diff -r a773af3e37d6 -r 9965099f51ad doc-src/Classes/Thy/document/Classes.tex --- a/doc-src/Classes/Thy/document/Classes.tex Mon Aug 27 22:22:42 2012 +0200 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,1561 +0,0 @@ -% -\begin{isabellebody}% -\def\isabellecontext{Classes}% -% -\isadelimtheory -% -\endisadelimtheory -% -\isatagtheory -\isacommand{theory}\isamarkupfalse% -\ Classes\isanewline -\isakeyword{imports}\ Main\ Setup\isanewline -\isakeyword{begin}% -\endisatagtheory -{\isafoldtheory}% -% -\isadelimtheory -% -\endisadelimtheory -% -\isamarkupsection{Introduction% -} -\isamarkuptrue% -% -\begin{isamarkuptext}% -Type classes were introduced by Wadler and Blott \cite{wadler89how} - into the Haskell language to allow for a reasonable implementation - of overloading\footnote{throughout this tutorial, we are referring - to classical Haskell 1.0 type classes, not considering later - additions in expressiveness}. As a canonical example, a polymorphic - equality function \isa{eq\ {\isaliteral{5C3C436F6C6F6E3E}{\isasymColon}}\ {\isaliteral{5C3C616C7068613E}{\isasymalpha}}\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ {\isaliteral{5C3C616C7068613E}{\isasymalpha}}\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ bool} which is overloaded on - different types for \isa{{\isaliteral{5C3C616C7068613E}{\isasymalpha}}}, which is achieved by splitting - introduction of the \isa{eq} function from its overloaded - definitions by means of \isa{class} and \isa{instance} - declarations: \footnote{syntax here is a kind of isabellized - Haskell} - - \begin{quote} - - \noindent\isa{class\ eq\ where} \\ - \hspace*{2ex}\isa{eq\ {\isaliteral{5C3C436F6C6F6E3E}{\isasymColon}}\ {\isaliteral{5C3C616C7068613E}{\isasymalpha}}\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ {\isaliteral{5C3C616C7068613E}{\isasymalpha}}\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ bool} - - \medskip\noindent\isa{instance\ nat\ {\isaliteral{5C3C436F6C6F6E3E}{\isasymColon}}\ eq\ where} \\ - \hspace*{2ex}\isa{eq\ {\isadigit{0}}\ {\isadigit{0}}\ {\isaliteral{3D}{\isacharequal}}\ True} \\ - \hspace*{2ex}\isa{eq\ {\isadigit{0}}\ {\isaliteral{5F}{\isacharunderscore}}\ {\isaliteral{3D}{\isacharequal}}\ False} \\ - \hspace*{2ex}\isa{eq\ {\isaliteral{5F}{\isacharunderscore}}\ {\isadigit{0}}\ {\isaliteral{3D}{\isacharequal}}\ False} \\ - \hspace*{2ex}\isa{eq\ {\isaliteral{28}{\isacharparenleft}}Suc\ n{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{28}{\isacharparenleft}}Suc\ m{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ eq\ n\ m} - - \medskip\noindent\isa{instance\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C616C7068613E}{\isasymalpha}}{\isaliteral{5C3C436F6C6F6E3E}{\isasymColon}}eq{\isaliteral{2C}{\isacharcomma}}\ {\isaliteral{5C3C626574613E}{\isasymbeta}}{\isaliteral{5C3C436F6C6F6E3E}{\isasymColon}}eq{\isaliteral{29}{\isacharparenright}}\ pair\ {\isaliteral{5C3C436F6C6F6E3E}{\isasymColon}}\ eq\ where} \\ - \hspace*{2ex}\isa{eq\ {\isaliteral{28}{\isacharparenleft}}x{\isadigit{1}}{\isaliteral{2C}{\isacharcomma}}\ y{\isadigit{1}}{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{28}{\isacharparenleft}}x{\isadigit{2}}{\isaliteral{2C}{\isacharcomma}}\ y{\isadigit{2}}{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ eq\ x{\isadigit{1}}\ x{\isadigit{2}}\ {\isaliteral{5C3C616E643E}{\isasymand}}\ eq\ y{\isadigit{1}}\ y{\isadigit{2}}} - - \medskip\noindent\isa{class\ ord\ extends\ eq\ where} \\ - \hspace*{2ex}\isa{less{\isaliteral{5F}{\isacharunderscore}}eq\ {\isaliteral{5C3C436F6C6F6E3E}{\isasymColon}}\ {\isaliteral{5C3C616C7068613E}{\isasymalpha}}\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ {\isaliteral{5C3C616C7068613E}{\isasymalpha}}\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ bool} \\ - \hspace*{2ex}\isa{less\ {\isaliteral{5C3C436F6C6F6E3E}{\isasymColon}}\ {\isaliteral{5C3C616C7068613E}{\isasymalpha}}\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ {\isaliteral{5C3C616C7068613E}{\isasymalpha}}\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ bool} - - \end{quote} - - \noindent Type variables are annotated with (finitely many) classes; - these annotations are assertions that a particular polymorphic type - provides definitions for overloaded functions. - - Indeed, type classes not only allow for simple overloading but form - a generic calculus, an instance of order-sorted algebra - \cite{nipkow-sorts93,Nipkow-Prehofer:1993,Wenzel:1997:TPHOL}. - - From a software engineering point of view, type classes roughly - correspond to interfaces in object-oriented languages like Java; so, - it is naturally desirable that type classes do not only provide - functions (class parameters) but also state specifications - implementations must obey. For example, the \isa{class\ eq} - above could be given the following specification, demanding that - \isa{class\ eq} is an equivalence relation obeying reflexivity, - symmetry and transitivity: - - \begin{quote} - - \noindent\isa{class\ eq\ where} \\ - \hspace*{2ex}\isa{eq\ {\isaliteral{5C3C436F6C6F6E3E}{\isasymColon}}\ {\isaliteral{5C3C616C7068613E}{\isasymalpha}}\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ {\isaliteral{5C3C616C7068613E}{\isasymalpha}}\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ bool} \\ - \isa{satisfying} \\ - \hspace*{2ex}\isa{refl{\isaliteral{3A}{\isacharcolon}}\ eq\ x\ x} \\ - \hspace*{2ex}\isa{sym{\isaliteral{3A}{\isacharcolon}}\ eq\ x\ y\ {\isaliteral{5C3C6C6F6E676C65667472696768746172726F773E}{\isasymlongleftrightarrow}}\ eq\ x\ y} \\ - \hspace*{2ex}\isa{trans{\isaliteral{3A}{\isacharcolon}}\ eq\ x\ y\ {\isaliteral{5C3C616E643E}{\isasymand}}\ eq\ y\ z\ {\isaliteral{5C3C6C6F6E6772696768746172726F773E}{\isasymlongrightarrow}}\ eq\ x\ z} - - \end{quote} - - \noindent From a theoretical point of view, type classes are - lightweight modules; Haskell type classes may be emulated by SML - functors \cite{classes_modules}. Isabelle/Isar offers a discipline - of type classes which brings all those aspects together: - - \begin{enumerate} - \item specifying abstract parameters together with - corresponding specifications, - \item instantiating those abstract parameters by a particular - type - \item in connection with a ``less ad-hoc'' approach to overloading, - \item with a direct link to the Isabelle module system: - locales \cite{kammueller-locales}. - \end{enumerate} - - \noindent Isar type classes also directly support code generation in - a Haskell like fashion. Internally, they are mapped to more - primitive Isabelle concepts \cite{Haftmann-Wenzel:2006:classes}. - - This tutorial demonstrates common elements of structured - specifications and abstract reasoning with type classes by the - algebraic hierarchy of semigroups, monoids and groups. Our - background theory is that of Isabelle/HOL \cite{isa-tutorial}, for - which some familiarity is assumed.% -\end{isamarkuptext}% -\isamarkuptrue% -% -\isamarkupsection{A simple algebra example \label{sec:example}% -} -\isamarkuptrue% -% -\isamarkupsubsection{Class definition% -} -\isamarkuptrue% -% -\begin{isamarkuptext}% -Depending on an arbitrary type \isa{{\isaliteral{5C3C616C7068613E}{\isasymalpha}}}, class \isa{semigroup} introduces a binary operator \isa{{\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C6F74696D65733E}{\isasymotimes}}{\isaliteral{29}{\isacharparenright}}} that is - assumed to be associative:% -\end{isamarkuptext}% -\isamarkuptrue% -% -\isadelimquote -% -\endisadelimquote -% -\isatagquote -\isacommand{class}\isamarkupfalse% -\ semigroup\ {\isaliteral{3D}{\isacharequal}}\isanewline -\ \ \isakeyword{fixes}\ mult\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{5C3C616C7068613E}{\isasymalpha}}\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ {\isaliteral{5C3C616C7068613E}{\isasymalpha}}\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ {\isaliteral{5C3C616C7068613E}{\isasymalpha}}{\isaliteral{22}{\isachardoublequoteclose}}\ \ \ \ {\isaliteral{28}{\isacharparenleft}}\isakeyword{infixl}\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{5C3C6F74696D65733E}{\isasymotimes}}{\isaliteral{22}{\isachardoublequoteclose}}\ {\isadigit{7}}{\isadigit{0}}{\isaliteral{29}{\isacharparenright}}\isanewline -\ \ \isakeyword{assumes}\ assoc{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{28}{\isacharparenleft}}x\ {\isaliteral{5C3C6F74696D65733E}{\isasymotimes}}\ y{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C6F74696D65733E}{\isasymotimes}}\ z\ {\isaliteral{3D}{\isacharequal}}\ x\ {\isaliteral{5C3C6F74696D65733E}{\isasymotimes}}\ {\isaliteral{28}{\isacharparenleft}}y\ {\isaliteral{5C3C6F74696D65733E}{\isasymotimes}}\ z{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}% -\endisatagquote -{\isafoldquote}% -% -\isadelimquote -% -\endisadelimquote -% -\begin{isamarkuptext}% -\noindent This \hyperlink{command.class}{\mbox{\isa{\isacommand{class}}}} specification consists of two parts: - the \qn{operational} part names the class parameter (\hyperlink{element.fixes}{\mbox{\isa{\isakeyword{fixes}}}}), the \qn{logical} part specifies properties on them - (\hyperlink{element.assumes}{\mbox{\isa{\isakeyword{assumes}}}}). The local \hyperlink{element.fixes}{\mbox{\isa{\isakeyword{fixes}}}} and \hyperlink{element.assumes}{\mbox{\isa{\isakeyword{assumes}}}} are lifted to the theory toplevel, yielding the global - parameter \isa{{\isaliteral{22}{\isachardoublequote}}mult\ {\isaliteral{5C3C436F6C6F6E3E}{\isasymColon}}\ {\isaliteral{5C3C616C7068613E}{\isasymalpha}}{\isaliteral{5C3C436F6C6F6E3E}{\isasymColon}}semigroup\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ {\isaliteral{5C3C616C7068613E}{\isasymalpha}}\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ {\isaliteral{5C3C616C7068613E}{\isasymalpha}}{\isaliteral{22}{\isachardoublequote}}} and the - global theorem \hyperlink{fact.semigroup.assoc:}{\mbox{\isa{semigroup{\isaliteral{2E}{\isachardot}}assoc{\isaliteral{3A}{\isacharcolon}}}}}~\isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C416E643E}{\isasymAnd}}x\ y\ z\ {\isaliteral{5C3C436F6C6F6E3E}{\isasymColon}}\ {\isaliteral{5C3C616C7068613E}{\isasymalpha}}{\isaliteral{5C3C436F6C6F6E3E}{\isasymColon}}semigroup{\isaliteral{2E}{\isachardot}}\ {\isaliteral{28}{\isacharparenleft}}x\ {\isaliteral{5C3C6F74696D65733E}{\isasymotimes}}\ y{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C6F74696D65733E}{\isasymotimes}}\ z\ {\isaliteral{3D}{\isacharequal}}\ x\ {\isaliteral{5C3C6F74696D65733E}{\isasymotimes}}\ {\isaliteral{28}{\isacharparenleft}}y\ {\isaliteral{5C3C6F74696D65733E}{\isasymotimes}}\ z{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequote}}}.