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\begin{isamarkuptext}%


32 
\begin{abstract}


33 
This document lists the main types, functions and syntax provided by theory \isa{Main}. It is meant as a quick overview of what is available. The sophisticated class structure is only hinted at. For details see \url{http://isabelle.in.tum.de/dist/library/HOL/}.


34 
\end{abstract}


35 


36 
\section{HOL}


37 


38 
The basic logic: \isa{x\ {\isacharequal}\ y}, \isa{True}, \isa{False}, \isa{{\isasymnot}\ P}, \isa{P\ {\isasymand}\ Q}, \isa{P\ {\isasymor}\ Q}, \isa{P\ {\isasymlongrightarrow}\ Q}, \isa{{\isasymforall}x{\isachardot}\ P}, \isa{{\isasymexists}x{\isachardot}\ P}, \isa{{\isasymexists}{\isacharbang}x{\isachardot}\ P}, \isa{THE\ x{\isachardot}\ P}.


39 
\smallskip


40 


41 
\begin{tabular}{@ {} l @ {~::~} l @ {}}


42 
\isa{undefined} & \isa{{\isacharprime}a}\\


43 
\isa{default} & \isa{{\isacharprime}a}\\


44 
\end{tabular}


45 


46 
\subsubsection*{Syntax}


47 


48 
\begin{supertabular}{@ {} l @ {\quad$\equiv$\quad} l l @ {}}


49 
\isa{x\ {\isasymnoteq}\ y} & \isa{{\isachardoublequote}{\isasymnot}\ {\isacharparenleft}x\ {\isacharequal}\ y{\isacharparenright}{\isachardoublequote}} & (\verb$~=$)\\


50 
\isa{{\isachardoublequote}P\ {\isasymlongleftrightarrow}\ Q{\isachardoublequote}} & \isa{P\ {\isacharequal}\ Q} \\


51 
\isa{if\ x\ then\ y\ else\ z} & \isa{{\isachardoublequote}If\ x\ y\ z{\isachardoublequote}}\\


52 
\isa{let\ x\ {\isacharequal}\ e\isactrlisub {\isadigit{1}}\ in\ e\isactrlisub {\isadigit{2}}} & \isa{{\isachardoublequote}Let\ e\isactrlisub {\isadigit{1}}\ {\isacharparenleft}{\isasymlambda}x{\isachardot}\ e\isactrlisub {\isadigit{2}}{\isacharparenright}{\isachardoublequote}}\\


53 
\end{supertabular}


54 


55 


56 
\section{Orderings}


57 


58 
A collection of classes defining basic orderings:


59 
preorder, partial order, linear order, dense linear order and wellorder.


60 
\smallskip


61 


62 
\begin{supertabular}{@ {} l @ {~::~} l l @ {}}


63 
\isa{op\ {\isasymle}} & \isa{{\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}a\ {\isasymRightarrow}\ bool} & (\verb$<=$)\\


64 
\isa{op\ {\isacharless}} & \isa{{\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}a\ {\isasymRightarrow}\ bool}\\


65 
\isa{Least} & \isa{{\isacharparenleft}{\isacharprime}a\ {\isasymRightarrow}\ bool{\isacharparenright}\ {\isasymRightarrow}\ {\isacharprime}a}\\


66 
\isa{min} & \isa{{\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}a}\\


67 
\isa{max} & \isa{{\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}a}\\


68 
\isa{top} & \isa{{\isacharprime}a}\\


69 
\isa{bot} & \isa{{\isacharprime}a}\\


70 
\isa{mono} & \isa{{\isacharparenleft}{\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}b{\isacharparenright}\ {\isasymRightarrow}\ bool}\\


71 
\isa{strict{\isacharunderscore}mono} & \isa{{\isacharparenleft}{\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}b{\isacharparenright}\ {\isasymRightarrow}\ bool}\\


72 
\end{supertabular}


73 


74 
\subsubsection*{Syntax}


75 


76 
\begin{supertabular}{@ {} l @ {\quad$\equiv$\quad} l l @ {}}


77 
\isa{{\isachardoublequote}x\ {\isasymge}\ y{\isachardoublequote}} & \isa{y\ {\isasymle}\ x} & (\verb$>=$)\\


78 
\isa{{\isachardoublequote}x\ {\isachargreater}\ y{\isachardoublequote}} & \isa{y\ {\isacharless}\ x}\\


79 
\isa{{\isasymforall}x{\isasymle}y{\isachardot}\ P} & \isa{{\isachardoublequote}{\isasymforall}x{\isachardot}\ x\ {\isasymle}\ y\ {\isasymlongrightarrow}\ P{\isachardoublequote}}\\


80 
\isa{{\isasymexists}x{\isasymle}y{\isachardot}\ P} & \isa{{\isachardoublequote}{\isasymexists}x{\isachardot}\ x\ {\isasymle}\ y\ {\isasymand}\ P{\isachardoublequote}}\\


81 
\multicolumn{2}{@ {}l@ {}}{Similarly for $<$, $\ge$ and $>$}\\


82 
\isa{LEAST\ x{\isachardot}\ P} & \isa{{\isachardoublequote}Least\ {\isacharparenleft}{\isasymlambda}x{\isachardot}\ P{\isacharparenright}{\isachardoublequote}}\\


83 
\end{supertabular}


84 


85 


86 
\section{Lattices}


87 


88 
Classes semilattice, lattice, distributive lattice and complete lattice (the


89 
latter in theory \isa{Set}).


90 


91 
\begin{tabular}{@ {} l @ {~::~} l @ {}}


92 
\isa{inf} & \isa{{\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}a}\\


93 
\isa{sup} & \isa{{\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}a}\\


94 
\isa{Inf} & \isa{{\isacharprime}a\ set\ {\isasymRightarrow}\ {\isacharprime}a}\\


95 
\isa{Sup} & \isa{{\isacharprime}a\ set\ {\isasymRightarrow}\ {\isacharprime}a}\\


96 
\end{tabular}


97 


98 
\subsubsection*{Syntax}


99 


100 
Available by loading theory \isa{Lattice{\isacharunderscore}Syntax} in directory \isa{Library}.


101 


102 
\begin{supertabular}{@ {} l @ {\quad$\equiv$\quad} l @ {}}


103 
\isa{{\isachardoublequote}x\ {\isasymsqsubseteq}\ y{\isachardoublequote}} & \isa{x\ {\isasymle}\ y}\\


104 
\isa{{\isachardoublequote}x\ {\isasymsqsubset}\ y{\isachardoublequote}} & \isa{x\ {\isacharless}\ y}\\


105 
\isa{{\isachardoublequote}x\ {\isasymsqinter}\ y{\isachardoublequote}} & \isa{inf\ x\ y}\\


106 
\isa{{\isachardoublequote}x\ {\isasymsqunion}\ y{\isachardoublequote}} & \isa{sup\ x\ y}\\


107 
\isa{{\isachardoublequote}{\isasymSqinter}\ A{\isachardoublequote}} & \isa{Sup\ A}\\


108 
\isa{{\isachardoublequote}{\isasymSqunion}\ A{\isachardoublequote}} & \isa{Inf\ A}\\


109 
\isa{{\isachardoublequote}{\isasymtop}{\isachardoublequote}} & \isa{top}\\


110 
\isa{{\isachardoublequote}{\isasymbottom}{\isachardoublequote}} & \isa{bot}\\


111 
\end{supertabular}


112 


113 


114 
\section{Set}


115 


116 
Sets are predicates: \isa{{\isachardoublequote}{\isacharprime}a\ set\ \ {\isacharequal}\ \ {\isacharprime}a\ {\isasymRightarrow}\ bool{\isachardoublequote}}


117 
\bigskip


118 


119 
\begin{supertabular}{@ {} l @ {~::~} l l @ {}}


120 
\isa{{\isacharbraceleft}{\isacharbraceright}} & \isa{{\isacharprime}a\ set}\\


121 
\isa{insert} & \isa{{\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}a\ set\ {\isasymRightarrow}\ {\isacharprime}a\ set}\\


122 
\isa{Collect} & \isa{{\isacharparenleft}{\isacharprime}a\ {\isasymRightarrow}\ bool{\isacharparenright}\ {\isasymRightarrow}\ {\isacharprime}a\ set}\\


123 
\isa{op\ {\isasymin}} & \isa{{\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}a\ set\ {\isasymRightarrow}\ bool} & (\texttt{:})\\


124 
\isa{op\ {\isasymunion}} & \isa{{\isacharprime}a\ set\ {\isasymRightarrow}\ {\isacharprime}a\ set\ {\isasymRightarrow}\ {\isacharprime}a\ set} & (\texttt{Un})\\


