa,\number\rail@boxht){\box\rail@box}}
-\advance\rail@y by \rail@boxhht
-\advance\rail@x by \rail@tmpa
-\rail@ex=\rail@x
-\advance\rail@x by \rail@boxrt
-}
-
-\def\rail@vlframe{
-\advance\rail@x by \rail@boxlf
-\rail@eline
-\advance\rail@tmpa by \rail@framesp
-\ifnum\rail@tmpa<\rail@boxht\rail@tmpa=\rail@boxht\fi
-\advance\rail@y by -\rail@boxhht
-\rail@put{\framebox(\number\rail@tmpa,\number\rail@boxht){\box\rail@box}}
-\advance\rail@y by \rail@boxhht
-\advance\rail@x by \rail@tmpa
-\rail@ex=\rail@x
-\advance\rail@x by \rail@boxrt
-\rail@vleline
-}
-
-\def\rail@vrframe{
-\advance\rail@x by \rail@boxlf
-\rail@vreline
-\advance\rail@tmpa by \rail@framesp
-\ifnum\rail@tmpa<\rail@boxht\rail@tmpa=\rail@boxht\fi
-\advance\rail@y by -\rail@boxhht
-\rail@put{\framebox(\number\rail@tmpa,\number\rail@boxht){\box\rail@box}}
-\advance\rail@y by \rail@boxhht
-\advance\rail@x by \rail@tmpa
-\rail@ex=\rail@x
-\advance\rail@x by \rail@boxrt
-}
-
-\def\rail@cframe{
-\rail@tmpb=\rail@tmpa
-\advance\rail@tmpb by \rail@framesp
-\ifnum\rail@tmpb<\rail@boxht\rail@tmpb=\rail@boxht\fi
-\advance\rail@tmpb by \rail@boxlf
-\advance\rail@tmpb by \rail@boxrt
-\rail@tmpc=\rail@mx
-\advance\rail@tmpc by -\rail@x
-\advance\rail@tmpc by -\rail@tmpb
-\divide\rail@tmpc by 2
-\ifnum\rail@tmpc>0
-\advance\rail@x by \rail@tmpc
-\fi
-\rail@frame
-}
-
-\def\rail@vlcframe{
-\rail@tmpb=\rail@tmpa
-\advance\rail@tmpb by \rail@framesp
-\ifnum\rail@tmpb<\rail@boxht\rail@tmpb=\rail@boxht\fi
-\advance\rail@tmpb by \rail@boxlf
-\advance\rail@tmpb by \rail@boxrt
-\rail@tmpc=\rail@mx
-\advance\rail@tmpc by -\rail@x
-\advance\rail@tmpc by -\rail@tmpb
-\divide\rail@tmpc by 2
-\ifnum\rail@tmpc>0
-\advance\rail@x by \rail@tmpc
-\fi
-\rail@vlframe
-}
-
-\def\rail@vrcframe{
-\rail@tmpb=\rail@tmpa
-\advance\rail@tmpb by \rail@framesp
-\ifnum\rail@tmpb<\rail@boxht\rail@tmpb=\rail@boxht\fi
-\advance\rail@tmpb by \rail@boxlf
-\advance\rail@tmpb by \rail@boxrt
-\rail@tmpc=\rail@mx
-\advance\rail@tmpc by -\rail@x
-\advance\rail@tmpc by -\rail@tmpb
-\divide\rail@tmpc by 2
-\ifnum\rail@tmpc>0
-\advance\rail@x by \rail@tmpc
-\fi
-\rail@vrframe
-}
-
-\def\rail@text{
-\advance\rail@x by \rail@textlf
-\advance\rail@y by \rail@textup
-\rail@put{\box\rail@box}
-\advance\rail@y by -\rail@textup
-\advance\rail@x by \rail@tmpa
-\advance\rail@x by \rail@textrt
-}
-
-% alternatives
-%
-% \rail@jx \rail@jy : current join point
-%
-% \rail@gx \rail@gy \rail@gex \rail@grx : global versions of \rail@x etc,
-%                                         to pass values over group closings
-%
-% \rail@mx : maximum x so far
-%
-% \rail@sy : starting \rail@y for alternatives
-%