% -\end{isamarkuptext}% -\isamarkuptrue% -% -\isamarkupsubsection{Class instantiation \label{sec:class_inst}% -} -\isamarkuptrue% -% -\begin{isamarkuptext}% -The concrete type \isa{int} is made a \isa{semigroup} instance - by providing a suitable definition for the class parameter \isa{{\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C6F74696D65733E}{\isasymotimes}}{\isaliteral{29}{\isacharparenright}}} and a proof for the specification of \hyperlink{fact.assoc}{\mbox{\isa{assoc}}}. This is - accomplished by the \hyperlink{command.instantiation}{\mbox{\isa{\isacommand{instantiation}}}} target:% -\end{isamarkuptext}% -\isamarkuptrue% -% -\isadelimquote -% -\endisadelimquote -% -\isatagquote -\isacommand{instantiation}\isamarkupfalse% -\ int\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ semigroup\isanewline -\isakeyword{begin}\isanewline -\isanewline -\isacommand{definition}\isamarkupfalse% -\isanewline -\ \ mult{\isaliteral{5F}{\isacharunderscore}}int{\isaliteral{5F}{\isacharunderscore}}def{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}i\ {\isaliteral{5C3C6F74696D65733E}{\isasymotimes}}\ j\ {\isaliteral{3D}{\isacharequal}}\ i\ {\isaliteral{2B}{\isacharplus}}\ {\isaliteral{28}{\isacharparenleft}}j{\isaliteral{5C3C436F6C6F6E3E}{\isasymColon}}int{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline -\isanewline -\isacommand{instance}\isamarkupfalse% -\ \isacommand{proof}\isamarkupfalse% -\isanewline -\ \ \isacommand{fix}\isamarkupfalse% -\ i\ j\ k\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ int\ \isacommand{have}\isamarkupfalse% -\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{28}{\isacharparenleft}}i\ {\isaliteral{2B}{\isacharplus}}\ j{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{2B}{\isacharplus}}\ k\ {\isaliteral{3D}{\isacharequal}}\ i\ {\isaliteral{2B}{\isacharplus}}\ {\isaliteral{28}{\isacharparenleft}}j\ {\isaliteral{2B}{\isacharplus}}\ k{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\ \isacommand{by}\isamarkupfalse% -\ simp\isanewline -\ \ \isacommand{then}\isamarkupfalse% -\ \isacommand{show}\isamarkupfalse% -\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{28}{\isacharparenleft}}i\ {\isaliteral{5C3C6F74696D65733E}{\isasymotimes}}\ j{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C6F74696D65733E}{\isasymotimes}}\ k\ {\isaliteral{3D}{\isacharequal}}\ i\ {\isaliteral{5C3C6F74696D65733E}{\isasymotimes}}\ {\isaliteral{28}{\isacharparenleft}}j\ {\isaliteral{5C3C6F74696D65733E}{\isasymotimes}}\ k{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline -\ \ \ \ \isacommand{unfolding}\isamarkupfalse% -\ mult{\isaliteral{5F}{\isacharunderscore}}int{\isaliteral{5F}{\isacharunderscore}}def\ \isacommand{{\isaliteral{2E}{\isachardot}}}\isamarkupfalse% -\isanewline -\isacommand{qed}\isamarkupfalse% -\isanewline -\isanewline -\isacommand{end}\isamarkupfalse% -% -\endisatagquote -{\isafoldquote}% -% -\isadelimquote -% -\endisadelimquote -% -\begin{isamarkuptext}% -\noindent \hyperlink{command.instantiation}{\mbox{\isa{\isacommand{instantiation}}}} defines class parameters at a - particular instance using common specification tools (here, - \hyperlink{command.definition}{\mbox{\isa{\isacommand{definition}}}}). The concluding \hyperlink{command.instance}{\mbox{\isa{\isacommand{instance}}}} opens a - proof that the given parameters actually conform to the class - specification. Note that the first proof step is the \hyperlink{method.default}{\mbox{\isa{default}}} method, which for such instance proofs maps to the \hyperlink{method.intro-classes}{\mbox{\isa{intro{\isaliteral{5F}{\isacharunderscore}}classes}}} method. This reduces an instance judgement to the - relevant primitive proof goals; typically it is the first method - applied in an instantiation proof. - - From now on, the type-checker will consider \isa{int} as a \isa{semigroup} automatically, i.e.\ any general results are immediately - available on concrete instances. - - \medskip Another instance of \isa{semigroup} yields the natural - numbers:% -\end{isamarkuptext}% -\isamarkuptrue% -% -\isadelimquote -% -\endisadelimquote -% -\isatagquote -\isacommand{instantiation}\isamarkupfalse% -\ nat\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ semigroup\isanewline -\isakeyword{begin}\isanewline -\isanewline -\isacommand{primrec}\isamarkupfalse% -\ mult{\isaliteral{5F}{\isacharunderscore}}nat\ \isakeyword{where}\isanewline -\ \ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{28}{\isacharparenleft}}{\isadigit{0}}{\isaliteral{5C3C436F6C6F6E3E}{\isasymColon}}nat{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C6F74696D65733E}{\isasymotimes}}\ n\ {\isaliteral{3D}{\isacharequal}}\ n{\isaliteral{22}{\isachardoublequoteclose}}\isanewline -\ \ {\isaliteral{7C}{\isacharbar}}\ {\isaliteral{22}{\isachardoublequoteopen}}Suc\ m\ {\isaliteral{5C3C6F74696D65733E}{\isasymotimes}}\ n\ {\isaliteral{3D}{\isacharequal}}\ Suc\ {\isaliteral{28}{\isacharparenleft}}m\ {\isaliteral{5C3C6F74696D65733E}{\isasymotimes}}\ n{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline -\isanewline -\isacommand{instance}\isamarkupfalse% -\ \isacommand{proof}\isamarkupfalse% -\isanewline -\ \ \isacommand{fix}\isamarkupfalse% -\ m\ n\ q\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ nat\ \isanewline -\ \ \isacommand{show}\isamarkupfalse% -\ {\isaliteral{22}{\isachardoublequoteopen}}m\ {\isaliteral{5C3C6F74696D65733E}{\isasymotimes}}\ n\ {\isaliteral{5C3C6F74696D65733E}{\isasymotimes}}\ q\ {\isaliteral{3D}{\isacharequal}}\ m\ {\isaliteral{5C3C6F74696D65733E}{\isasymotimes}}\ {\isaliteral{28}{\isacharparenleft}}n\ {\isaliteral{5C3C6F74696D65733E}{\isasymotimes}}\ q{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline -\ \ \ \ \isacommand{by}\isamarkupfalse% -\ {\isaliteral{28}{\isacharparenleft}}induct\ m{\isaliteral{29}{\isacharparenright}}\ auto\isanewline -\isacommand{qed}\isamarkupfalse% -\isanewline -\isanewline -\isacommand{end}\isamarkupfalse% -% -\endisatagquote -{\isafoldquote}% -% -\isadelimquote -% -\endisadelimquote -% -\begin{isamarkuptext}% -\noindent Note the occurence of the name \isa{mult{\isaliteral{5F}{\isacharunderscore}}nat} in the - primrec declaration; by default, the local name of a class operation - \isa{f} to be instantiated on type constructor \isa{{\isaliteral{5C3C6B617070613E}{\isasymkappa}}} is - mangled as \isa{f{\isaliteral{5F}{\isacharunderscore}}{\isaliteral{5C3C6B617070613E}{\isasymkappa}}}. In case of uncertainty, these names may be - inspected using the \hyperlink{command.print-context}{\mbox{\isa{\isacommand{print{\isaliteral{5F}{\isacharunderscore}}context}}}} command or the - corresponding ProofGeneral button.% -\end{isamarkuptext}% -\isamarkuptrue% -% -\isamarkupsubsection{Lifting and parametric types% -} -\isamarkuptrue% -% -\begin{isamarkuptext}% -Overloaded definitions given at a class instantiation may include - recursion over the syntactic structure of types. As a canonical - example, we model product semigroups using our simple algebra:% -\end{isamarkuptext}% -\isamarkuptrue% -% -\isadelimquote -% -\endisadelimquote -% -\isatagquote -\isacommand{instantiation}\isamarkupfalse% -\ prod\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{28}{\isacharparenleft}}semigroup{\isaliteral{2C}{\isacharcomma}}\ semigroup{\isaliteral{29}{\isacharparenright}}\ semigroup\isanewline -\isakeyword{begin}\isanewline -\isanewline -\isacommand{definition}\isamarkupfalse% -\isanewline -\ \ mult{\isaliteral{5F}{\isacharunderscore}}prod{\isaliteral{5F}{\isacharunderscore}}def{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}p\isaliteral{5C3C5E697375623E}{}\isactrlisub {\isadigit{1}}\ {\isaliteral{5C3C6F74696D65733E}{\isasymotimes}}\ p\isaliteral{5C3C5E697375623E}{}\isactrlisub {\isadigit{2}}\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{28}{\isacharparenleft}}fst\ p\isaliteral{5C3C5E697375623E}{}\isactrlisub {\isadigit{1}}\ {\isaliteral{5C3C6F74696D65733E}{\isasymotimes}}\ fst\ p\isaliteral{5C3C5E697375623E}{}\isactrlisub {\isadigit{2}}{\isaliteral{2C}{\isacharcomma}}\ snd\ p\isaliteral{5C3C5E697375623E}{}\isactrlisub {\isadigit{1}}\ {\isaliteral{5C3C6F74696D65733E}{\isasymotimes}}\ snd\ p\isaliteral{5C3C5E697375623E}{}\isactrlisub {\isadigit{2}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline -\isanewline -\isacommand{instance}\isamarkupfalse% -\ \isacommand{proof}\isamarkupfalse% -\isanewline -\ \ \isacommand{fix}\isamarkupfalse% -\ p\isaliteral{5C3C5E697375623E}{}\isactrlisub {\isadigit{1}}\ p\isaliteral{5C3C5E697375623E}{}\isactrlisub {\isadigit{2}}\ p\isaliteral{5C3C5E697375623E}{}\isactrlisub {\isadigit{3}}\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{5C3C616C7068613E}{\isasymalpha}}{\isaliteral{5C3C436F6C6F6E3E}{\isasymColon}}semigroup\ {\isaliteral{5C3C74696D65733E}{\isasymtimes}}\ {\isaliteral{5C3C626574613E}{\isasymbeta}}{\isaliteral{5C3C436F6C6F6E3E}{\isasymColon}}semigroup{\isaliteral{22}{\isachardoublequoteclose}}\isanewline -\ \ \isacommand{show}\isamarkupfalse% -\ {\isaliteral{22}{\isachardoublequoteopen}}p\isaliteral{5C3C5E697375623E}{}\isactrlisub {\isadigit{1}}\ {\isaliteral{5C3C6F74696D65733E}{\isasymotimes}}\ p\isaliteral{5C3C5E697375623E}{}\isactrlisub {\isadigit{2}}\ {\isaliteral{5C3C6F74696D65733E}{\isasymotimes}}\ p\isaliteral{5C3C5E697375623E}{}\isactrlisub {\isadigit{3}}\ {\isaliteral{3D}{\isacharequal}}\ p\isaliteral{5C3C5E697375623E}{}\isactrlisub {\isadigit{1}}\ {\isaliteral{5C3C6F74696D65733E}{\isasymotimes}}\ {\isaliteral{28}{\isacharparenleft}}p\isaliteral{5C3C5E697375623E}{}\isactrlisub {\isadigit{2}}\ {\isaliteral{5C3C6F74696D65733E}{\isasymotimes}}\ p\isaliteral{5C3C5E697375623E}{}\isactrlisub {\isadigit{3}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline -\ \ \ \ \isacommand{unfolding}\isamarkupfalse% -\ mult{\isaliteral{5F}{\isacharunderscore}}prod{\isaliteral{5F}{\isacharunderscore}}def\ \isacommand{by}\isamarkupfalse% -\ {\isaliteral{28}{\isacharparenleft}}simp\ add{\isaliteral{3A}{\isacharcolon}}\ assoc{\isaliteral{29}{\isacharparenright}}\isanewline -\isacommand{qed}\isamarkupfalse% -\ \ \ \ \ \ \isanewline -\isanewline -\isacommand{end}\isamarkupfalse% -% -\endisatagquote -{\isafoldquote}% -% -\isadelimquote -% -\endisadelimquote -% -\begin{isamarkuptext}% -\noindent Associativity of product semigroups is established using - the definition of \isa{{\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C6F74696D65733E}{\isasymotimes}}{\isaliteral{29}{\isacharparenright}}} on products and the hypothetical - associativity of the type components; these hypotheses are - legitimate due to the \isa{semigroup} constraints imposed on the - type components by the \hyperlink{command.instance}{\mbox{\isa{\isacommand{instance}}}} proposition. Indeed, - this pattern often occurs with parametric types and type classes.% -\end{isamarkuptext}% -\isamarkuptrue% -% -\isamarkupsubsection{Subclassing% -} -\isamarkuptrue% -% -\begin{isamarkuptext}% -We define a subclass \isa{monoidl} (a semigroup with a left-hand - neutral) by extending \isa{semigroup} with one additional - parameter \isa{neutral} together with its characteristic property:% -\end{isamarkuptext}% -\isamarkuptrue% -% -\isadelimquote -% -\endisadelimquote -% -\isatagquote -\isacommand{class}\isamarkupfalse% -\ monoidl\ {\isaliteral{3D}{\isacharequal}}\ semigroup\ {\isaliteral{2B}{\isacharplus}}\isanewline -\ \ \isakeyword{fixes}\ neutral\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{5C3C616C7068613E}{\isasymalpha}}{\isaliteral{22}{\isachardoublequoteclose}}\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{5C3C6F6E653E}{\isasymone}}{\isaliteral{22}{\isachardoublequoteclose}}{\isaliteral{29}{\isacharparenright}}\isanewline -\ \ \isakeyword{assumes}\ neutl{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{5C3C6F6E653E}{\isasymone}}\ {\isaliteral{5C3C6F74696D65733E}{\isasymotimes}}\ x\ {\isaliteral{3D}{\isacharequal}}\ x{\isaliteral{22}{\isachardoublequoteclose}}% -\endisatagquote -{\isafoldquote}% -% -\isadelimquote -% -\endisadelimquote -% -\begin{isamarkuptext}% -\noindent Again, we prove some instances, by providing suitable - parameter definitions and proofs for the additional specifications. - Observe that instantiations for types with the same arity may be - simultaneous:% -\end{isamarkuptext}% -\isamarkuptrue% -% -\isadelimquote -% -\endisadelimquote -% -\isatagquote -\isacommand{instantiation}\isamarkupfalse% -\ nat\ \isakeyword{and}\ int\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ monoidl\isanewline -\isakeyword{begin}\isanewline -\isanewline -\isacommand{definition}\isamarkupfalse% -\isanewline -\ \ neutral{\isaliteral{5F}{\isacharunderscore}}nat{\isaliteral{5F}{\isacharunderscore}}def{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{5C3C6F6E653E}{\isasymone}}\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{28}{\isacharparenleft}}{\isadigit{0}}{\isaliteral{5C3C436F6C6F6E3E}{\isasymColon}}nat{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline -\isanewline -\isacommand{definition}\isamarkupfalse% -\isanewline -\ \ neutral{\isaliteral{5F}{\isacharunderscore}}int{\isaliteral{5F}{\isacharunderscore}}def{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{5C3C6F6E653E}{\isasymone}}\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{28}{\isacharparenleft}}{\isadigit{0}}{\isaliteral{5C3C436F6C6F6E3E}{\isasymColon}}int{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline -\isanewline -\isacommand{instance}\isamarkupfalse% -\ \isacommand{proof}\isamarkupfalse% -\isanewline -\ \ \isacommand{fix}\isamarkupfalse% -\ n\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ nat\isanewline -\ \ \isacommand{show}\isamarkupfalse% -\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{5C3C6F6E653E}{\isasymone}}\ {\isaliteral{5C3C6F74696D65733E}{\isasymotimes}}\ n\ {\isaliteral{3D}{\isacharequal}}\ n{\isaliteral{22}{\isachardoublequoteclose}}\isanewline -\ \ \ \ \isacommand{unfolding}\isamarkupfalse% -\ neutral{\isaliteral{5F}{\isacharunderscore}}nat{\isaliteral{5F}{\isacharunderscore}}def\ \isacommand{by}\isamarkupfalse% -\ simp\isanewline -\isacommand{next}\isamarkupfalse% -\isanewline -\ \ \isacommand{fix}\isamarkupfalse% -\ k\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ int\isanewline -\ \ \isacommand{show}\isamarkupfalse% -\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{5C3C6F6E653E}{\isasymone}}\ {\isaliteral{5C3C6F74696D65733E}{\isasymotimes}}\ k\ {\isaliteral{3D}{\isacharequal}}\ k{\isaliteral{22}{\isachardoublequoteclose}}\isanewline -\ \ \ \ \isacommand{unfolding}\isamarkupfalse% -\ neutral{\isaliteral{5F}{\isacharunderscore}}int{\isaliteral{5F}{\isacharunderscore}}def\ mult{\isaliteral{5F}{\isacharunderscore}}int{\isaliteral{5F}{\isacharunderscore}}def\ \isacommand{by}\isamarkupfalse% -\ simp\isanewline -\isacommand{qed}\isamarkupfalse% -\isanewline -\isanewline -\isacommand{end}\isamarkupfalse% -\isanewline -\isanewline -\isacommand{instantiation}\isamarkupfalse% -\ prod\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{28}{\isacharparenleft}}monoidl{\isaliteral{2C}{\isacharcomma}}\ monoidl{\isaliteral{29}{\isacharparenright}}\ monoidl\isanewline -\isakeyword{begin}\isanewline -\isanewline -\isacommand{definition}\isamarkupfalse% -\isanewline -\ \ neutral{\isaliteral{5F}{\isacharunderscore}}prod{\isaliteral{5F}{\isacharunderscore}}def{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{5C3C6F6E653E}{\isasymone}}\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C6F6E653E}{\isasymone}}{\isaliteral{2C}{\isacharcomma}}\ {\isaliteral{5C3C6F6E653E}{\isasymone}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline -\isanewline -\isacommand{instance}\isamarkupfalse% -\ \isacommand{proof}\isamarkupfalse% -\isanewline -\ \ \isacommand{fix}\isamarkupfalse% -\ p\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{5C3C616C7068613E}{\isasymalpha}}{\isaliteral{5C3C436F6C6F6E3E}{\isasymColon}}monoidl\ {\isaliteral{5C3C74696D65733E}{\isasymtimes}}\ {\isaliteral{5C3C626574613E}{\isasymbeta}}{\isaliteral{5C3C436F6C6F6E3E}{\isasymColon}}monoidl{\isaliteral{22}{\isachardoublequoteclose}}\isanewline -\ \ \isacommand{show}\isamarkupfalse% -\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{5C3C6F6E653E}{\isasymone}}\ {\isaliteral{5C3C6F74696D65733E}{\isasymotimes}}\ p\ {\isaliteral{3D}{\isacharequal}}\ p{\isaliteral{22}{\isachardoublequoteclose}}\isanewline -\ \ \ \ \isacommand{unfolding}\isamarkupfalse% -\ neutral{\isaliteral{5F}{\isacharunderscore}}prod{\isaliteral{5F}{\isacharunderscore}}def\ mult{\isaliteral{5F}{\isacharunderscore}}prod{\isaliteral{5F}{\isacharunderscore}}def\ \isacommand{by}\isamarkupfalse% -\ {\isaliteral{28}{\isacharparenleft}}simp\ add{\isaliteral{3A}{\isacharcolon}}\ neutl{\isaliteral{29}{\isacharparenright}}\isanewline -\isacommand{qed}\isamarkupfalse% -\isanewline -\isanewline -\isacommand{end}\isamarkupfalse% -% -\endisatagquote -{\isafoldquote}% -% -\isadelimquote -% -\endisadelimquote -% -\begin{isamarkuptext}% -\noindent Fully-fledged monoids are modelled by another subclass, - which does not add new parameters but tightens the specification:% -\end{isamarkuptext}% -\isamarkuptrue% -% -\isadelimquote -% -\endisadelimquote -% -\isatagquote -\isacommand{class}\isamarkupfalse% -\ monoid\ {\isaliteral{3D}{\isacharequal}}\ monoidl\ {\isaliteral{2B}{\isacharplus}}\isanewline -\ \ \isakeyword{assumes}\ neutr{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}x\ {\isaliteral{5C3C6F74696D65733E}{\isasymotimes}}\ {\isaliteral{5C3C6F6E653E}{\isasymone}}\ {\isaliteral{3D}{\isacharequal}}\ x{\isaliteral{22}{\isachardoublequoteclose}}\isanewline -\isanewline -\isacommand{instantiation}\isamarkupfalse% -\ nat\ \isakeyword{and}\ int\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ monoid\ \isanewline -\isakeyword{begin}\isanewline -\isanewline -\isacommand{instance}\isamarkupfalse% -\ \isacommand{proof}\isamarkupfalse% -\isanewline -\ \ \isacommand{fix}\isamarkupfalse% -\ n\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ nat\isanewline -\ \ \isacommand{show}\isamarkupfalse% -\ {\isaliteral{22}{\isachardoublequoteopen}}n\ {\isaliteral{5C3C6F74696D65733E}{\isasymotimes}}\ {\isaliteral{5C3C6F6E653E}{\isasymone}}\ {\isaliteral{3D}{\isacharequal}}\ n{\isaliteral{22}{\isachardoublequoteclose}}\isanewline -\ \ \ \ \isacommand{unfolding}\isamarkupfalse% -\ neutral{\isaliteral{5F}{\isacharunderscore}}nat{\isaliteral{5F}{\isacharunderscore}}def\ \isacommand{by}\isamarkupfalse% -\ {\isaliteral{28}{\isacharparenleft}}induct\ n{\isaliteral{29}{\isacharparenright}}\ simp{\isaliteral{5F}{\isacharunderscore}}all\isanewline -\isacommand{next}\isamarkupfalse% -\isanewline -\ \ \isacommand{fix}\isamarkupfalse% -\ k\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ int\isanewline -\ \ \isacommand{show}\isamarkupfalse% -\ {\isaliteral{22}{\isachardoublequoteopen}}k\ {\isaliteral{5C3C6F74696D65733E}{\isasymotimes}}\ {\isaliteral{5C3C6F6E653E}{\isasymone}}\ {\isaliteral{3D}{\isacharequal}}\ k{\isaliteral{22}{\isachardoublequoteclose}}\isanewline -\ \ \ \ \isacommand{unfolding}\isamarkupfalse% -\ neutral{\isaliteral{5F}{\isacharunderscore}}int{\isaliteral{5F}{\isacharunderscore}}def\ mult{\isaliteral{5F}{\isacharunderscore}}int{\isaliteral{5F}{\isacharunderscore}}def\ \isacommand{by}\isamarkupfalse% -\ simp\isanewline -\isacommand{qed}\isamarkupfalse% -\isanewline -\isanewline -\isacommand{end}\isamarkupfalse% -\isanewline -\isanewline -\isacommand{instantiation}\isamarkupfalse% -\ prod\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{28}{\isacharparenleft}}monoid{\isaliteral{2C}{\isacharcomma}}\ monoid{\isaliteral{29}{\isacharparenright}}\ monoid\isanewline -\isakeyword{begin}\isanewline -\isanewline -\isacommand{instance}\isamarkupfalse% -\ \isacommand{proof}\isamarkupfalse% -\ \isanewline -\ \ \isacommand{fix}\isamarkupfalse% -\ p\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{5C3C616C7068613E}{\isasymalpha}}{\isaliteral{5C3C436F6C6F6E3E}{\isasymColon}}monoid\ {\isaliteral{5C3C74696D65733E}{\isasymtimes}}\ {\isaliteral{5C3C626574613E}{\isasymbeta}}{\isaliteral{5C3C436F6C6F6E3E}{\isasymColon}}monoid{\isaliteral{22}{\isachardoublequoteclose}}\isanewline -\ \ \isacommand{show}\isamarkupfalse% -\ {\isaliteral{22}{\isachardoublequoteopen}}p\ {\isaliteral{5C3C6F74696D65733E}{\isasymotimes}}\ {\isaliteral{5C3C6F6E653E}{\isasymone}}\ {\isaliteral{3D}{\isacharequal}}\ p{\isaliteral{22}{\isachardoublequoteclose}}\isanewline -\ \ \ \ \isacommand{unfolding}\isamarkupfalse% -\ neutral{\isaliteral{5F}{\isacharunderscore}}prod{\isaliteral{5F}{\isacharunderscore}}def\ mult{\isaliteral{5F}{\isacharunderscore}}prod{\isaliteral{5F}{\isacharunderscore}}def\ \isacommand{by}\isamarkupfalse% -\ {\isaliteral{28}{\isacharparenleft}}simp\ add{\isaliteral{3A}{\isacharcolon}}\ neutr{\isaliteral{29}{\isacharparenright}}\isanewline -\isacommand{qed}\isamarkupfalse% -\isanewline -\isanewline -\isacommand{end}\isamarkupfalse% -% -\endisatagquote -{\isafoldquote}% -% -\isadelimquote -% -\endisadelimquote -% -\begin{isamarkuptext}% -\noindent To finish our small algebra example, we add a \isa{group} class with a corresponding instance:% -\end{isamarkuptext}% -\isamarkuptrue% -% -\isadelimquote -% -\endisadelimquote -% -\isatagquote -\isacommand{class}\isamarkupfalse% -\ group\ {\isaliteral{3D}{\isacharequal}}\ monoidl\ {\isaliteral{2B}{\isacharplus}}\isanewline -\ \ \isakeyword{fixes}\ inverse\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{5C3C616C7068613E}{\isasymalpha}}\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ {\isaliteral{5C3C616C7068613E}{\isasymalpha}}{\isaliteral{22}{\isachardoublequoteclose}}\ \ \ \ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{28}{\isacharparenleft}}{\isaliteral{5F}{\isacharunderscore}}{\isaliteral{5C3C6469763E}{\isasymdiv}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\ {\isaliteral{5B}{\isacharbrackleft}}{\isadigit{1}}{\isadigit{0}}{\isadigit{0}}{\isadigit{0}}{\isaliteral{5D}{\isacharbrackright}}\ {\isadigit{9}}{\isadigit{9}}{\isadigit{9}}{\isaliteral{29}{\isacharparenright}}\isanewline -\ \ \isakeyword{assumes}\ invl{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}x{\isaliteral{5C3C6469763E}{\isasymdiv}}\ {\isaliteral{5C3C6F74696D65733E}{\isasymotimes}}\ x\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{5C3C6F6E653E}{\isasymone}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline -\isanewline -\isacommand{instantiation}\isamarkupfalse% -\ int\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ group\isanewline -\isakeyword{begin}\isanewline -\isanewline -\isacommand{definition}\isamarkupfalse% -\isanewline -\ \ inverse{\isaliteral{5F}{\isacharunderscore}}int{\isaliteral{5F}{\isacharunderscore}}def{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}i{\isaliteral{5C3C6469763E}{\isasymdiv}}\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{2D}{\isacharminus}}\ {\isaliteral{28}{\isacharparenleft}}i{\isaliteral{5C3C436F6C6F6E3E}{\isasymColon}}int{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline -\isanewline -\isacommand{instance}\isamarkupfalse% -\ \isacommand{proof}\isamarkupfalse% -\isanewline -\ \ \isacommand{fix}\isamarkupfalse% -\ i\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ int\isanewline -\ \ \isacommand{have}\isamarkupfalse% -\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{2D}{\isacharminus}}i\ {\isaliteral{2B}{\isacharplus}}\ i\ {\isaliteral{3D}{\isacharequal}}\ {\isadigit{0}}{\isaliteral{22}{\isachardoublequoteclose}}\ \isacommand{by}\isamarkupfalse% -\ simp\isanewline -\ \ \isacommand{then}\isamarkupfalse% -\ \isacommand{show}\isamarkupfalse% -\ {\isaliteral{22}{\isachardoublequoteopen}}i{\isaliteral{5C3C6469763E}{\isasymdiv}}\ {\isaliteral{5C3C6F74696D65733E}{\isasymotimes}}\ i\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{5C3C6F6E653E}{\isasymone}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline -\ \ \ \ \isacommand{unfolding}\isamarkupfalse% -\ mult{\isaliteral{5F}{\isacharunderscore}}int{\isaliteral{5F}{\isacharunderscore}}def\ neutral{\isaliteral{5F}{\isacharunderscore}}int{\isaliteral{5F}{\isacharunderscore}}def\ inverse{\isaliteral{5F}{\isacharunderscore}}int{\isaliteral{5F}{\isacharunderscore}}def\ \isacommand{{\isaliteral{2E}{\isachardot}}}\isamarkupfalse% -\isanewline -\isacommand{qed}\isamarkupfalse% -\isanewline -\isanewline -\isacommand{end}\isamarkupfalse% -% -\endisatagquote -{\isafoldquote}% -% -\isadelimquote -% -\endisadelimquote -% -\isamarkupsection{Type classes as locales% -} -\isamarkuptrue% -% -\isamarkupsubsection{A look behind the scenes% -} -\isamarkuptrue% -% -\begin{isamarkuptext}% -The example above gives an impression how Isar type classes work in - practice. As stated in the introduction, classes also provide a - link to Isar's locale system. Indeed, the logical core of a class - is nothing other than a locale:% -\end{isamarkuptext}% -\isamarkuptrue% -% -\isadelimquote -% -\endisadelimquote -% -\isatagquote -\isacommand{class}\isamarkupfalse% -\ idem\ {\isaliteral{3D}{\isacharequal}}\isanewline -\ \ \isakeyword{fixes}\ f\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{5C3C616C7068613E}{\isasymalpha}}\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ {\isaliteral{5C3C616C7068613E}{\isasymalpha}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline -\ \ \isakeyword{assumes}\ idem{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}f\ {\isaliteral{28}{\isacharparenleft}}f\ x{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ f\ x{\isaliteral{22}{\isachardoublequoteclose}}% -\endisatagquote -{\isafoldquote}% -% -\isadelimquote -% -\endisadelimquote -% -\begin{isamarkuptext}% -\noindent essentially introduces the locale% -\end{isamarkuptext}% -\isamarkuptrue% -\ % -\isadeliminvisible -% -\endisadeliminvisible -% -\isataginvisible -% -\endisataginvisible -{\isafoldinvisible}% -% -\isadeliminvisible -% -\endisadeliminvisible -% -\isadelimquote -% -\endisadelimquote -% -\isatagquote -\isacommand{locale}\isamarkupfalse% -\ idem\ {\isaliteral{3D}{\isacharequal}}\isanewline -\ \ \isakeyword{fixes}\ f\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{5C3C616C7068613E}{\isasymalpha}}\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ {\isaliteral{5C3C616C7068613E}{\isasymalpha}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline -\ \ \isakeyword{assumes}\ idem{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}f\ {\isaliteral{28}{\isacharparenleft}}f\ x{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ f\ x{\isaliteral{22}{\isachardoublequoteclose}}% -\endisatagquote -{\isafoldquote}% -% -\isadelimquote -% -\endisadelimquote -% -\begin{isamarkuptext}% -\noindent together with corresponding constant(s):% -\end{isamarkuptext}% -\isamarkuptrue% -% -\isadelimquote -% -\endisadelimquote -% -\isatagquote -\isacommand{consts}\isamarkupfalse% -\ f\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{5C3C616C7068613E}{\isasymalpha}}\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ {\isaliteral{5C3C616C7068613E}{\isasymalpha}}{\isaliteral{22}{\isachardoublequoteclose}}% -\endisatagquote -{\isafoldquote}% -% -\isadelimquote -% -\endisadelimquote -% -\begin{isamarkuptext}% -\noindent The connection to the type system