125 
\isa{op\ {\isasyminter}} & \isa{{\isacharprime}a\ set\ {\isasymRightarrow}\ {\isacharprime}a\ set\ {\isasymRightarrow}\ {\isacharprime}a\ set} & (\texttt{Int})\\


126 
\isa{UNION} & \isa{{\isacharprime}a\ set\ {\isasymRightarrow}\ {\isacharparenleft}{\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}b\ set{\isacharparenright}\ {\isasymRightarrow}\ {\isacharprime}b\ set}\\


127 
\isa{INTER} & \isa{{\isacharprime}a\ set\ {\isasymRightarrow}\ {\isacharparenleft}{\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}b\ set{\isacharparenright}\ {\isasymRightarrow}\ {\isacharprime}b\ set}\\


128 
\isa{Union} & \isa{{\isacharprime}a\ set\ set\ {\isasymRightarrow}\ {\isacharprime}a\ set}\\


129 
\isa{Inter} & \isa{{\isacharprime}a\ set\ set\ {\isasymRightarrow}\ {\isacharprime}a\ set}\\


130 
\isa{Pow} & \isa{{\isacharprime}a\ set\ {\isasymRightarrow}\ {\isacharprime}a\ set\ set}\\


131 
\isa{UNIV} & \isa{{\isacharprime}a\ set}\\


132 
\isa{op\ {\isacharbackquote}} & \isa{{\isacharparenleft}{\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}b{\isacharparenright}\ {\isasymRightarrow}\ {\isacharprime}a\ set\ {\isasymRightarrow}\ {\isacharprime}b\ set}\\


133 
\isa{Ball} & \isa{{\isacharprime}a\ set\ {\isasymRightarrow}\ {\isacharparenleft}{\isacharprime}a\ {\isasymRightarrow}\ bool{\isacharparenright}\ {\isasymRightarrow}\ bool}\\


134 
\isa{Bex} & \isa{{\isacharprime}a\ set\ {\isasymRightarrow}\ {\isacharparenleft}{\isacharprime}a\ {\isasymRightarrow}\ bool{\isacharparenright}\ {\isasymRightarrow}\ bool}\\


135 
\end{supertabular}


136 


137 
\subsubsection*{Syntax}


138 


139 
\begin{supertabular}{@ {} l @ {\quad$\equiv$\quad} l l @ {}}


140 
\isa{{\isacharbraceleft}x\isactrlisub {\isadigit{1}}{\isacharcomma}{\isasymdots}{\isacharcomma}x\isactrlisub n{\isacharbraceright}} & \isa{insert\ x\isactrlisub {\isadigit{1}}\ {\isacharparenleft}{\isasymdots}\ {\isacharparenleft}insert\ x\isactrlisub n\ {\isacharbraceleft}{\isacharbraceright}{\isacharparenright}{\isasymdots}{\isacharparenright}}\\


141 
\isa{x\ {\isasymnotin}\ A} & \isa{{\isachardoublequote}{\isasymnot}{\isacharparenleft}x\ {\isasymin}\ A{\isacharparenright}{\isachardoublequote}}\\


142 
\isa{A\ {\isasymsubseteq}\ B} & \isa{{\isachardoublequote}A\ {\isasymle}\ B{\isachardoublequote}}\\


143 
\isa{A\ {\isasymsubset}\ B} & \isa{{\isachardoublequote}A\ {\isacharless}\ B{\isachardoublequote}}\\


144 
\isa{{\isachardoublequote}A\ {\isasymsupseteq}\ B{\isachardoublequote}} & \isa{{\isachardoublequote}B\ {\isasymle}\ A{\isachardoublequote}}\\


145 
\isa{{\isachardoublequote}A\ {\isasymsupset}\ B{\isachardoublequote}} & \isa{{\isachardoublequote}B\ {\isacharless}\ A{\isachardoublequote}}\\


146 
\isa{{\isacharbraceleft}x{\isachardot}\ P{\isacharbraceright}} & \isa{{\isachardoublequote}Collect\ {\isacharparenleft}{\isasymlambda}x{\isachardot}\ P{\isacharparenright}{\isachardoublequote}}\\


147 
\isa{{\isasymUnion}x{\isasymin}I{\isachardot}\ A} & \isa{{\isachardoublequote}UNION\ I\ {\isacharparenleft}{\isasymlambda}x{\isachardot}\ A{\isacharparenright}{\isachardoublequote}} & (\texttt{UN})\\


148 
\isa{{\isasymUnion}x{\isachardot}\ A} & \isa{{\isachardoublequote}UNION\ UNIV\ {\isacharparenleft}{\isasymlambda}x{\isachardot}\ A{\isacharparenright}{\isachardoublequote}}\\


149 
\isa{{\isasymInter}x{\isasymin}I{\isachardot}\ A} & \isa{{\isachardoublequote}INTER\ I\ {\isacharparenleft}{\isasymlambda}x{\isachardot}\ A{\isacharparenright}{\isachardoublequote}} & (\texttt{INT})\\


150 
\isa{{\isasymInter}x{\isachardot}\ A} & \isa{{\isachardoublequote}INTER\ UNIV\ {\isacharparenleft}{\isasymlambda}x{\isachardot}\ A{\isacharparenright}{\isachardoublequote}}\\


151 
\isa{{\isasymforall}x{\isasymin}A{\isachardot}\ P} & \isa{{\isachardoublequote}Ball\ A\ {\isacharparenleft}{\isasymlambda}x{\isachardot}\ P{\isacharparenright}{\isachardoublequote}}\\


152 
\isa{{\isasymexists}x{\isasymin}A{\isachardot}\ P} & \isa{{\isachardoublequote}Bex\ A\ {\isacharparenleft}{\isasymlambda}x{\isachardot}\ P{\isacharparenright}{\isachardoublequote}}\\


153 
\isa{range\ f} & \isa{{\isachardoublequote}f\ {\isacharbackquote}\ UNIV{\isachardoublequote}}\\


154 
\end{supertabular}


155 


156 


157 
\section{Fun}


158 


159 
\begin{supertabular}{@ {} l @ {~::~} l @ {}}


160 
\isa{id} & \isa{{\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}a}\\


161 
\isa{op\ {\isasymcirc}} & \isa{{\isacharparenleft}{\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}b{\isacharparenright}\ {\isasymRightarrow}\ {\isacharparenleft}{\isacharprime}c\ {\isasymRightarrow}\ {\isacharprime}a{\isacharparenright}\ {\isasymRightarrow}\ {\isacharprime}c\ {\isasymRightarrow}\ {\isacharprime}b}\\


162 
\isa{inj{\isacharunderscore}on} & \isa{{\isacharparenleft}{\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}b{\isacharparenright}\ {\isasymRightarrow}\ {\isacharprime}a\ set\ {\isasymRightarrow}\ bool}\\


163 
\isa{inj} & \isa{{\isacharparenleft}{\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}b{\isacharparenright}\ {\isasymRightarrow}\ bool}\\


164 
\isa{surj} & \isa{{\isacharparenleft}{\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}b{\isacharparenright}\ {\isasymRightarrow}\ bool}\\


165 
\isa{bij} & \isa{{\isacharparenleft}{\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}b{\isacharparenright}\ {\isasymRightarrow}\ bool}\\


166 
\isa{bij{\isacharunderscore}betw} & \isa{{\isacharparenleft}{\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}b{\isacharparenright}\ {\isasymRightarrow}\ {\isacharprime}a\ set\ {\isasymRightarrow}\ {\isacharprime}b\ set\ {\isasymRightarrow}\ bool}\\


167 
\isa{fun{\isacharunderscore}upd} & \isa{{\isacharparenleft}{\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}b{\isacharparenright}\ {\isasymRightarrow}\ {\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}b\ {\isasymRightarrow}\ {\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}b}\\


168 
\end{supertabular}


169 


170 
\subsubsection*{Syntax}


171 


172 
\begin{tabular}{@ {} l @ {\quad$\equiv$\quad} l @ {}}


173 
\isa{f{\isacharparenleft}x\ {\isacharcolon}{\isacharequal}\ y{\isacharparenright}} & \isa{{\isachardoublequote}fun{\isacharunderscore}upd\ f\ x\ y{\isachardoublequote}}\\


174 
\isa{f{\isacharparenleft}x\isactrlisub {\isadigit{1}}{\isacharcolon}{\isacharequal}y\isactrlisub {\isadigit{1}}{\isacharcomma}{\isasymdots}{\isacharcomma}x\isactrlisub n{\isacharcolon}{\isacharequal}y\isactrlisub n{\isacharparenright}} & \isa{f{\isacharparenleft}x\isactrlisub {\isadigit{1}}{\isacharcolon}{\isacharequal}y\isactrlisub {\isadigit{1}}{\isacharparenright}{\isasymdots}{\isacharparenleft}x\isactrlisub n{\isacharcolon}{\isacharequal}y\isactrlisub n{\isacharparenright}}\\