-% \rail@jput : put at (\rail@jx,\rail@jy)
-%
-% \rail@joval[PART] : put \oval[PART] with adjust
-
-\newcount\rail@jx
-\newcount\rail@jy
-
-\newcount\rail@gx
-\newcount\rail@gy
-\newcount\rail@gex
-\newcount\rail@grx
-
-\newcount\rail@sy
-\newcount\rail@mx
-
-\def\rail@jput{
-\put(\number\rail@jx,\number\rail@jy)
-}
-
-\def\rail@joval[#1]{
-\advance\rail@jx by \rail@joinadj
-\rail@jput{\oval(\number\rail@joinsz,\number\rail@joinsz)[#1]}
-\advance\rail@jx by -\rail@joinadj
-}
-
-% \rail@barsplit : incoming split for '|'
-%
-% \rail@plussplit : incoming split for '+'
-%
-
-\def\rail@barsplit{
-\advance\rail@jy by -\rail@joinhsz
-\rail@joval[tr]
-\advance\rail@jx by \rail@joinhsz
-}
-
-\def\rail@plussplit{
-\advance\rail@jy by -\rail@joinhsz
-\advance\rail@jx by \rail@joinsz
-\rail@joval[tl]
-\advance\rail@jx by -\rail@joinhsz
-}
-
-% \rail@alt{SPLIT} : start alternatives with incoming split SPLIT
-%
-
-\def\rail@alt#1{
-\rail@sy=\rail@y
-\rail@jx=\rail@x
-\rail@jy=\rail@y
-\advance\rail@x by \rail@joinsz
-\rail@mx=0
-\let\rail@list=\@empty
-\let\rail@comma=\@empty
-\let\rail@split=#1
-\begingroup
-\rail@sx=\rail@x
-\rail@rx=0
-}
-
-% \rail@nextalt{FIX}{Y} : start next alternative at vertical position Y
-%                         and fix-up FIX
-%
-
-\def\rail@nextalt#1#2{
-\global\rail@gx=\rail@x
-\global\rail@gy=\rail@y
-\global\rail@gex=\rail@ex
-\global\rail@grx=\rail@rx
-\endgroup
-#1
-\ifnum\rail@gx>\rail@mx\rail@mx=\rail@gx\fi
-\ifnum\rail@grx>\rail@mx\rail@mx=\rail@grx\fi
-\edef\rail@list{\rail@list\rail@comma\number\rail@gex:\number\rail@gy}
-\def\rail@comma{,}
-\rail@split
-\let\rail@split=\@empty
-\rail@sety{#2}
-\rail@tmpa=\rail@jy
-\advance\rail@tmpa by -\rail@y
-\advance\rail@tmpa by -\rail@joinhsz
-\rail@jput{\line(0,-1){\number\rail@tmpa}}
-\rail@jy=\rail@y
-\advance\rail@jy by \rail@joinhsz
-\advance\rail@jx by \rail@joinhsz
-\rail@joval[bl]
-\advance\rail@jx by -\rail@joinhsz
-\rail@ex=\rail@x
-\begingroup
-\rail@sx=\rail@x
-\rail@rx=0
-}
-
-% \rail@barjoin : outgoing join for first '|' alternative
-%
-% \rail@plusjoin : outgoing join for first '+' alternative
-%
-% \rail@altjoin : join for subsequent alternative
-%
-
-\def\rail@barjoin{
-\ifnum\rail@y<\rail@sy
-\global\rail@gex=\rail@jx
-\else
-\global\rail@gex=\rail@ex
-\fi
-\advance\rail@jy by -\rail@joinhsz
-\rail@joval[tl]
-\advance\rail@jx by -\rail@joinhsz
-\ifnum\rail@y<\rail@sy
-\rail@altjoin
-\fi
-}
-
-\def\rail@plusjoin{
-\global\rail@gex=\rail@ex
-\advance\rail@jy by -\rail@joinhsz
-\advance\rail@jx by -\rail@joinsz
-\rail@joval[tr]
-\advance\rail@jx by \rail@joinhsz
-}
-
-\def\rail@altjoin{
-\rail@eline
-\rail@tmpa=\rail@jy
-\advance\rail@tmpa by -\rail@y