is done by means - of a primitive type class% -\end{isamarkuptext}% -\isamarkuptrue% -\ % -\isadeliminvisible -% -\endisadeliminvisible -% -\isataginvisible -% -\endisataginvisible -{\isafoldinvisible}% -% -\isadeliminvisible -% -\endisadeliminvisible -% -\isadelimquote -% -\endisadelimquote -% -\isatagquote -\isacommand{classes}\isamarkupfalse% -\ idem\ {\isaliteral{3C}{\isacharless}}\ type% -\endisatagquote -{\isafoldquote}% -% -\isadelimquote -% -\endisadelimquote -% -\isadeliminvisible -% -\endisadeliminvisible -% -\isataginvisible -% -\endisataginvisible -{\isafoldinvisible}% -% -\isadeliminvisible -% -\endisadeliminvisible -% -\begin{isamarkuptext}% -\noindent together with a corresponding interpretation:% -\end{isamarkuptext}% -\isamarkuptrue% -% -\isadelimquote -% -\endisadelimquote -% -\isatagquote -\isacommand{interpretation}\isamarkupfalse% -\ idem{\isaliteral{5F}{\isacharunderscore}}class{\isaliteral{3A}{\isacharcolon}}\isanewline -\ \ idem\ {\isaliteral{22}{\isachardoublequoteopen}}f\ {\isaliteral{5C3C436F6C6F6E3E}{\isasymColon}}\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C616C7068613E}{\isasymalpha}}{\isaliteral{5C3C436F6C6F6E3E}{\isasymColon}}idem{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ {\isaliteral{5C3C616C7068613E}{\isasymalpha}}{\isaliteral{22}{\isachardoublequoteclose}}% -\endisatagquote -{\isafoldquote}% -% -\isadelimquote -% -\endisadelimquote -% -\begin{isamarkuptext}% -\noindent This gives you the full power of the Isabelle module system; - conclusions in locale \isa{idem} are implicitly propagated - to class \isa{idem}.% -\end{isamarkuptext}% -\isamarkuptrue% -\ % -\isadeliminvisible -% -\endisadeliminvisible -% -\isataginvisible -% -\endisataginvisible -{\isafoldinvisible}% -% -\isadeliminvisible -% -\endisadeliminvisible -% -\isamarkupsubsection{Abstract reasoning% -} -\isamarkuptrue% -% -\begin{isamarkuptext}% -Isabelle locales enable reasoning at a general level, while results - are implicitly transferred to all instances. For example, we can - now establish the \isa{left{\isaliteral{5F}{\isacharunderscore}}cancel} lemma for groups, which - states that the function \isa{{\isaliteral{28}{\isacharparenleft}}x\ {\isaliteral{5C3C6F74696D65733E}{\isasymotimes}}{\isaliteral{29}{\isacharparenright}}} is injective:% -\end{isamarkuptext}% -\isamarkuptrue% -% -\isadelimquote -% -\endisadelimquote -% -\isatagquote -\isacommand{lemma}\isamarkupfalse% -\ {\isaliteral{28}{\isacharparenleft}}\isakeyword{in}\ group{\isaliteral{29}{\isacharparenright}}\ left{\isaliteral{5F}{\isacharunderscore}}cancel{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}x\ {\isaliteral{5C3C6F74696D65733E}{\isasymotimes}}\ y\ {\isaliteral{3D}{\isacharequal}}\ x\ {\isaliteral{5C3C6F74696D65733E}{\isasymotimes}}\ z\ {\isaliteral{5C3C6C6F6E676C65667472696768746172726F773E}{\isasymlongleftrightarrow}}\ y\ {\isaliteral{3D}{\isacharequal}}\ z{\isaliteral{22}{\isachardoublequoteclose}}\isanewline -\isacommand{proof}\isamarkupfalse% -\isanewline -\ \ \isacommand{assume}\isamarkupfalse% -\ {\isaliteral{22}{\isachardoublequoteopen}}x\ {\isaliteral{5C3C6F74696D65733E}{\isasymotimes}}\ y\ {\isaliteral{3D}{\isacharequal}}\ x\ {\isaliteral{5C3C6F74696D65733E}{\isasymotimes}}\ z{\isaliteral{22}{\isachardoublequoteclose}}\isanewline -\ \ \isacommand{then}\isamarkupfalse% -\ \isacommand{have}\isamarkupfalse% -\ {\isaliteral{22}{\isachardoublequoteopen}}x{\isaliteral{5C3C6469763E}{\isasymdiv}}\ {\isaliteral{5C3C6F74696D65733E}{\isasymotimes}}\ {\isaliteral{28}{\isacharparenleft}}x\ {\isaliteral{5C3C6F74696D65733E}{\isasymotimes}}\ y{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ x{\isaliteral{5C3C6469763E}{\isasymdiv}}\ {\isaliteral{5C3C6F74696D65733E}{\isasymotimes}}\ {\isaliteral{28}{\isacharparenleft}}x\ {\isaliteral{5C3C6F74696D65733E}{\isasymotimes}}\ z{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\ \isacommand{by}\isamarkupfalse% -\ simp\isanewline -\ \ \isacommand{then}\isamarkupfalse% -\ \isacommand{have}\isamarkupfalse% -\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{28}{\isacharparenleft}}x{\isaliteral{5C3C6469763E}{\isasymdiv}}\ {\isaliteral{5C3C6F74696D65733E}{\isasymotimes}}\ x{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C6F74696D65733E}{\isasymotimes}}\ y\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{28}{\isacharparenleft}}x{\isaliteral{5C3C6469763E}{\isasymdiv}}\ {\isaliteral{5C3C6F74696D65733E}{\isasymotimes}}\ x{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C6F74696D65733E}{\isasymotimes}}\ z{\isaliteral{22}{\isachardoublequoteclose}}\ \isacommand{using}\isamarkupfalse% -\ assoc\ \isacommand{by}\isamarkupfalse% -\ simp\isanewline -\ \ \isacommand{then}\isamarkupfalse% -\ \isacommand{show}\isamarkupfalse% -\ {\isaliteral{22}{\isachardoublequoteopen}}y\ {\isaliteral{3D}{\isacharequal}}\ z{\isaliteral{22}{\isachardoublequoteclose}}\ \isacommand{using}\isamarkupfalse% -\ neutl\ \isakeyword{and}\ invl\ \isacommand{by}\isamarkupfalse% -\ simp\isanewline -\isacommand{next}\isamarkupfalse% -\isanewline -\ \ \isacommand{assume}\isamarkupfalse% -\ {\isaliteral{22}{\isachardoublequoteopen}}y\ {\isaliteral{3D}{\isacharequal}}\ z{\isaliteral{22}{\isachardoublequoteclose}}\isanewline -\ \ \isacommand{then}\isamarkupfalse% -\ \isacommand{show}\isamarkupfalse% -\ {\isaliteral{22}{\isachardoublequoteopen}}x\ {\isaliteral{5C3C6F74696D65733E}{\isasymotimes}}\ y\ {\isaliteral{3D}{\isacharequal}}\ x\ {\isaliteral{5C3C6F74696D65733E}{\isasymotimes}}\ z{\isaliteral{22}{\isachardoublequoteclose}}\ \isacommand{by}\isamarkupfalse% -\ simp\isanewline -\isacommand{qed}\isamarkupfalse% -% -\endisatagquote -{\isafoldquote}% -% -\isadelimquote -% -\endisadelimquote -% -\begin{isamarkuptext}% -\noindent Here the \qt{\hyperlink{keyword.in}{\mbox{\isa{\isakeyword{in}}}} \isa{group}} target - specification indicates that the result is recorded within that - context for later use. This local theorem is also lifted to the - global one \hyperlink{fact.group.left-cancel:}{\mbox{\isa{group{\isaliteral{2E}{\isachardot}}left{\isaliteral{5F}{\isacharunderscore}}cancel{\isaliteral{3A}{\isacharcolon}}}}} \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C416E643E}{\isasymAnd}}x\ y\ z\ {\isaliteral{5C3C436F6C6F6E3E}{\isasymColon}}\ {\isaliteral{5C3C616C7068613E}{\isasymalpha}}{\isaliteral{5C3C436F6C6F6E3E}{\isasymColon}}group{\isaliteral{2E}{\isachardot}}\ x\ {\isaliteral{5C3C6F74696D65733E}{\isasymotimes}}\ y\ {\isaliteral{3D}{\isacharequal}}\ x\ {\isaliteral{5C3C6F74696D65733E}{\isasymotimes}}\ z\ {\isaliteral{5C3C6C6F6E676C65667472696768746172726F773E}{\isasymlongleftrightarrow}}\ y\ {\isaliteral{3D}{\isacharequal}}\ z{\isaliteral{22}{\isachardoublequote}}}. Since type \isa{int} has been - made an instance of \isa{group} before, we may refer to that - fact as well: \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C416E643E}{\isasymAnd}}x\ y\ z\ {\isaliteral{5C3C436F6C6F6E3E}{\isasymColon}}\ int{\isaliteral{2E}{\isachardot}}\ x\ {\isaliteral{5C3C6F74696D65733E}{\isasymotimes}}\ y\ {\isaliteral{3D}{\isacharequal}}\ x\ {\isaliteral{5C3C6F74696D65733E}{\isasymotimes}}\ z\ {\isaliteral{5C3C6C6F6E676C65667472696768746172726F773E}{\isasymlongleftrightarrow}}\ y\ {\isaliteral{3D}{\isacharequal}}\ z{\isaliteral{22}{\isachardoublequote}}}.% -\end{isamarkuptext}% -\isamarkuptrue% -% -\isamarkupsubsection{Derived definitions% -} -\isamarkuptrue% -% -\begin{isamarkuptext}% -Isabelle locales are targets which support local definitions:% -\end{isamarkuptext}% -\isamarkuptrue% -% -\isadelimquote -% -\endisadelimquote -% -\isatagquote -\isacommand{primrec}\isamarkupfalse% -\ {\isaliteral{28}{\isacharparenleft}}\isakeyword{in}\ monoid{\isaliteral{29}{\isacharparenright}}\ pow{\isaliteral{5F}{\isacharunderscore}}nat\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}nat\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ {\isaliteral{5C3C616C7068613E}{\isasymalpha}}\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ {\isaliteral{5C3C616C7068613E}{\isasymalpha}}{\isaliteral{22}{\isachardoublequoteclose}}\ \isakeyword{where}\isanewline -\ \ {\isaliteral{22}{\isachardoublequoteopen}}pow{\isaliteral{5F}{\isacharunderscore}}nat\ {\isadigit{0}}\ x\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{5C3C6F6E653E}{\isasymone}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline -\ \ {\isaliteral{7C}{\isacharbar}}\ {\isaliteral{22}{\isachardoublequoteopen}}pow{\isaliteral{5F}{\isacharunderscore}}nat\ {\isaliteral{28}{\isacharparenleft}}Suc\ n{\isaliteral{29}{\isacharparenright}}\ x\ {\isaliteral{3D}{\isacharequal}}\ x\ {\isaliteral{5C3C6F74696D65733E}{\isasymotimes}}\ pow{\isaliteral{5F}{\isacharunderscore}}nat\ n\ x{\isaliteral{22}{\isachardoublequoteclose}}% -\endisatagquote -{\isafoldquote}% -% -\isadelimquote -% -\endisadelimquote -% -\begin{isamarkuptext}% -\noindent If the locale \isa{group} is also a class, this local - definition is propagated onto a global definition of \isa{{\isaliteral{22}{\isachardoublequote}}pow{\isaliteral{5F}{\isacharunderscore}}nat\ {\isaliteral{5C3C436F6C6F6E3E}{\isasymColon}}\ nat\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ {\isaliteral{5C3C616C7068613E}{\isasymalpha}}{\isaliteral{5C3C436F6C6F6E3E}{\isasymColon}}monoid\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ {\isaliteral{5C3C616C7068613E}{\isasymalpha}}{\isaliteral{5C3C436F6C6F6E3E}{\isasymColon}}monoid{\isaliteral{22}{\isachardoublequote}}} with corresponding theorems - - \isa{pow{\isaliteral{5F}{\isacharunderscore}}nat\ {\isadigit{0}}\ x\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{5C3C6F6E653E}{\isasymone}}\isasep\isanewline% -pow{\isaliteral{5F}{\isacharunderscore}}nat\ {\isaliteral{28}{\isacharparenleft}}Suc\ n{\isaliteral{29}{\isacharparenright}}\ x\ {\isaliteral{3D}{\isacharequal}}\ x\ {\isaliteral{5C3C6F74696D65733E}{\isasymotimes}}\ pow{\isaliteral{5F}{\isacharunderscore}}nat\ n\ x}. - - \noindent As you can see from this example, for local definitions - you may use any specification tool which works together with - locales, such as Krauss's recursive function package - \cite{krauss2006}.