175 
\end{tabular}


176 


177 


178 
\section{Fixed Points}


179 


180 
Theory: \isa{Inductive}.


181 


182 
Least and greatest fixed points in a complete lattice \isa{{\isacharprime}a}:


183 


184 
\begin{tabular}{@ {} l @ {~::~} l @ {}}


185 
\isa{lfp} & \isa{{\isacharparenleft}{\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}a{\isacharparenright}\ {\isasymRightarrow}\ {\isacharprime}a}\\


186 
\isa{gfp} & \isa{{\isacharparenleft}{\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}a{\isacharparenright}\ {\isasymRightarrow}\ {\isacharprime}a}\\


187 
\end{tabular}


188 


189 
Note that in particular sets (\isa{{\isacharprime}a\ {\isasymRightarrow}\ bool}) are complete lattices.


190 


191 
\section{Sum\_Type}


192 


193 
Type constructor \isa{{\isacharplus}}.


194 


195 
\begin{tabular}{@ {} l @ {~::~} l @ {}}


196 
\isa{Inl} & \isa{{\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}a\ {\isacharplus}\ {\isacharprime}b}\\


197 
\isa{Inr} & \isa{{\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}b\ {\isacharplus}\ {\isacharprime}a}\\


198 
\isa{op\ {\isacharless}{\isacharplus}{\isachargreater}} & \isa{{\isacharprime}a\ set\ {\isasymRightarrow}\ {\isacharprime}b\ set\ {\isasymRightarrow}\ {\isacharparenleft}{\isacharprime}a\ {\isacharplus}\ {\isacharprime}b{\isacharparenright}\ set}


199 
\end{tabular}


200 


201 


202 
\section{Product\_Type}


203 


204 
Types \isa{unit} and \isa{{\isasymtimes}}.


205 


206 
\begin{supertabular}{@ {} l @ {~::~} l @ {}}


207 
\isa{{\isacharparenleft}{\isacharparenright}} & \isa{unit}\\


208 
\isa{Pair} & \isa{{\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}b\ {\isasymRightarrow}\ {\isacharprime}a\ {\isasymtimes}\ {\isacharprime}b}\\


209 
\isa{fst} & \isa{{\isacharprime}a\ {\isasymtimes}\ {\isacharprime}b\ {\isasymRightarrow}\ {\isacharprime}a}\\


210 
\isa{snd} & \isa{{\isacharprime}a\ {\isasymtimes}\ {\isacharprime}b\ {\isasymRightarrow}\ {\isacharprime}b}\\


211 
\isa{split} & \isa{{\isacharparenleft}{\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}b\ {\isasymRightarrow}\ {\isacharprime}c{\isacharparenright}\ {\isasymRightarrow}\ {\isacharprime}a\ {\isasymtimes}\ {\isacharprime}b\ {\isasymRightarrow}\ {\isacharprime}c}\\


212 
\isa{curry} & \isa{{\isacharparenleft}{\isacharprime}a\ {\isasymtimes}\ {\isacharprime}b\ {\isasymRightarrow}\ {\isacharprime}c{\isacharparenright}\ {\isasymRightarrow}\ {\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}b\ {\isasymRightarrow}\ {\isacharprime}c}\\


213 
\isa{Sigma} & \isa{{\isacharprime}a\ set\ {\isasymRightarrow}\ {\isacharparenleft}{\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}b\ set{\isacharparenright}\ {\isasymRightarrow}\ {\isacharparenleft}{\isacharprime}a\ {\isasymtimes}\ {\isacharprime}b{\isacharparenright}\ set}\\


214 
\end{supertabular}


215 


216 
\subsubsection*{Syntax}


217 


218 
\begin{tabular}{@ {} l @ {\quad$\equiv$\quad} ll @ {}}


219 
\isa{{\isacharparenleft}a{\isacharcomma}\ b{\isacharparenright}} & \isa{{\isachardoublequote}Pair\ a\ b{\isachardoublequote}}\\


220 
\isa{{\isasymlambda}{\isacharparenleft}x{\isacharcomma}\ y{\isacharparenright}{\isachardot}\ t} & \isa{{\isachardoublequote}split\ {\isacharparenleft}{\isasymlambda}x\ y{\isachardot}\ t{\isacharparenright}{\isachardoublequote}}\\


221 
\isa{A\ {\isasymtimes}\ B} & \isa{Sigma\ A\ {\isacharparenleft}{\isasymlambda}\_{\isachardot}\ B{\isacharparenright}} & (\verb$<*>$)


222 
\end{tabular}


223 


224 
Pairs may be nested. Nesting to the right is printed as a tuple,


225 
e.g.\ \mbox{\isa{{\isacharparenleft}a{\isacharcomma}\ b{\isacharcomma}\ c{\isacharparenright}}} is really \mbox{\isa{{\isacharparenleft}a{\isacharcomma}\ {\isacharparenleft}b{\isacharcomma}\ c{\isacharparenright}{\isacharparenright}}.}


226 
Pattern matching with pairs and tuples extends to all binders,


227 
e.g.\ \mbox{\isa{{\isasymforall}{\isacharparenleft}x{\isacharcomma}\ y{\isacharparenright}{\isasymin}A{\isachardot}\ P},} \isa{{\isacharbraceleft}{\isacharparenleft}x{\isacharcomma}\ y{\isacharparenright}{\isachardot}\ P{\isacharbraceright}}, etc.


228 


229 


230 
\section{Relation}


231 


232 
\begin{supertabular}{@ {} l @ {~::~} l @ {}}


233 
\isa{converse} & \isa{{\isacharparenleft}{\isacharprime}a\ {\isasymtimes}\ {\isacharprime}b{\isacharparenright}\ set\ {\isasymRightarrow}\ {\isacharparenleft}{\isacharprime}b\ {\isasymtimes}\ {\isacharprime}a{\isacharparenright}\ set}\\


234 
\isa{op\ O} & \isa{{\isacharparenleft}{\isacharprime}a\ {\isasymtimes}\ {\isacharprime}b{\isacharparenright}\ set\ {\isasymRightarrow}\ {\isacharparenleft}{\isacharprime}c\ {\isasymtimes}\ {\isacharprime}a{\isacharparenright}\ set\ {\isasymRightarrow}\ {\isacharparenleft}{\isacharprime}c\ {\isasymtimes}\ {\isacharprime}b{\isacharparenright}\ set}\\


235 
\isa{op\ {\isacharbackquote}{\isacharbackquote}} & \isa{{\isacharparenleft}{\isacharprime}a\ {\isasymtimes}\ {\isacharprime}b{\isacharparenright}\ set\ {\isasymRightarrow}\ {\isacharprime}a\ set\ {\isasymRightarrow}\ {\isacharprime}b\ set}\\


236 
\isa{inv{\isacharunderscore}image} & \isa{{\isacharparenleft}{\isacharprime}a\ {\isasymtimes}\ {\isacharprime}a{\isacharparenright}\ set\ {\isasymRightarrow}\ {\isacharparenleft}{\isacharprime}b\ {\isasymRightarrow}\ {\isacharprime}a{\isacharparenright}\ {\isasymRightarrow}\ {\isacharparenleft}{\isacharprime}b\ {\isasymtimes}\ {\isacharprime}b{\isacharparenright}\ set}\\


237 
\isa{Id{\isacharunderscore}on} & \isa{{\isacharprime}a\ set\ {\isasymRightarrow}\ {\isacharparenleft}{\isacharprime}a\ {\isasymtimes}\ {\isacharprime}a{\isacharparenright}\ set}\\


238 
\isa{Id} & \isa{{\isacharparenleft}{\isacharprime}a\ {\isasymtimes}\ {\isacharprime}a{\isacharparenright}\ set}\\


239 
\isa{Domain} & \isa{{\isacharparenleft}{\isacharprime}a\ {\isasymtimes}\ {\isacharprime}b{\isacharparenright}\ set\ {\isasymRightarrow}\ {\isacharprime}a\ set}\\


240 
\isa{Range} & \isa{{\isacharparenleft}{\isacharprime}a\ {\isasymtimes}\ {\isacharprime}b{\isacharparenright}\ set\ {\isasymRightarrow}\ {\isacharprime}b\ set}\\


241 
\isa{Field} & \isa{{\isacharparenleft}{\isacharprime}a\ {\isasymtimes}\ {\isacharprime}a{\isacharparenright}\ set\ {\isasymRightarrow}\ {\isacharprime}a\ set}\\