-\advance\rail@tmpa by -\rail@joinhsz
-\rail@jput{\line(0,-1){\number\rail@tmpa}}
-\rail@jy=\rail@y
-\advance\rail@jy by \rail@joinhsz
-\advance\rail@jx by -\rail@joinhsz
-\rail@joval[br]
-\advance\rail@jx by \rail@joinhsz
-}
-
-% \rail@eltsplit EX:Y; : split EX:Y into \rail@ex \rail@y
-%
-% \rail@endalt{JOIN} : end alternatives with outgoing join JOIN
-
-\def\rail@eltsplit#1:#2;{\rail@ex=#1\rail@y=#2}
-
-\def\rail@endalt#1{
-\global\rail@gx=\rail@x
-\global\rail@gy=\rail@y
-\global\rail@gex=\rail@ex
-\global\rail@grx=\rail@rx
-\endgroup
-\ifnum\rail@gx>\rail@mx\rail@mx=\rail@gx\fi
-\ifnum\rail@grx>\rail@mx\rail@mx=\rail@grx\fi
-\edef\rail@list{\rail@list\rail@comma\number\rail@gex:\number\rail@gy}
-\rail@x=\rail@mx
-\rail@jx=\rail@x
-\rail@jy=\rail@sy
-\advance\rail@jx by \rail@joinsz
-\let\rail@join=#1
-\@for\rail@elt:=\rail@list\do{
-\expandafter\rail@eltsplit\rail@elt;
-\rail@join
-\let\rail@join=\rail@altjoin
-}
-\rail@x=\rail@mx
-\rail@y=\rail@sy
-\rail@ex=\rail@gex
-\advance\rail@x by \rail@joinsz
-}
-
-% \rail@bar : start '|' alternatives
-%
-% \rail@nextbar : next '|' alternative
-%
-% \rail@endbar : end '|' alternatives
-%
-
-\def\rail@bar{
-\rail@alt\rail@barsplit
-}
-
-\def\rail@nextbar{
-\rail@nextalt\relax
-}
-
-\def\rail@endbar{
-\rail@endalt\rail@barjoin
-}
-
-% \rail@plus : start '+' alternatives
-%
-% \rail@nextplus: next '+' alternative
-%
-% \rail@endplus : end '+' alternatives
-%
-
-\def\rail@plus{
-\rail@alt\rail@plussplit
-}
-
-\def\rail@nextplus{
-\rail@nextalt\rail@fixplus
-}
-
-\def\rail@fixplus{
-\ifnum\rail@gy<\rail@sy
-\begingroup
-\rail@x=\rail@gx
-\rail@y=\rail@gy
-\rail@ex=\rail@gex
-\rail@rx=\rail@grx
-\ifnum\rail@x<\rail@rx
-\rail@x=\rail@rx
-\fi
-\rail@eline
-\rail@jx=\rail@x
-\rail@jy=\rail@y
-\advance\rail@jy by \rail@joinhsz
-\rail@joval[br]
-\advance\rail@jx by \rail@joinhsz
-\rail@tmpa=\rail@sy
-\advance\rail@tmpa by -\rail@joinhsz
-\advance\rail@tmpa by -\rail@jy
-\rail@jput{\line(0,1){\number\rail@tmpa}}
-\rail@jy=\rail@sy
-\advance\rail@jy by -\rail@joinhsz
-\advance\rail@jx by \rail@joinhsz
-\rail@joval[tl]
-\advance\rail@jy by \rail@joinhsz
-\global\rail@gx=\rail@jx
-\global\rail@gy=\rail@jy
-\global\rail@gex=\rail@gx
-\global\rail@grx=\rail@rx
-\endgroup
-\fi
-}
-
-\def\rail@endplus{
-\rail@endalt\rail@plusjoin
-}
-
-% \rail@cr{Y} : carriage return to vertical position Y
-
-\def\rail@cr#1{
-\rail@tmpa=\rail@sx
-\advance\rail@tmpa by \rail@joinsz
-\ifnum\rail@x<\rail@tmpa\rail@x=\rail@tmpa\fi
-\rail@eline
-\rail@jx=\rail@x
-\rail@jy=\rail@y
-\advance\rail@x by \rail@joinsz
-\ifnum\rail@x>\rail@rx\rail@rx=\rail@x\fi
-\advance\rail@jy by -\rail@joinhsz