% -\end{isamarkuptext}% -\isamarkuptrue% -% -\isamarkupsubsection{A functor analogy% -} -\isamarkuptrue% -% -\begin{isamarkuptext}% -We introduced Isar classes by analogy to type classes in functional - programming; if we reconsider this in the context of what has been - said about type classes and locales, we can drive this analogy - further by stating that type classes essentially correspond to - functors that have a canonical interpretation as type classes. - There is also the possibility of other interpretations. For - example, \isa{list}s also form a monoid with \isa{append} and - \isa{{\isaliteral{5B}{\isacharbrackleft}}{\isaliteral{5D}{\isacharbrackright}}} as operations, but it seems inappropriate to apply to - lists the same operations as for genuinely algebraic types. In such - a case, we can simply make a particular interpretation of monoids - for lists:% -\end{isamarkuptext}% -\isamarkuptrue% -% -\isadelimquote -% -\endisadelimquote -% -\isatagquote -\isacommand{interpretation}\isamarkupfalse% -\ list{\isaliteral{5F}{\isacharunderscore}}monoid{\isaliteral{3A}{\isacharcolon}}\ monoid\ append\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{5B}{\isacharbrackleft}}{\isaliteral{5D}{\isacharbrackright}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline -\ \ \isacommand{proof}\isamarkupfalse% -\ \isacommand{qed}\isamarkupfalse% -\ auto% -\endisatagquote -{\isafoldquote}% -% -\isadelimquote -% -\endisadelimquote -% -\begin{isamarkuptext}% -\noindent This enables us to apply facts on monoids - to lists, e.g. \isa{{\isaliteral{5B}{\isacharbrackleft}}{\isaliteral{5D}{\isacharbrackright}}\ {\isaliteral{40}{\isacharat}}\ x\ {\isaliteral{3D}{\isacharequal}}\ x}. - - When using this interpretation pattern, it may also - be appropriate to map derived definitions accordingly:% -\end{isamarkuptext}% -\isamarkuptrue% -% -\isadelimquote -% -\endisadelimquote -% -\isatagquote -\isacommand{primrec}\isamarkupfalse% -\ replicate\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}nat\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ {\isaliteral{5C3C616C7068613E}{\isasymalpha}}\ list\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ {\isaliteral{5C3C616C7068613E}{\isasymalpha}}\ list{\isaliteral{22}{\isachardoublequoteclose}}\ \isakeyword{where}\isanewline -\ \ {\isaliteral{22}{\isachardoublequoteopen}}replicate\ {\isadigit{0}}\ {\isaliteral{5F}{\isacharunderscore}}\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{5B}{\isacharbrackleft}}{\isaliteral{5D}{\isacharbrackright}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline -\ \ {\isaliteral{7C}{\isacharbar}}\ {\isaliteral{22}{\isachardoublequoteopen}}replicate\ {\isaliteral{28}{\isacharparenleft}}Suc\ n{\isaliteral{29}{\isacharparenright}}\ xs\ {\isaliteral{3D}{\isacharequal}}\ xs\ {\isaliteral{40}{\isacharat}}\ replicate\ n\ xs{\isaliteral{22}{\isachardoublequoteclose}}\isanewline -\isanewline -\isacommand{interpretation}\isamarkupfalse% -\ list{\isaliteral{5F}{\isacharunderscore}}monoid{\isaliteral{3A}{\isacharcolon}}\ monoid\ append\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{5B}{\isacharbrackleft}}{\isaliteral{5D}{\isacharbrackright}}{\isaliteral{22}{\isachardoublequoteclose}}\ \isakeyword{where}\isanewline -\ \ {\isaliteral{22}{\isachardoublequoteopen}}monoid{\isaliteral{2E}{\isachardot}}pow{\isaliteral{5F}{\isacharunderscore}}nat\ append\ {\isaliteral{5B}{\isacharbrackleft}}{\isaliteral{5D}{\isacharbrackright}}\ {\isaliteral{3D}{\isacharequal}}\ replicate{\isaliteral{22}{\isachardoublequoteclose}}\isanewline -\isacommand{proof}\isamarkupfalse% -\ {\isaliteral{2D}{\isacharminus}}\isanewline -\ \ \isacommand{interpret}\isamarkupfalse% -\ monoid\ append\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{5B}{\isacharbrackleft}}{\isaliteral{5D}{\isacharbrackright}}{\isaliteral{22}{\isachardoublequoteclose}}\ \isacommand{{\isaliteral{2E}{\isachardot}}{\isaliteral{2E}{\isachardot}}}\isamarkupfalse% -\isanewline -\ \ \isacommand{show}\isamarkupfalse% -\ {\isaliteral{22}{\isachardoublequoteopen}}monoid{\isaliteral{2E}{\isachardot}}pow{\isaliteral{5F}{\isacharunderscore}}nat\ append\ {\isaliteral{5B}{\isacharbrackleft}}{\isaliteral{5D}{\isacharbrackright}}\ {\isaliteral{3D}{\isacharequal}}\ replicate{\isaliteral{22}{\isachardoublequoteclose}}\isanewline -\ \ \isacommand{proof}\isamarkupfalse% -\isanewline -\ \ \ \ \isacommand{fix}\isamarkupfalse% -\ n\isanewline -\ \ \ \ \isacommand{show}\isamarkupfalse% -\ {\isaliteral{22}{\isachardoublequoteopen}}monoid{\isaliteral{2E}{\isachardot}}pow{\isaliteral{5F}{\isacharunderscore}}nat\ append\ {\isaliteral{5B}{\isacharbrackleft}}{\isaliteral{5D}{\isacharbrackright}}\ n\ {\isaliteral{3D}{\isacharequal}}\ replicate\ n{\isaliteral{22}{\isachardoublequoteclose}}\isanewline -\ \ \ \ \ \ \isacommand{by}\isamarkupfalse% -\ {\isaliteral{28}{\isacharparenleft}}induct\ n{\isaliteral{29}{\isacharparenright}}\ auto\isanewline -\ \ \isacommand{qed}\isamarkupfalse% -\isanewline -\isacommand{qed}\isamarkupfalse% -\ intro{\isaliteral{5F}{\isacharunderscore}}locales% -\endisatagquote -{\isafoldquote}% -% -\isadelimquote -% -\endisadelimquote -% -\begin{isamarkuptext}% -\noindent This pattern is also helpful to reuse abstract - specifications on the \emph{same} type. For example, think of a - class \isa{preorder}; for type \isa{nat}, there are at least two - possible instances: the natural order or the order induced by the - divides relation. But only one of these instances can be used for - \hyperlink{command.instantiation}{\mbox{\isa{\isacommand{instantiation}}}}; using the locale behind the class \isa{preorder}, it is still possible to utilise the same abstract - specification again using \hyperlink{command.interpretation}{\mbox{\isa{\isacommand{interpretation}}}}.% -\end{isamarkuptext}% -\isamarkuptrue% -% -\isamarkupsubsection{Additional subclass relations% -} -\isamarkuptrue% -% -\begin{isamarkuptext}% -Any \isa{group} is also a \isa{monoid}; this can be made - explicit by claiming an additional subclass relation, together with - a proof of the logical difference:% -\end{isamarkuptext}% -\isamarkuptrue% -% -\isadelimquote -% -\endisadelimquote -% -\isatagquote -\isacommand{subclass}\isamarkupfalse% -\ {\isaliteral{28}{\isacharparenleft}}\isakeyword{in}\ group{\isaliteral{29}{\isacharparenright}}\ monoid\isanewline -\isacommand{proof}\isamarkupfalse% -\isanewline -\ \ \isacommand{fix}\isamarkupfalse% -\ x\isanewline -\ \ \isacommand{from}\isamarkupfalse% -\ invl\ \isacommand{have}\isamarkupfalse% -\ {\isaliteral{22}{\isachardoublequoteopen}}x{\isaliteral{5C3C6469763E}{\isasymdiv}}\ {\isaliteral{5C3C6F74696D65733E}{\isasymotimes}}\ x\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{5C3C6F6E653E}{\isasymone}}{\isaliteral{22}{\isachardoublequoteclose}}\ \isacommand{by}\isamarkupfalse% -\ simp\isanewline -\ \ \isacommand{with}\isamarkupfalse% -\ assoc\ {\isaliteral{5B}{\isacharbrackleft}}symmetric{\isaliteral{5D}{\isacharbrackright}}\ neutl\ invl\ \isacommand{have}\isamarkupfalse% -\ {\isaliteral{22}{\isachardoublequoteopen}}x{\isaliteral{5C3C6469763E}{\isasymdiv}}\ {\isaliteral{5C3C6F74696D65733E}{\isasymotimes}}\ {\isaliteral{28}{\isacharparenleft}}x\ {\isaliteral{5C3C6F74696D65733E}{\isasymotimes}}\ {\isaliteral{5C3C6F6E653E}{\isasymone}}{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ x{\isaliteral{5C3C6469763E}{\isasymdiv}}\ {\isaliteral{5C3C6F74696D65733E}{\isasymotimes}}\ x{\isaliteral{22}{\isachardoublequoteclose}}\ \isacommand{by}\isamarkupfalse% -\ simp\isanewline -\ \ \isacommand{with}\isamarkupfalse% -\ left{\isaliteral{5F}{\isacharunderscore}}cancel\ \isacommand{show}\isamarkupfalse% -\ {\isaliteral{22}{\isachardoublequoteopen}}x\ {\isaliteral{5C3C6F74696D65733E}{\isasymotimes}}\ {\isaliteral{5C3C6F6E653E}{\isasymone}}\ {\isaliteral{3D}{\isacharequal}}\ x{\isaliteral{22}{\isachardoublequoteclose}}\ \isacommand{by}\isamarkupfalse% -\ simp\isanewline -\isacommand{qed}\isamarkupfalse% -% -\endisatagquote -{\isafoldquote}% -% -\isadelimquote -% -\endisadelimquote -% -\begin{isamarkuptext}% -The logical proof is carried out on the locale level. Afterwards it - is propagated to the type system, making \isa{group} an instance - of \isa{monoid} by adding an additional edge to the graph of - subclass relations (\figref{fig:subclass}). - - \begin{figure}[htbp] - \begin{center} - \small - \unitlength 0.6mm - \begin{picture}(40,60)(0,0) - \put(20,60){\makebox(0,0){\isa{semigroup}}} - \put(20,40){\makebox(0,0){\isa{monoidl}}} - \put(00,20){\makebox(0,0){\isa{monoid}}} - \put(40,00){\makebox(0,0){\isa{group}}} - \put(20,55){\vector(0,-1){10}} - \put(15,35){\vector(-1,-1){10}} - \put(25,35){\vector(1,-3){10}} - \end{picture} - \hspace{8em} - \begin{picture}(40,60)(0,0) - \put(20,60){\makebox(0,0){\isa{semigroup}}} - \put(20,40){\makebox(0,0){\isa{monoidl}}} - \put(00,20){\makebox(0,0){\isa{monoid}}} - \put(40,00){\makebox(0,0){\isa{group}}} - \put(20,55){\vector(0,-1){10}} - \put(15,35){\vector(-1,-1){10}} - \put(05,15){\vector(3,-1){30}} - \end{picture} - \caption{Subclass relationship of monoids and groups: - before and after establishing the relationship - \isa{group\ {\isaliteral{5C3C73756273657465713E}{\isasymsubseteq}}\ monoid}; transitive edges are left out.