242 
\isa{refl{\isacharunderscore}on} & \isa{{\isacharprime}a\ set\ {\isasymRightarrow}\ {\isacharparenleft}{\isacharprime}a\ {\isasymtimes}\ {\isacharprime}a{\isacharparenright}\ set\ {\isasymRightarrow}\ bool}\\


243 
\isa{refl} & \isa{{\isacharparenleft}{\isacharprime}a\ {\isasymtimes}\ {\isacharprime}a{\isacharparenright}\ set\ {\isasymRightarrow}\ bool}\\


244 
\isa{sym} & \isa{{\isacharparenleft}{\isacharprime}a\ {\isasymtimes}\ {\isacharprime}a{\isacharparenright}\ set\ {\isasymRightarrow}\ bool}\\


245 
\isa{antisym} & \isa{{\isacharparenleft}{\isacharprime}a\ {\isasymtimes}\ {\isacharprime}a{\isacharparenright}\ set\ {\isasymRightarrow}\ bool}\\


246 
\isa{trans} & \isa{{\isacharparenleft}{\isacharprime}a\ {\isasymtimes}\ {\isacharprime}a{\isacharparenright}\ set\ {\isasymRightarrow}\ bool}\\


247 
\isa{irrefl} & \isa{{\isacharparenleft}{\isacharprime}a\ {\isasymtimes}\ {\isacharprime}a{\isacharparenright}\ set\ {\isasymRightarrow}\ bool}\\


248 
\isa{total{\isacharunderscore}on} & \isa{{\isacharprime}a\ set\ {\isasymRightarrow}\ {\isacharparenleft}{\isacharprime}a\ {\isasymtimes}\ {\isacharprime}a{\isacharparenright}\ set\ {\isasymRightarrow}\ bool}\\


249 
\isa{total} & \isa{{\isacharparenleft}{\isacharprime}a\ {\isasymtimes}\ {\isacharprime}a{\isacharparenright}\ set\ {\isasymRightarrow}\ bool}\\


250 
\end{supertabular}


251 


252 
\subsubsection*{Syntax}


253 


254 
\begin{tabular}{@ {} l @ {\quad$\equiv$\quad} l l @ {}}


255 
\isa{r{\isasyminverse}} & \isa{{\isachardoublequote}converse\ r{\isachardoublequote}} & (\verb$^1$)


256 
\end{tabular}


257 


258 
\section{Equiv\_Relations}


259 


260 
\begin{supertabular}{@ {} l @ {~::~} l @ {}}


261 
\isa{equiv} & \isa{{\isacharprime}a\ set\ {\isasymRightarrow}\ {\isacharparenleft}{\isacharprime}a\ {\isasymtimes}\ {\isacharprime}a{\isacharparenright}\ set\ {\isasymRightarrow}\ bool}\\


262 
\isa{op\ {\isacharslash}{\isacharslash}} & \isa{{\isacharprime}a\ set\ {\isasymRightarrow}\ {\isacharparenleft}{\isacharprime}a\ {\isasymtimes}\ {\isacharprime}a{\isacharparenright}\ set\ {\isasymRightarrow}\ {\isacharprime}a\ set\ set}\\


263 
\isa{congruent} & \isa{{\isacharparenleft}{\isacharprime}a\ {\isasymtimes}\ {\isacharprime}a{\isacharparenright}\ set\ {\isasymRightarrow}\ {\isacharparenleft}{\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}b{\isacharparenright}\ {\isasymRightarrow}\ bool}\\


264 
\isa{congruent{\isadigit{2}}} & \isa{{\isacharparenleft}{\isacharprime}a\ {\isasymtimes}\ {\isacharprime}a{\isacharparenright}\ set\ {\isasymRightarrow}\ {\isacharparenleft}{\isacharprime}b\ {\isasymtimes}\ {\isacharprime}b{\isacharparenright}\ set\ {\isasymRightarrow}\ {\isacharparenleft}{\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}b\ {\isasymRightarrow}\ {\isacharprime}c{\isacharparenright}\ {\isasymRightarrow}\ bool}\\


265 
%@ {const Equiv_Relations.} & @ {term_type_only Equiv_Relations. ""}\\


266 
\end{supertabular}


267 


268 
\subsubsection*{Syntax}


269 


270 
\begin{tabular}{@ {} l @ {\quad$\equiv$\quad} l @ {}}


271 
\isa{f\ respects\ r} & \isa{{\isachardoublequote}congruent\ r\ f{\isachardoublequote}}\\


272 
\isa{f\ respects{\isadigit{2}}\ r} & \isa{{\isachardoublequote}congruent{\isadigit{2}}\ r\ r\ f{\isachardoublequote}}\\


273 
\end{tabular}


274 


275 


276 
\section{Transitive\_Closure}


277 


278 
\begin{tabular}{@ {} l @ {~::~} l @ {}}


279 
\isa{rtrancl} & \isa{{\isacharparenleft}{\isacharprime}a\ {\isasymtimes}\ {\isacharprime}a{\isacharparenright}\ set\ {\isasymRightarrow}\ {\isacharparenleft}{\isacharprime}a\ {\isasymtimes}\ {\isacharprime}a{\isacharparenright}\ set}\\


280 
\isa{trancl} & \isa{{\isacharparenleft}{\isacharprime}a\ {\isasymtimes}\ {\isacharprime}a{\isacharparenright}\ set\ {\isasymRightarrow}\ {\isacharparenleft}{\isacharprime}a\ {\isasymtimes}\ {\isacharprime}a{\isacharparenright}\ set}\\


281 
\isa{reflcl} & \isa{{\isacharparenleft}{\isacharprime}a\ {\isasymtimes}\ {\isacharprime}a{\isacharparenright}\ set\ {\isasymRightarrow}\ {\isacharparenleft}{\isacharprime}a\ {\isasymtimes}\ {\isacharprime}a{\isacharparenright}\ set}\\


282 
\end{tabular}


283 


284 
\subsubsection*{Syntax}


285 


286 
\begin{tabular}{@ {} l @ {\quad$\equiv$\quad} l l @ {}}


287 
\isa{r\isactrlsup {\isacharasterisk}} & \isa{{\isachardoublequote}rtrancl\ r{\isachardoublequote}} & (\verb$^*$)\\


288 
\isa{r\isactrlsup {\isacharplus}} & \isa{{\isachardoublequote}trancl\ r{\isachardoublequote}} & (\verb$^+$)\\


289 
\isa{r\isactrlsup {\isacharequal}} & \isa{{\isachardoublequote}reflcl\ r{\isachardoublequote}} & (\verb$^=$)


290 
\end{tabular}


291 


292 


293 
\section{Algebra}


294 


295 
Theories \isa{OrderedGroup}, \isa{Ring{\isacharunderscore}and{\isacharunderscore}Field} and \isa{Divides} define a large collection of classes describing common algebraic


296 
structures from semigroups up to fields. Everything is done in terms of


297 
overloaded operators:


298 


299 
\begin{supertabular}{@ {} l @ {~::~} l l @ {}}


300 
\isa{{\isadigit{0}}} & \isa{{\isacharprime}a}\\


301 
\isa{{\isadigit{1}}} & \isa{{\isacharprime}a}\\


302 
\isa{op\ {\isacharplus}} & \isa{{\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}a}\\


303 
\isa{op\ {\isacharminus}} & \isa{{\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}a}\\


304 
\isa{uminus} & \isa{{\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}a} & (\verb$$)\\


305 
\isa{op\ {\isacharasterisk}} & \isa{{\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}a}\\


306 
\isa{inverse} & \isa{{\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}a}\\


307 
\isa{op\ {\isacharslash}} & \isa{{\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}a}\\


308 
\isa{abs} & \isa{{\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}a}\\


309 
\isa{sgn} & \isa{{\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}a}\\


310 
\isa{op\ dvd} & \isa{{\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}a\ {\isasymRightarrow}\ bool}\\


311 
\isa{op\ div} & \isa{{\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}a}\\


312 
\isa{op\ mod} & \isa{{\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}a}\\


313 
\end{supertabular}


314 


315 
\subsubsection*{Syntax}


316 


317 
\begin{tabular}{@ {} l @ {\quad$\equiv$\quad} l @ {}}


318 
\isa{{\isasymbar}x{\isasymbar}} & \isa{{\isachardoublequote}abs\ x{\isachardoublequote}}


319 
\end{tabular}


320 


321 


322 
\section{Nat}


323 


324 
\isa{\isacommand{datatype}\ nat\ {\isacharequal}\ {\isadigit{0}}\ {\isacharbar}\ Suc\ nat}