-\rail@joval[tr]
-\advance\rail@jx by \rail@joinhsz
-\rail@sety{#1}
-\rail@tmpa=\rail@jy
-\advance\rail@tmpa by -\rail@y
-\advance\rail@tmpa by -\rail@boxsp
-\advance\rail@tmpa by -\rail@joinhsz
-\rail@jput{\line(0,-1){\number\rail@tmpa}}
-\rail@jy=\rail@y
-\advance\rail@jy by \rail@boxsp
-\advance\rail@jy by \rail@joinhsz
-\advance\rail@jx by -\rail@joinhsz
-\rail@joval[br]
-\advance\rail@jy by -\rail@joinhsz
-\rail@tmpa=\rail@jx
-\advance\rail@tmpa by -\rail@sx
-\advance\rail@tmpa by -\rail@joinhsz
-\rail@jput{\line(-1,0){\number\rail@tmpa}}
-\rail@jx=\rail@sx
-\advance\rail@jx by \rail@joinhsz
-\advance\rail@jy by -\rail@joinhsz
-\rail@joval[tl]
-\advance\rail@jx by -\rail@joinhsz
-\rail@tmpa=\rail@boxsp
-\advance\rail@tmpa by -\rail@joinsz
-\rail@jput{\line(0,-1){\number\rail@tmpa}}
-\advance\rail@jy by -\rail@tmpa
-\advance\rail@jx by \rail@joinhsz
-\rail@joval[bl]
-\rail@x=\rail@jx
-\rail@ex=\rail@x
-}
-
-% default setup for Isabelle
-\newenvironment{railoutput}%
-{\begin{list}{}{\rail@param}\def\rail@expand{\relax}\makeatletter}{\makeatother\end{list}}
-
-\def\rail@termfont{\isabellestyle{tt}}
-\def\rail@nontfont{\isabellestyle{it}}
-\def\rail@namefont{\isabellestyle{it}}
diff --git a/thys/MDP-Algorithms/output/outline/root.bib b/thys/MDP-Algorithms/output/outline/root.bib
deleted file mode 100644
--- a/thys/MDP-Algorithms/output/outline/root.bib
+++ /dev/null
@@ -1,13 +0,0 @@
-@book{Puterman,
-  author    = {Martin L. Puterman},
-  title     = {Markov Decision Processes: Discrete Stochastic Dynamic Programming},
-  series    = {Wiley Series in Probability and Statistics},
-  publisher = {Wiley},
-  year      = {1994},
-  url       = {https://doi.org/10.1002/9780470316887},
-  doi       = {10.1002/9780470316887},
-  isbn      = {978-0-47161977-2},
-  timestamp = {Mon, 22 Jul 2019 15:00:49 +0200},
-  biburl    = {https://dblp.org/rec/books/wi/Puterman94.bib},
-  bibsource = {dblp computer science bibliography, https://dblp.org}
-}
diff --git a/thys/MDP-Algorithms/output/outline/root.tex b/thys/MDP-Algorithms/output/outline/root.tex
deleted file mode 100644
--- a/thys/MDP-Algorithms/output/outline/root.tex
+++ /dev/null
@@ -1,69 +0,0 @@
-\documentclass[11pt,a4paper]{article}
-\usepackage{isabelle,isabellesym}
-
-% further packages required for unusual symbols (see also
-% isabellesym.sty), use only when needed
-
-\usepackage{amssymb, amsmath, amsfonts}
-  %for \<leadsto>, \<box>, \<diamond>, \<sqsupset>, \<mho>, \<Join>,
-  %\<lhd>, \<lesssim>, \<greatersim>, \<lessapprox>, \<greaterapprox>,
-  %\<triangleq>, \<yen>, \<lozenge>
-
-%\usepackage{eurosym}
-  %for \<euro>
-
-\usepackage[only,bigsqcap]{stmaryrd}