325 
\bigskip


326 


327 
\begin{tabular}{@ {} lllllll @ {}}


328 
\isa{op\ {\isacharplus}} &


329 
\isa{op\ {\isacharminus}} &


330 
\isa{op\ {\isacharasterisk}} &


331 
\isa{op\ {\isacharcircum}} &


332 
\isa{op\ div}&


333 
\isa{op\ mod}&


334 
\isa{op\ dvd}\\


335 
\isa{op\ {\isasymle}} &


336 
\isa{op\ {\isacharless}} &


337 
\isa{min} &


338 
\isa{max} &


339 
\isa{Min} &


340 
\isa{Max}\\


341 
\end{tabular}


342 


343 
\begin{tabular}{@ {} l @ {~::~} l @ {}}


344 
\isa{of{\isacharunderscore}nat} & \isa{nat\ {\isasymRightarrow}\ {\isacharprime}a}


345 
\end{tabular}


346 


347 
\section{Int}


348 


349 
Type \isa{int}


350 
\bigskip


351 


352 
\begin{tabular}{@ {} llllllll @ {}}


353 
\isa{op\ {\isacharplus}} &


354 
\isa{op\ {\isacharminus}} &


355 
\isa{uminus} &


356 
\isa{op\ {\isacharasterisk}} &


357 
\isa{op\ {\isacharcircum}} &


358 
\isa{op\ div}&


359 
\isa{op\ mod}&


360 
\isa{op\ dvd}\\


361 
\isa{op\ {\isasymle}} &


362 
\isa{op\ {\isacharless}} &


363 
\isa{min} &


364 
\isa{max} &


365 
\isa{Min} &


366 
\isa{Max}\\


367 
\isa{abs} &


368 
\isa{sgn}\\


369 
\end{tabular}


370 


371 
\begin{tabular}{@ {} l @ {~::~} l l @ {}}


372 
\isa{nat} & \isa{int\ {\isasymRightarrow}\ nat}\\


373 
\isa{of{\isacharunderscore}int} & \isa{int\ {\isasymRightarrow}\ {\isacharprime}a}\\


374 
\isa{{\isasymint}} & \isa{{\isacharprime}a\ set} & (\verb$Ints$)


375 
\end{tabular}


376 


377 
\subsubsection*{Syntax}


378 


379 
\begin{tabular}{@ {} l @ {\quad$\equiv$\quad} l @ {}}


380 
\isa{int} & \isa{{\isachardoublequote}of{\isacharunderscore}nat{\isachardoublequote}}\\


381 
\end{tabular}


382 


383 


384 
\section{Finite\_Set}


385 


386 


387 
\begin{supertabular}{@ {} l @ {~::~} l @ {}}


388 
\isa{finite} & \isa{{\isacharprime}a\ set\ {\isasymRightarrow}\ bool}\\


389 
\isa{card} & \isa{{\isacharprime}a\ set\ {\isasymRightarrow}\ nat}\\


390 
\isa{fold} & \isa{{\isacharparenleft}{\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}b\ {\isasymRightarrow}\ {\isacharprime}b{\isacharparenright}\ {\isasymRightarrow}\ {\isacharprime}b\ {\isasymRightarrow}\ {\isacharprime}a\ set\ {\isasymRightarrow}\ {\isacharprime}b}\\


391 
\isa{fold{\isacharunderscore}image} & \isa{{\isacharparenleft}{\isacharprime}b\ {\isasymRightarrow}\ {\isacharprime}b\ {\isasymRightarrow}\ {\isacharprime}b{\isacharparenright}\ {\isasymRightarrow}\ {\isacharparenleft}{\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}b{\isacharparenright}\ {\isasymRightarrow}\ {\isacharprime}b\ {\isasymRightarrow}\ {\isacharprime}a\ set\ {\isasymRightarrow}\ {\isacharprime}b}\\


392 
\isa{setsum} & \isa{{\isacharparenleft}{\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}b{\isacharparenright}\ {\isasymRightarrow}\ {\isacharprime}a\ set\ {\isasymRightarrow}\ {\isacharprime}b}\\


393 
\isa{setprod} & \isa{{\isacharparenleft}{\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}b{\isacharparenright}\ {\isasymRightarrow}\ {\isacharprime}a\ set\ {\isasymRightarrow}\ {\isacharprime}b}\\


394 
\end{supertabular}


395 


396 


397 
\subsubsection*{Syntax}


398 


399 
\begin{supertabular}{@ {} l @ {\quad$\equiv$\quad} l l @ {}}


400 
\isa{{\isasymSum}A} & \isa{{\isachardoublequote}setsum\ {\isacharparenleft}{\isasymlambda}x{\isachardot}\ x{\isacharparenright}\ A{\isachardoublequote}} & (\verb$SUM$)\\


401 
\isa{{\isasymSum}x{\isasymin}A{\isachardot}\ t} & \isa{{\isachardoublequote}setsum\ {\isacharparenleft}{\isasymlambda}x{\isachardot}\ t{\isacharparenright}\ A{\isachardoublequote}}\\


402 
\isa{{\isachardoublequote}{\isasymSum}x{\isacharbar}P{\isachardot}\ t{\isachardoublequote}} & \isa{{\isasymSum}x{\isasymin}{\isacharbraceleft}x{\isachardot}\ P{\isacharbraceright}{\isachardot}\ t}\\


403 
\multicolumn{2}{@ {}l@ {}}{Similarly for \isa{{\isasymProd}} instead of \isa{{\isasymSum}}} & (\verb$PROD$)\\


404 
\end{supertabular}


405 


406 


407 
\section{Wellfounded}


408 


409 
\begin{supertabular}{@ {} l @ {~::~} l @ {}}


410 
\isa{wf} & \isa{{\isacharparenleft}{\isacharprime}a\ {\isasymtimes}\ {\isacharprime}a{\isacharparenright}\ set\ {\isasymRightarrow}\ bool}\\


411 
\isa{acyclic} & \isa{{\isacharparenleft}{\isacharprime}a\ {\isasymtimes}\ {\isacharprime}a{\isacharparenright}\ set\ {\isasymRightarrow}\ bool}\\


412 
\isa{acc} & \isa{{\isacharparenleft}{\isacharprime}a\ {\isasymtimes}\ {\isacharprime}a{\isacharparenright}\ set\ {\isasymRightarrow}\ {\isacharprime}a\ set}\\


413 
\isa{measure} & \isa{{\isacharparenleft}{\isacharprime}a\ {\isasymRightarrow}\ nat{\isacharparenright}\ {\isasymRightarrow}\ {\isacharparenleft}{\isacharprime}a\ {\isasymtimes}\ {\isacharprime}a{\isacharparenright}\ set}\\


414 
\isa{op\ {\isacharless}{\isacharasterisk}lex{\isacharasterisk}{\isachargreater}} & \isa{{\isacharparenleft}{\isacharprime}a\ {\isasymtimes}\ {\isacharprime}a{\isacharparenright}\ set\ {\isasymRightarrow}\ {\isacharparenleft}{\isacharprime}b\ {\isasymtimes}\ {\isacharprime}b{\isacharparenright}\ set\ {\isasymRightarrow}\ {\isacharparenleft}{\isacharparenleft}{\isacharprime}a\ {\isasymtimes}\ {\isacharprime}b{\isacharparenright}\ {\isasymtimes}\ {\isacharprime}a\ {\isasymtimes}\ {\isacharprime}b{\isacharparenright}\ set}\\


415 
\isa{op\ {\isacharless}{\isacharasterisk}mlex{\isacharasterisk}{\isachargreater}} & \isa{{\isacharparenleft}{\isacharprime}a\ {\isasymRightarrow}\ nat{\isacharparenright}\ {\isasymRightarrow}\ {\isacharparenleft}{\isacharprime}a\ {\isasymtimes}\ {\isacharprime}a{\isacharparenright}\ set\ {\isasymRightarrow}\ {\isacharparenleft}{\isacharprime}a\ {\isasymtimes}\ {\isacharprime}a{\isacharparenright}\ set}\\


416 
\isa{less{\isacharunderscore}than} & \isa{{\isacharparenleft}nat\ {\isasymtimes}\ nat{\isacharparenright}\ set}\\


417 
\isa{pred{\isacharunderscore}nat} & \isa{{\isacharparenleft}nat\ {\isasymtimes}\ nat{\isacharparenright}\ set}\\


418 
\end{supertabular}


419 


420 


421 
\section{SetInterval}


422 


423 
\begin{supertabular}{@ {} l @ {~::~} l @ {}}


424 
\isa{lessThan} & \isa{{\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}a\ set}\\