-  %for \<Sqinter>
-
-%\usepackage{eufrak}
-  %for \<AA> ... \<ZZ>, \<aa> ... \<zz> (also included in amssymb)
-
-%\usepackage{textcomp}
-  %for \<onequarter>, \<onehalf>, \<threequarters>, \<degree>, \<cent>,
-  %\<currency>
-
-% this should be the last package used
-\usepackage{pdfsetup}
-
-% urls in roman style, theory text in math-similar italics
-\urlstyle{rm}
-\isabellestyle{it}
-
-% for uniform font size
-%\renewcommand{\isastyle}{\isastyleminor}
-
-
-\begin{document}
-
-\title{Verified Algorithms for Solving Markov Decision Processes}
-\author{Maximilian Schäffeler and Mohammad Abdulaziz}
-\maketitle
-
-\abstract{
-We present a formalization of algorithms for solving Markov Decision Processes (MDPs) with formal guarantees on the optimality of their
-solutions.
-In particular we build on our analysis of the Bellman operator for discounted infinite horizon MDPs.
-From the iterator rule on the Bellman operator we directly derive executable value iteration and policy iteration algorithms to iteratively solve finite MDPs.
-We also prove correct optimized versions of value iteration that use matrix splittings to improve the convergence rate. In particular, we formally verify Gauss-Seidel value iteration and modified policy iteration.
-The algorithms are evaluated on two standard examples from the literature, namely, inventory management and gridworld.
-Our formalization covers most of chapter 6 in Puterman's book \cite{Puterman}.
-}
-
-\tableofcontents
-
-% sane default for proof documents
-\parindent 0pt\parskip 0.5ex
-
-% generated text of all theories
-\input{session}
-
-% optional bibliography
-\bibliographystyle{abbrv}
-\bibliography{root}
-
-\end{document}
-
-%%% Local Variables:
-%%% mode: latex
-%%% TeX-master: t
-%%% End:
diff --git a/thys/MDP-Algorithms/output/outline/session.tex b/thys/MDP-Algorithms/output/outline/session.tex
deleted file mode 100644
--- a/thys/MDP-Algorithms/output/outline/session.tex
+++ /dev/null
@@ -1,13 +0,0 @@
-\input{Value_Iteration.tex}
-\input{Policy_Iteration.tex}
-\input{Modified_Policy_Iteration.tex}
-\input{Matrix_Util.tex}
-\input{Blinfun_Matrix.tex}
-\input{Splitting_Methods.tex}
-\input{Algorithms.tex}
-\input{Code_DP.tex}
-\input{Code_Mod.tex}
-\input{Code_Real_Approx_By_Float_Fix.tex}
-\input{Code_Inventory.tex}
-\input{Code_Gridworld.tex}
-\input{Examples.tex}
diff --git a/thys/MDP-Algorithms/output/outline/session_graph.pdf b/thys/MDP-Algorithms/output/outline/session_graph.pdf
deleted file mode 100644
index b03b483f5536b1b7b1f0209d0e8065bb41570191..e69de29bb2d1d6434b8b29ae775ad8c2e48c5391
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