425 
\isa{atMost} & \isa{{\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}a\ set}\\


426 
\isa{greaterThan} & \isa{{\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}a\ set}\\


427 
\isa{atLeast} & \isa{{\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}a\ set}\\


428 
\isa{greaterThanLessThan} & \isa{{\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}a\ set}\\


429 
\isa{atLeastLessThan} & \isa{{\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}a\ set}\\


430 
\isa{greaterThanAtMost} & \isa{{\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}a\ set}\\


431 
\isa{atLeastAtMost} & \isa{{\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}a\ set}\\


432 
\end{supertabular}


433 


434 
\subsubsection*{Syntax}


435 


436 
\begin{supertabular}{@ {} l @ {\quad$\equiv$\quad} l @ {}}


437 
\isa{{\isacharbraceleft}{\isachardot}{\isachardot}{\isacharless}y{\isacharbraceright}} & \isa{{\isachardoublequote}lessThan\ y{\isachardoublequote}}\\


438 
\isa{{\isacharbraceleft}{\isachardot}{\isachardot}y{\isacharbraceright}} & \isa{{\isachardoublequote}atMost\ y{\isachardoublequote}}\\


439 
\isa{{\isacharbraceleft}x{\isacharless}{\isachardot}{\isachardot}{\isacharbraceright}} & \isa{{\isachardoublequote}greaterThan\ x{\isachardoublequote}}\\


440 
\isa{{\isacharbraceleft}x{\isachardot}{\isachardot}{\isacharbraceright}} & \isa{{\isachardoublequote}atLeast\ x{\isachardoublequote}}\\


441 
\isa{{\isacharbraceleft}x{\isacharless}{\isachardot}{\isachardot}{\isacharless}y{\isacharbraceright}} & \isa{{\isachardoublequote}greaterThanLessThan\ x\ y{\isachardoublequote}}\\


442 
\isa{{\isacharbraceleft}x{\isachardot}{\isachardot}{\isacharless}y{\isacharbraceright}} & \isa{{\isachardoublequote}atLeastLessThan\ x\ y{\isachardoublequote}}\\


443 
\isa{{\isacharbraceleft}x{\isacharless}{\isachardot}{\isachardot}y{\isacharbraceright}} & \isa{{\isachardoublequote}greaterThanAtMost\ x\ y{\isachardoublequote}}\\


444 
\isa{{\isacharbraceleft}x{\isachardot}{\isachardot}y{\isacharbraceright}} & \isa{{\isachardoublequote}atLeastAtMost\ x\ y{\isachardoublequote}}\\


445 
\isa{{\isasymUnion}\ i{\isasymle}n{\isachardot}\ A} & \isa{{\isachardoublequote}{\isasymUnion}\ i\ {\isasymin}\ {\isacharbraceleft}{\isachardot}{\isachardot}n{\isacharbraceright}{\isachardot}\ A{\isachardoublequote}}\\


446 
\isa{{\isasymUnion}\ i{\isacharless}n{\isachardot}\ A} & \isa{{\isachardoublequote}{\isasymUnion}\ i\ {\isasymin}\ {\isacharbraceleft}{\isachardot}{\isachardot}{\isacharless}n{\isacharbraceright}{\isachardot}\ A{\isachardoublequote}}\\


447 
\multicolumn{2}{@ {}l@ {}}{Similarly for \isa{{\isasymInter}} instead of \isa{{\isasymUnion}}}\\


448 
\isa{{\isasymSum}x\ {\isacharequal}\ a{\isachardot}{\isachardot}b{\isachardot}\ t} & \isa{{\isachardoublequote}setsum\ {\isacharparenleft}{\isasymlambda}x{\isachardot}\ t{\isacharparenright}\ {\isacharbraceleft}a{\isachardot}{\isachardot}b{\isacharbraceright}{\isachardoublequote}}\\


449 
\isa{{\isasymSum}x\ {\isacharequal}\ a{\isachardot}{\isachardot}{\isacharless}b{\isachardot}\ t} & \isa{{\isachardoublequote}setsum\ {\isacharparenleft}{\isasymlambda}x{\isachardot}\ t{\isacharparenright}\ {\isacharbraceleft}a{\isachardot}{\isachardot}{\isacharless}b{\isacharbraceright}{\isachardoublequote}}\\


450 
\isa{{\isasymSum}x{\isasymle}b{\isachardot}\ t} & \isa{{\isachardoublequote}setsum\ {\isacharparenleft}{\isasymlambda}x{\isachardot}\ t{\isacharparenright}\ {\isacharbraceleft}{\isachardot}{\isachardot}b{\isacharbraceright}{\isachardoublequote}}\\


451 
\isa{{\isasymSum}x{\isacharless}b{\isachardot}\ t} & \isa{{\isachardoublequote}setsum\ {\isacharparenleft}{\isasymlambda}x{\isachardot}\ t{\isacharparenright}\ {\isacharbraceleft}{\isachardot}{\isachardot}{\isacharless}b{\isacharbraceright}{\isachardoublequote}}\\


452 
\multicolumn{2}{@ {}l@ {}}{Similarly for \isa{{\isasymProd}} instead of \isa{{\isasymSum}}}\\


453 
\end{supertabular}


454 


455 


456 
\section{Power}


457 


458 
\begin{tabular}{@ {} l @ {~::~} l @ {}}


459 
\isa{op\ {\isacharcircum}} & \isa{{\isacharprime}a\ {\isasymRightarrow}\ nat\ {\isasymRightarrow}\ {\isacharprime}a}


460 
\end{tabular}


461 


462 


463 
\section{Iterated Functions and Relations}


464 


465 
Theory: \isa{Relation{\isacharunderscore}Power}


466 


467 
Iterated functions \ \isa{{\isachardoublequote}{\isacharparenleft}f{\isacharcolon}{\isacharcolon}{\isacharprime}a{\isasymRightarrow}{\isacharprime}a{\isacharparenright}\ {\isacharcircum}\ n{\isachardoublequote}} \


468 
and relations \ \isa{{\isachardoublequote}{\isacharparenleft}r{\isacharcolon}{\isacharcolon}{\isacharparenleft}{\isacharprime}a{\isasymtimes}{\isacharprime}a{\isacharparenright}set{\isacharparenright}\ {\isacharcircum}\ n{\isachardoublequote}}.


469 


470 


471 
\section{Option}


472 


473 
\isa{\isacommand{datatype}\ {\isacharprime}a\ option\ {\isacharequal}\ None\ {\isacharbar}\ Some\ {\isacharprime}a}


474 
\bigskip


475 


476 
\begin{tabular}{@ {} l @ {~::~} l @ {}}


477 
\isa{the} & \isa{{\isacharprime}a\ option\ {\isasymRightarrow}\ {\isacharprime}a}\\


478 
\isa{Option{\isachardot}map} & \isa{{\isachardoublequote}{\isacharparenleft}{\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}b{\isacharparenright}\ {\isasymRightarrow}\ {\isacharprime}a\ option\ {\isasymRightarrow}\ {\isacharprime}b\ option{\isachardoublequote}}\\


479 
\isa{Option{\isachardot}set} & \isa{{\isacharprime}a\ option\ {\isasymRightarrow}\ {\isacharprime}a\ set}


480 
\end{tabular}


481 


482 
\section{List}


483 


484 
\isa{\isacommand{datatype}\ {\isacharprime}a\ list\ {\isacharequal}\ {\isacharbrackleft}{\isacharbrackright}\ {\isacharbar}\ op\ {\isacharhash}\ {\isacharprime}a\ {\isacharparenleft}{\isacharprime}a\ list{\isacharparenright}}


485 
\bigskip


486 


487 
\begin{supertabular}{@ {} l @ {~::~} l @ {}}


488 
\isa{op\ {\isacharat}} & \isa{{\isacharprime}a\ list\ {\isasymRightarrow}\ {\isacharprime}a\ list\ {\isasymRightarrow}\ {\isacharprime}a\ list}\\


489 
\isa{butlast} & \isa{{\isacharprime}a\ list\ {\isasymRightarrow}\ {\isacharprime}a\ list}\\


490 
\isa{concat} & \isa{{\isacharprime}a\ list\ list\ {\isasymRightarrow}\ {\isacharprime}a\ list}\\


491 
\isa{distinct} & \isa{{\isacharprime}a\ list\ {\isasymRightarrow}\ bool}\\


492 
\isa{drop} & \isa{nat\ {\isasymRightarrow}\ {\isacharprime}a\ list\ {\isasymRightarrow}\ {\isacharprime}a\ list}\\


493 
\isa{dropWhile} & \isa{{\isacharparenleft}{\isacharprime}a\ {\isasymRightarrow}\ bool{\isacharparenright}\ {\isasymRightarrow}\ {\isacharprime}a\ list\ {\isasymRightarrow}\ {\isacharprime}a\ list}\\


494 
\isa{filter} & \isa{{\isacharparenleft}{\isacharprime}a\ {\isasymRightarrow}\ bool{\isacharparenright}\ {\isasymRightarrow}\ {\isacharprime}a\ list\ {\isasymRightarrow}\ {\isacharprime}a\ list}\\


495 
\isa{foldl} & \isa{{\isacharparenleft}{\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}b\ {\isasymRightarrow}\ {\isacharprime}a{\isacharparenright}\ {\isasymRightarrow}\ {\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}b\ list\ {\isasymRightarrow}\ {\isacharprime}a}\\


496 
\isa{foldr} & \isa{{\isacharparenleft}{\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}b\ {\isasymRightarrow}\ {\isacharprime}b{\isacharparenright}\ {\isasymRightarrow}\ {\isacharprime}a\ list\ {\isasymRightarrow}\ {\isacharprime}b\ {\isasymRightarrow}\ {\isacharprime}b}\\


497 
\isa{hd} & \isa{{\isacharprime}a\ list\ {\isasymRightarrow}\ {\isacharprime}a}\\


498 
\isa{last} & \isa{{\isacharprime}a\ list\ {\isasymRightarrow}\ {\isacharprime}a}\\


499 
\isa{length} & \isa{{\isacharprime}a\ list\ {\isasymRightarrow}\ nat}\\


500 
\isa{lenlex} & \isa{{\isacharparenleft}{\isacharprime}a\ {\isasymtimes}\ {\isacharprime}a{\isacharparenright}\ set\ {\isasymRightarrow}\ {\isacharparenleft}{\isacharprime}a\ list\ {\isasymtimes}\ {\isacharprime}a\ list{\isacharparenright}\ set}\\


501 
\isa{lex} & \isa{{\isacharparenleft}{\isacharprime}a\ {\isasymtimes}\ {\isacharprime}a{\isacharparenright}\ set\ {\isasymRightarrow}\ {\isacharparenleft}{\isacharprime}a\ list\ {\isasymtimes}\ {\isacharprime}a\ list{\isacharparenright}\ set}\\


502 
\isa{lexn} & \isa{{\isacharparenleft}{\isacharprime}a\ {\isasymtimes}\ {\isacharprime}a{\isacharparenright}\ set\ {\isasymRightarrow}\ nat\ {\isasymRightarrow}\ {\isacharparenleft}{\isacharprime}a\ list\ {\isasymtimes}\ {\isacharprime}a\ list{\isacharparenright}\ set}\\


503 
\isa{lexord} & \isa{{\isacharparenleft}{\isacharprime}a\ {\isasymtimes}\ {\isacharprime}a{\isacharparenright}\ set\ {\isasymRightarrow}\ {\isacharparenleft}{\isacharprime}a\ list\ {\isasymtimes}\ {\isacharprime}a\ list{\isacharparenright}\ set}\\


504 
\isa{listrel} & \isa{{\isacharparenleft}{\isacharprime}a\ {\isasymtimes}\ {\isacharprime}a{\isacharparenright}\ set\ {\isasymRightarrow}\ {\isacharparenleft}{\isacharprime}a\ list\ {\isasymtimes}\ {\isacharprime}a\ list{\isacharparenright}\ set}\\


505 
\isa{lists} & \isa{{\isacharprime}a\ set\ {\isasymRightarrow}\ {\isacharprime}a\ list\ set}\\


506 
\isa{listset} & \isa{{\isacharprime}a\ set\ list\ {\isasymRightarrow}\ {\isacharprime}a\ list\ set}\\


507 
\isa{listsum} & \isa{{\isacharprime}a\ list\ {\isasymRightarrow}\ {\isacharprime}a}\\


508 
\isa{list{\isacharunderscore}all{\isadigit{2}}} & \isa{{\isacharparenleft}{\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}b\ {\isasymRightarrow}\ bool{\isacharparenright}\ {\isasymRightarrow}\ {\isacharprime}a\ list\ {\isasymRightarrow}\ {\isacharprime}b\ list\ {\isasymRightarrow}\ bool}\\


509 
\isa{list{\isacharunderscore}update} & \isa{{\isacharprime}a\ list\ {\isasymRightarrow}\ nat\ {\isasymRightarrow}\ {\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}a\ list}\\


510 
\isa{map} & \isa{{\isacharparenleft}{\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}b{\isacharparenright}\ {\isasymRightarrow}\ {\isacharprime}a\ list\ {\isasymRightarrow}\ {\isacharprime}b\ list}\\


511 
\isa{measures} & \isa{{\isacharparenleft}{\isacharprime}a\ {\isasymRightarrow}\ nat{\isacharparenright}\ list\ {\isasymRightarrow}\ {\isacharparenleft}{\isacharprime}a\ {\isasymtimes}\ {\isacharprime}a{\isacharparenright}\ set}\\


512 
\isa{remdups} & \isa{{\isacharprime}a\ list\ {\isasymRightarrow}\ {\isacharprime}a\ list}\\


513 
\isa{removeAll} & \isa{{\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}a\ list\ {\isasymRightarrow}\ {\isacharprime}a\ list}\\


514 
\isa{remove{\isadigit{1}}} & \isa{{\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}a\ list\ {\isasymRightarrow}\ {\isacharprime}a\ list}\\


515 
\isa{replicate} & \isa{nat\ {\isasymRightarrow}\ {\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}a\ list}\\


516 
\isa{rev} & \isa{{\isacharprime}a\ list\ {\isasymRightarrow}\ {\isacharprime}a\ list}\\


517 
\isa{rotate} & \isa{nat\ {\isasymRightarrow}\ {\isacharprime}a\ list\ {\isasymRightarrow}\ {\isacharprime}a\ list}\\


518 
\isa{rotate{\isadigit{1}}} & \isa{{\isacharprime}a\ list\ {\isasymRightarrow}\ {\isacharprime}a\ list}\\


519 
\isa{set} & \isa{{\isacharprime}a\ list\ {\isasymRightarrow}\ {\isacharprime}a\ set}\\


520 
\isa{sort} & \isa{{\isacharprime}a\ list\ {\isasymRightarrow}\ {\isacharprime}a\ list}\\


521 
\isa{sorted} & \isa{{\isacharprime}a\ list\ {\isasymRightarrow}\ bool}\\


522 
\isa{splice} & \isa{{\isacharprime}a\ list\ {\isasymRightarrow}\ {\isacharprime}a\ list\ {\isasymRightarrow}\ {\isacharprime}a\ list}\\


523 
\isa{sublist} & \isa{{\isacharprime}a\ list\ {\isasymRightarrow}\ {\isacharparenleft}nat\ {\isasymRightarrow}\ bool{\isacharparenright}\ {\isasymRightarrow}\ {\isacharprime}a\ list}\\


524 
\isa{take} & \isa{nat\ {\isasymRightarrow}\ {\isacharprime}a\ list\ {\isasymRightarrow}\ {\isacharprime}a\ list}\\


525 
\isa{takeWhile} & \isa{{\isacharparenleft}{\isacharprime}a\ {\isasymRightarrow}\ bool{\isacharparenright}\ {\isasymRightarrow}\ {\isacharprime}a\ list\ {\isasymRightarrow}\ {\isacharprime}a\ list}\\


526 
\isa{tl} & \isa{{\isacharprime}a\ list\ {\isasymRightarrow}\ {\isacharprime}a\ list}\\


527 
\isa{upt} & \isa{nat\ {\isasymRightarrow}\ nat\ {\isasymRightarrow}\ nat\ list}\\


528 
\isa{upto} & \isa{{\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}a\ list}\\


529 
\isa{zip} & \isa{{\isacharprime}a\ list\ {\isasymRightarrow}\ {\isacharprime}b\ list\ {\isasymRightarrow}\ {\isacharparenleft}{\isacharprime}a\ {\isasymtimes}\ {\isacharprime}b{\isacharparenright}\ list}\\


530 
\end{supertabular}


531 


532 
\subsubsection*{Syntax}


533 


534 
\begin{supertabular}{@ {} l @ {\quad$\equiv$\quad} l @ {}}


535 
\isa{{\isacharbrackleft}x\isactrlisub {\isadigit{1}}{\isacharcomma}{\isasymdots}{\isacharcomma}x\isactrlisub n{\isacharbrackright}} & \isa{x\isactrlisub {\isadigit{1}}\ {\isacharhash}\ {\isasymdots}\ {\isacharhash}\ x\isactrlisub n\ {\isacharhash}\ {\isacharbrackleft}{\isacharbrackright}}\\


536 
\isa{{\isacharbrackleft}m{\isachardot}{\isachardot}{\isacharless}n{\isacharbrackright}} & \isa{{\isachardoublequote}upt\ m\ n{\isachardoublequote}}\\


537 
\isa{{\isacharbrackleft}i{\isachardot}{\isachardot}j{\isacharbrackright}} & \isa{{\isachardoublequote}upto\ i\ j{\isachardoublequote}}\\


538 
\isa{{\isacharbrackleft}e{\isachardot}\ x\ {\isasymleftarrow}\ xs{\isacharbrackright}} & \isa{map\ {\isacharparenleft}{\isasymlambda}x{\isachardot}\ e{\isacharparenright}\ xs}\\


539 
\isa{{\isacharbrackleft}x{\isasymleftarrow}xs\ {\isachardot}\ b{\isacharbrackright}} & \isa{{\isachardoublequote}filter\ {\isacharparenleft}{\isasymlambda}x{\isachardot}\ b{\isacharparenright}\ xs{\isachardoublequote}} \\


540 
\isa{xs{\isacharbrackleft}n\ {\isacharcolon}{\isacharequal}\ x{\isacharbrackright}} & \isa{{\isachardoublequote}list{\isacharunderscore}update\ xs\ n\ x{\isachardoublequote}}\\


541 
\isa{{\isasymSum}x{\isasymleftarrow}xs{\isachardot}\ e} & \isa{{\isachardoublequote}listsum\ {\isacharparenleft}map\ {\isacharparenleft}{\isasymlambda}x{\isachardot}\ e{\isacharparenright}\ xs{\isacharparenright}{\isachardoublequote}}\\


542 
\end{supertabular}


543 
\medskip


544 


545 
List comprehension: \isa{{\isacharbrackleft}e{\isachardot}\ q\isactrlisub {\isadigit{1}}{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ q\isactrlisub n{\isacharbrackright}} where each


546 
qualifier \isa{q\isactrlisub i} is either a generator \mbox{\isa{pat\ {\isasymleftarrow}\ e}} or a


547 
guard, i.e.\ boolean expression.


548 


549 
\section{Map}


550 


551 
Maps model partial functions and are often used as finite tables. However,


552 
the domain of a map may be infinite.


553 


554 
\isa{{\isacharprime}a\ {\isasymrightharpoonup}\ {\isacharprime}b\ \ {\isacharequal}\ \ {\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}b\ option}


555 
\bigskip


556 


557 
\begin{supertabular}{@ {} l @ {~::~} l @ {}}


558 
\isa{Map{\isachardot}empty} & \isa{{\isacharprime}a\ {\isasymrightharpoonup}\ {\isacharprime}b}\\


559 
\isa{op\ {\isacharplus}{\isacharplus}} & \isa{{\isacharparenleft}{\isacharprime}a\ {\isasymrightharpoonup}\ {\isacharprime}b{\isacharparenright}\ {\isasymRightarrow}\ {\isacharparenleft}{\isacharprime}a\ {\isasymrightharpoonup}\ {\isacharprime}b{\isacharparenright}\ {\isasymRightarrow}\ {\isacharprime}a\ {\isasymrightharpoonup}\ {\isacharprime}b}\\


560 
\isa{op\ {\isasymcirc}\isactrlsub m} & \isa{{\isacharparenleft}{\isacharprime}a\ {\isasymrightharpoonup}\ {\isacharprime}b{\isacharparenright}\ {\isasymRightarrow}\ {\isacharparenleft}{\isacharprime}c\ {\isasymrightharpoonup}\ {\isacharprime}a{\isacharparenright}\ {\isasymRightarrow}\ {\isacharprime}c\ {\isasymrightharpoonup}\ {\isacharprime}b}\\


561 
\isa{op\ {\isacharbar}{\isacharbackquote}} & \isa{{\isacharparenleft}{\isacharprime}a\ {\isasymrightharpoonup}\ {\isacharprime}b{\isacharparenright}\ {\isasymRightarrow}\ {\isacharprime}a\ set\ {\isasymRightarrow}\ {\isacharprime}a\ {\isasymrightharpoonup}\ {\isacharprime}b}\\


562 
\isa{dom} & \isa{{\isacharparenleft}{\isacharprime}a\ {\isasymrightharpoonup}\ {\isacharprime}b{\isacharparenright}\ {\isasymRightarrow}\ {\isacharprime}a\ set}\\


563 
\isa{ran} & \isa{{\isacharparenleft}{\isacharprime}a\ {\isasymrightharpoonup}\ {\isacharprime}b{\isacharparenright}\ {\isasymRightarrow}\ {\isacharprime}b\ set}\\


564 
\isa{op\ {\isasymsubseteq}\isactrlsub m} & \isa{{\isacharparenleft}{\isacharprime}a\ {\isasymrightharpoonup}\ {\isacharprime}b{\isacharparenright}\ {\isasymRightarrow}\ {\isacharparenleft}{\isacharprime}a\ {\isasymrightharpoonup}\ {\isacharprime}b{\isacharparenright}\ {\isasymRightarrow}\ bool}\\


565 
\isa{map{\isacharunderscore}of} & \isa{{\isacharparenleft}{\isacharprime}a\ {\isasymtimes}\ {\isacharprime}b{\isacharparenright}\ list\ {\isasymRightarrow}\ {\isacharprime}a\ {\isasymrightharpoonup}\ {\isacharprime}b}\\


566 
\isa{map{\isacharunderscore}upds} & \isa{{\isacharparenleft}{\isacharprime}a\ {\isasymrightharpoonup}\ {\isacharprime}b{\isacharparenright}\ {\isasymRightarrow}\ {\isacharprime}a\ list\ {\isasymRightarrow}\ {\isacharprime}b\ list\ {\isasymRightarrow}\ {\isacharprime}a\ {\isasymrightharpoonup}\ {\isacharprime}b}\\


567 
\end{supertabular}


568 


569 
\subsubsection*{Syntax}


570 


571 
\begin{tabular}{@ {} l @ {\quad$\equiv$\quad} l @ {}}


572 
\isa{Map{\isachardot}empty} & \isa{{\isasymlambda}x{\isachardot}\ None}\\


573 
\isa{m{\isacharparenleft}x\ {\isasymmapsto}\ y{\isacharparenright}} & \isa{{\isachardoublequote}m{\isacharparenleft}x{\isacharcolon}{\isacharequal}Some\ y{\isacharparenright}{\isachardoublequote}}\\


574 
\isa{m{\isacharparenleft}x\isactrlisub {\isadigit{1}}{\isasymmapsto}y\isactrlisub {\isadigit{1}}{\isacharcomma}{\isasymdots}{\isacharcomma}x\isactrlisub n{\isasymmapsto}y\isactrlisub n{\isacharparenright}} & \isa{{\isachardoublequote}m{\isacharparenleft}x\isactrlisub {\isadigit{1}}{\isasymmapsto}y\isactrlisub {\isadigit{1}}{\isacharparenright}{\isasymdots}{\isacharparenleft}x\isactrlisub n{\isasymmapsto}y\isactrlisub n{\isacharparenright}{\isachardoublequote}}\\


575 
\isa{{\isacharbrackleft}x\isactrlisub {\isadigit{1}}{\isasymmapsto}y\isactrlisub {\isadigit{1}}{\isacharcomma}{\isasymdots}{\isacharcomma}x\isactrlisub n{\isasymmapsto}y\isactrlisub n{\isacharbrackright}} & \isa{{\isachardoublequote}Map{\isachardot}empty{\isacharparenleft}x\isactrlisub {\isadigit{1}}{\isasymmapsto}y\isactrlisub {\isadigit{1}}{\isacharcomma}{\isasymdots}{\isacharcomma}x\isactrlisub n{\isasymmapsto}y\isactrlisub n{\isacharparenright}{\isachardoublequote}}\\


576 
\isa{m{\isacharparenleft}xs\ {\isacharbrackleft}{\isasymmapsto}{\isacharbrackright}\ ys{\isacharparenright}} & \isa{{\isachardoublequote}map{\isacharunderscore}upds\ m\ xs\ ys{\isachardoublequote}}\\


577 
\end{tabular}%


578 
\end{isamarkuptext}%


579 
\isamarkuptrue%


580 
%


581 
\isadelimtheory


582 
%


583 
\endisadelimtheory


584 
%


585 
\isatagtheory


586 
%


587 
\endisatagtheory


588 
{\isafoldtheory}%


589 
%


590 
\isadelimtheory


591 
%


592 
\endisadelimtheory


593 
\end{isabellebody}%


594 
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596 
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597 
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