} - \label{fig:subclass} - \end{center} - \end{figure} - - For illustration, a derived definition in \isa{group} using \isa{pow{\isaliteral{5F}{\isacharunderscore}}nat}% -\end{isamarkuptext}% -\isamarkuptrue% -% -\isadelimquote -% -\endisadelimquote -% -\isatagquote -\isacommand{definition}\isamarkupfalse% -\ {\isaliteral{28}{\isacharparenleft}}\isakeyword{in}\ group{\isaliteral{29}{\isacharparenright}}\ pow{\isaliteral{5F}{\isacharunderscore}}int\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}int\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ {\isaliteral{5C3C616C7068613E}{\isasymalpha}}\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ {\isaliteral{5C3C616C7068613E}{\isasymalpha}}{\isaliteral{22}{\isachardoublequoteclose}}\ \isakeyword{where}\isanewline -\ \ {\isaliteral{22}{\isachardoublequoteopen}}pow{\isaliteral{5F}{\isacharunderscore}}int\ k\ x\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{28}{\isacharparenleft}}if\ k\ {\isaliteral{3E}{\isachargreater}}{\isaliteral{3D}{\isacharequal}}\ {\isadigit{0}}\isanewline -\ \ \ \ then\ pow{\isaliteral{5F}{\isacharunderscore}}nat\ {\isaliteral{28}{\isacharparenleft}}nat\ k{\isaliteral{29}{\isacharparenright}}\ x\isanewline -\ \ \ \ else\ {\isaliteral{28}{\isacharparenleft}}pow{\isaliteral{5F}{\isacharunderscore}}nat\ {\isaliteral{28}{\isacharparenleft}}nat\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{2D}{\isacharminus}}\ k{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}\ x{\isaliteral{29}{\isacharparenright}}{\isaliteral{5C3C6469763E}{\isasymdiv}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}% -\endisatagquote -{\isafoldquote}% -% -\isadelimquote -% -\endisadelimquote -% -\begin{isamarkuptext}% -\noindent yields the global definition of \isa{{\isaliteral{22}{\isachardoublequote}}pow{\isaliteral{5F}{\isacharunderscore}}int\ {\isaliteral{5C3C436F6C6F6E3E}{\isasymColon}}\ int\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ {\isaliteral{5C3C616C7068613E}{\isasymalpha}}{\isaliteral{5C3C436F6C6F6E3E}{\isasymColon}}group\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ {\isaliteral{5C3C616C7068613E}{\isasymalpha}}{\isaliteral{5C3C436F6C6F6E3E}{\isasymColon}}group{\isaliteral{22}{\isachardoublequote}}} with the corresponding theorem \isa{pow{\isaliteral{5F}{\isacharunderscore}}int\ k\ x\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{28}{\isacharparenleft}}if\ {\isadigit{0}}\ {\isaliteral{5C3C6C653E}{\isasymle}}\ k\ then\ pow{\isaliteral{5F}{\isacharunderscore}}nat\ {\isaliteral{28}{\isacharparenleft}}nat\ k{\isaliteral{29}{\isacharparenright}}\ x\ else\ {\isaliteral{28}{\isacharparenleft}}pow{\isaliteral{5F}{\isacharunderscore}}nat\ {\isaliteral{28}{\isacharparenleft}}nat\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{2D}{\isacharminus}}\ k{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}\ x{\isaliteral{29}{\isacharparenright}}{\isaliteral{5C3C6469763E}{\isasymdiv}}{\isaliteral{29}{\isacharparenright}}}.% -\end{isamarkuptext}% -\isamarkuptrue% -% -\isamarkupsubsection{A note on syntax% -} -\isamarkuptrue% -% -\begin{isamarkuptext}% -As a convenience, class context syntax allows references to local - class operations and their global counterparts uniformly; type - inference resolves ambiguities. For example:% -\end{isamarkuptext}% -\isamarkuptrue% -% -\isadelimquote -% -\endisadelimquote -% -\isatagquote -\isacommand{context}\isamarkupfalse% -\ semigroup\isanewline -\isakeyword{begin}\isanewline -\isanewline -\isacommand{term}\isamarkupfalse% -\ {\isaliteral{22}{\isachardoublequoteopen}}x\ {\isaliteral{5C3C6F74696D65733E}{\isasymotimes}}\ y{\isaliteral{22}{\isachardoublequoteclose}}\ % -\isamarkupcmt{example 1% -} -\isanewline -\isacommand{term}\isamarkupfalse% -\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{28}{\isacharparenleft}}x{\isaliteral{5C3C436F6C6F6E3E}{\isasymColon}}nat{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C6F74696D65733E}{\isasymotimes}}\ y{\isaliteral{22}{\isachardoublequoteclose}}\ % -\isamarkupcmt{example 2% -} -\isanewline -\isanewline -\isacommand{end}\isamarkupfalse% -\isanewline -\isanewline -\isacommand{term}\isamarkupfalse% -\ {\isaliteral{22}{\isachardoublequoteopen}}x\ {\isaliteral{5C3C6F74696D65733E}{\isasymotimes}}\ y{\isaliteral{22}{\isachardoublequoteclose}}\ % -\isamarkupcmt{example 3% -} -% -\endisatagquote -{\isafoldquote}% -% -\isadelimquote -% -\endisadelimquote -% -\begin{isamarkuptext}% -\noindent Here in example 1, the term refers to the local class - operation \isa{mult\ {\isaliteral{5B}{\isacharbrackleft}}{\isaliteral{5C3C616C7068613E}{\isasymalpha}}{\isaliteral{5D}{\isacharbrackright}}}, whereas in example 2 the type - constraint enforces the global class operation \isa{mult\ {\isaliteral{5B}{\isacharbrackleft}}nat{\isaliteral{5D}{\isacharbrackright}}}. - In the global context in example 3, the reference is to the - polymorphic global class operation \isa{mult\ {\isaliteral{5B}{\isacharbrackleft}}{\isaliteral{3F}{\isacharquery}}{\isaliteral{5C3C616C7068613E}{\isasymalpha}}\ {\isaliteral{5C3C436F6C6F6E3E}{\isasymColon}}\ semigroup{\isaliteral{5D}{\isacharbrackright}}}.% -\end{isamarkuptext}% -\isamarkuptrue% -% -\isamarkupsection{Further issues% -} -\isamarkuptrue% -% -\isamarkupsubsection{Type classes and code generation% -} -\isamarkuptrue% -% -\begin{isamarkuptext}% -Turning back to the first motivation for type classes, namely - overloading, it is obvious that overloading stemming from \hyperlink{command.class}{\mbox{\isa{\isacommand{class}}}} statements and \hyperlink{command.instantiation}{\mbox{\isa{\isacommand{instantiation}}}} targets naturally - maps to Haskell type classes. The code generator framework - \cite{isabelle-codegen} takes this into account. If the target - language (e.g.~SML) lacks type classes, then they are implemented by - an explicit dictionary construction. As example, let's go back to - the power function:% -\end{isamarkuptext}% -\isamarkuptrue% -% -\isadelimquote -% -\endisadelimquote -% -\isatagquote -\isacommand{definition}\isamarkupfalse% -\ example\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ int\ \isakeyword{where}\isanewline -\ \ {\isaliteral{22}{\isachardoublequoteopen}}example\ {\isaliteral{3D}{\isacharequal}}\ pow{\isaliteral{5F}{\isacharunderscore}}int\ {\isadigit{1}}{\isadigit{0}}\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{2D}{\isacharminus}}{\isadigit{2}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}% -\endisatagquote -{\isafoldquote}% -% -\isadelimquote -% -\endisadelimquote -% -\begin{isamarkuptext}% -\noindent This maps to Haskell as follows:% -\end{isamarkuptext}% -\isamarkuptrue% -% -\isadeliminvisible -% -\endisadeliminvisible -% -\isataginvisible -% -\endisataginvisible -{\isafoldinvisible}% -% -\isadeliminvisible -% -\endisadeliminvisible -% -\isadelimquotetypewriter -% -\endisadelimquotetypewriter -% -\isatagquotetypewriter -% -\begin{isamarkuptext}% -module\ Example\ where\ {\isaliteral{7B}{\isacharbraceleft}}\isanewline -\isanewline -import\ Prelude\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{28}{\isacharparenleft}}{\isaliteral{3D}{\isacharequal}}{\isaliteral{3D}{\isacharequal}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{2C}{\isacharcomma}}\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{2F}{\isacharslash}}{\isaliteral{3D}{\isacharequal}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{2C}{\isacharcomma}}\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{3C}{\isacharless}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{2C}{\isacharcomma}}\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{3C}{\isacharless}}{\isaliteral{3D}{\isacharequal}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{2C}{\isacharcomma}}\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{3E}{\isachargreater}}{\isaliteral{3D}{\isacharequal}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{2C}{\isacharcomma}}\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{3E}{\isachargreater}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{2C}{\isacharcomma}}\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{2B}{\isacharplus}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{2C}{\isacharcomma}}\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{2D}{\isacharminus}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{2C}{\isacharcomma}}\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{2A}{\isacharasterisk}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{2C}{\isacharcomma}}\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{2F}{\isacharslash}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{2C}{\isacharcomma}}\isanewline -\ \ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{2A}{\isacharasterisk}}{\isaliteral{2A}{\isacharasterisk}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{2C}{\isacharcomma}}\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{3E}{\isachargreater}}{\isaliteral{3E}{\isachargreater}}{\isaliteral{3D}{\isacharequal}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{2C}{\isacharcomma}}\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{3E}{\isachargreater}}{\isaliteral{3E}{\isachargreater}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{2C}{\isacharcomma}}\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{3D}{\isacharequal}}{\isaliteral{3C}{\isacharless}}{\isaliteral{3C}{\isacharless}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{2C}{\isacharcomma}}\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{26}{\isacharampersand}}{\isaliteral{26}{\isacharampersand}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{2C}{\isacharcomma}}\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{7C}{\isacharbar}}{\isaliteral{7C}{\isacharbar}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{2C}{\isacharcomma}}\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{5E}{\isacharcircum}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{2C}{\isacharcomma}}\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{5E}{\isacharcircum}}{\isaliteral{5E}{\isacharcircum}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{2C}{\isacharcomma}}\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{2E}{\isachardot}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{2C}{\isacharcomma}}\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{24}{\isachardollar}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{2C}{\isacharcomma}}\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{24}{\isachardollar}}{\isaliteral{21}{\isacharbang}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{2C}{\isacharcomma}}\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{2B}{\isacharplus}}{\isaliteral{2B}{\isacharplus}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{2C}{\isacharcomma}}\isanewline -\ \ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{21}{\isacharbang}}{\isaliteral{21}{\isacharbang}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{2C}{\isacharcomma}}\ Eq{\isaliteral{2C}{\isacharcomma}}\ error{\isaliteral{2C}{\isacharcomma}}\ id{\isaliteral{2C}{\isacharcomma}}\ return{\isaliteral{2C}{\isacharcomma}}\ not{\isaliteral{2C}{\isacharcomma}}\ fst{\isaliteral{2C}{\isacharcomma}}\ snd{\isaliteral{2C}{\isacharcomma}}\ map{\isaliteral{2C}{\isacharcomma}}\ filter{\isaliteral{2C}{\isacharcomma}}\ concat{\isaliteral{2C}{\isacharcomma}}\isanewline -\ \ concatMap{\isaliteral{2C}{\isacharcomma}}\ reverse{\isaliteral{2C}{\isacharcomma}}\ zip{\isaliteral{2C}{\isacharcomma}}\ null{\isaliteral{2C}{\isacharcomma}}\ takeWhile{\isaliteral{2C}{\isacharcomma}}\ dropWhile{\isaliteral{2C}{\isacharcomma}}\ all{\isaliteral{2C}{\isacharcomma}}\ any{\isaliteral{2C}{\isacharcomma}}\ Integer{\isaliteral{2C}{\isacharcomma}}\isanewline -\ \ negate{\isaliteral{2C}{\isacharcomma}}\ abs{\isaliteral{2C}{\isacharcomma}}\ divMod{\isaliteral{2C}{\isacharcomma}}\ String{\isaliteral{2C}{\isacharcomma}}\ Bool{\isaliteral{28}{\isacharparenleft}}True{\isaliteral{2C}{\isacharcomma}}\ False{\isaliteral{29}{\isacharparenright}}{\isaliteral{2C}{\isacharcomma}}\ Maybe{\isaliteral{28}{\isacharparenleft}}Nothing{\isaliteral{2C}{\isacharcomma}}\ Just{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{3B}{\isacharsemicolon}}\isanewline -import\ qualified\ Prelude{\isaliteral{3B}{\isacharsemicolon}}\isanewline -\isanewline -data\ Nat\ {\isaliteral{3D}{\isacharequal}}\ Zero{\isaliteral{5F}{\isacharunderscore}}nat\ {\isaliteral{7C}{\isacharbar}}\ Suc\ Nat{\isaliteral{3B}{\isacharsemicolon}}\isanewline -\isanewline -data\ Num\ {\isaliteral{3D}{\isacharequal}}\ One\ {\isaliteral{7C}{\isacharbar}}\ Bit{\isadigit{0}}\ Num\ {\isaliteral{7C}{\isacharbar}}\ Bit{\isadigit{1}}\ Num{\isaliteral{3B}{\isacharsemicolon}}\isanewline -\isanewline -apsnd\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ forall\ a\ b\ c{\isaliteral{2E}{\isachardot}}\ {\isaliteral{28}{\isacharparenleft}}a\ {\isaliteral{2D}{\isacharminus}}{\isaliteral{3E}{\isachargreater}}\ b{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{2D}{\isacharminus}}{\isaliteral{3E}{\isachargreater}}\ {\isaliteral{28}{\isacharparenleft}}c{\isaliteral{2C}{\isacharcomma}}\ a{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{2D}{\isacharminus}}{\isaliteral{3E}{\isachargreater}}\ {\isaliteral{28}{\isacharparenleft}}c{\isaliteral{2C}{\isacharcomma}}\ b{\isaliteral{29}{\isacharparenright}}{\isaliteral{3B}{\isacharsemicolon}}\isanewline -apsnd\ f\ {\isaliteral{28}{\isacharparenleft}}x{\isaliteral{2C}{\isacharcomma}}\ y{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{28}{\isacharparenleft}}x{\isaliteral{2C}{\isacharcomma}}\ f\ y{\isaliteral{29}{\isacharparenright}}{\isaliteral{3B}{\isacharsemicolon}}\isanewline -\isanewline -sgn{\isaliteral{5F}{\isacharunderscore}}int\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ Integer\ {\isaliteral{2D}{\isacharminus}}{\isaliteral{3E}{\isachargreater}}\ Integer{\isaliteral{3B}{\isacharsemicolon}}\isanewline -sgn{\isaliteral{5F}{\isacharunderscore}}int\ i\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{28}{\isacharparenleft}}if\ i\ {\isaliteral{3D}{\isacharequal}}{\isaliteral{3D}{\isacharequal}}\ {\isadigit{0}}\ then\ {\isadigit{0}}\ else\ {\isaliteral{28}{\isacharparenleft}}if\ {\isadigit{0}}\ {\isaliteral{3C}{\isacharless}}\ i\ then\ {\isadigit{1}}\ else\ negate\ {\isadigit{1}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{3B}{\isacharsemicolon}}\isanewline -\isanewline -abs{\isaliteral{5F}{\isacharunderscore}}int\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ Integer\ {\isaliteral{2D}{\isacharminus}}{\isaliteral{3E}{\isachargreater}}\ Integer{\isaliteral{3B}{\isacharsemicolon}}\isanewline -abs{\isaliteral{5F}{\isacharunderscore}}int\ i\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{28}{\isacharparenleft}}if\ i\ {\isaliteral{3C}{\isacharless}}\ {\isadigit{0}}\ then\ negate\ i\ else\ i{\isaliteral{29}{\isacharparenright}}{\isaliteral{3B}{\isacharsemicolon}}\isanewline -\isanewline -divmod{\isaliteral{5F}{\isacharunderscore}}int\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ Integer\ {\isaliteral{2D}{\isacharminus}}{\isaliteral{3E}{\isachargreater}}\ Integer\ {\isaliteral{2D}{\isacharminus}}{\isaliteral{3E}{\isachargreater}}\ {\isaliteral{28}{\isacharparenleft}}Integer{\isaliteral{2C}{\isacharcomma}}\ Integer{\isaliteral{29}{\isacharparenright}}{\isaliteral{3B}{\isacharsemicolon}}\isanewline -divmod{\isaliteral{5F}{\isacharunderscore}}int\ k\ l\ {\isaliteral{3D}{\isacharequal}}\isanewline -\ \ {\isaliteral{28}{\isacharparenleft}}if\ k\ {\isaliteral{3D}{\isacharequal}}{\isaliteral{3D}{\isacharequal}}\ {\isadigit{0}}\ then\ {\isaliteral{28}{\isacharparenleft}}{\isadigit{0}}{\isaliteral{2C}{\isacharcomma}}\ {\isadigit{0}}{\isaliteral{29}{\isacharparenright}}\isanewline -\ \ \ \ else\ {\isaliteral{28}{\isacharparenleft}}if\ l\ {\isaliteral{3D}{\isacharequal}}{\isaliteral{3D}{\isacharequal}}\ {\isadigit{0}}\ then\ {\isaliteral{28}{\isacharparenleft}}{\isadigit{0}}{\isaliteral{2C}{\isacharcomma}}\ k{\isaliteral{29}{\isacharparenright}}\isanewline -\ \ \ \ \ \ \ \ \ \ \ else\ apsnd\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C}{\isacharbackslash}}\ a\ {\isaliteral{2D}{\isacharminus}}{\isaliteral{3E}{\isachargreater}}\ sgn{\isaliteral{5F}{\isacharunderscore}}int\ l\ {\isaliteral{2A}{\isacharasterisk}}\ a{\isaliteral{29}{\isacharparenright}}\isanewline -\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isaliteral{28}{\isacharparenleft}}if\ sgn{\isaliteral{5F}{\isacharunderscore}}int\ k\ {\isaliteral{3D}{\isacharequal}}{\isaliteral{3D}{\isacharequal}}\ sgn{\isaliteral{5F}{\isacharunderscore}}int\ l\ then\ divMod\ {\isaliteral{28}{\isacharparenleft}}abs\ k{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{28}{\isacharparenleft}}abs\ l{\isaliteral{29}{\isacharparenright}}\isanewline -\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ else\ let\ {\isaliteral{7B}{\isacharbraceleft}}\isanewline -\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isaliteral{28}{\isacharparenleft}}r{\isaliteral{2C}{\isacharcomma}}\ s{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ divMod\ {\isaliteral{28}{\isacharparenleft}}abs\ k{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{28}{\isacharparenleft}}abs\ l{\isaliteral{29}{\isacharparenright}}{\isaliteral{3B}{\isacharsemicolon}}\isanewline -\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isaliteral{7D}{\isacharbraceright}}\ in\ {\isaliteral{28}{\isacharparenleft}}if\ s\ {\isaliteral{3D}{\isacharequal}}{\isaliteral{3D}{\isacharequal}}\ {\isadigit{0}}\ then\ {\isaliteral{28}{\isacharparenleft}}negate\ r{\isaliteral{2C}{\isacharcomma}}\ {\isadigit{0}}{\isaliteral{29}{\isacharparenright}}\isanewline -\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ else\ {\isaliteral{28}{\isacharparenleft}}negate\ r\ {\isaliteral{2D}{\isacharminus}}\ {\isadigit{1}}{\isaliteral{2C}{\isacharcomma}}\ abs{\isaliteral{5F}{\isacharunderscore}}int\ l\ {\isaliteral{2D}{\isacharminus}}\ s{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{3B}{\isacharsemicolon}}\isanewline -\isanewline -plus{\isaliteral{5F}{\isacharunderscore}}nat\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ Nat\ {\isaliteral{2D}{\isacharminus}}{\isaliteral{3E}{\isachargreater}}\ Nat\ {\isaliteral{2D}{\isacharminus}}{\isaliteral{3E}{\isachargreater}}\ Nat{\isaliteral{3B}{\isacharsemicolon}}\isanewline -plus{\isaliteral{5F}{\isacharunderscore}}nat\ {\isaliteral{28}{\isacharparenleft}}Suc\ m{\isaliteral{29}{\isacharparenright}}\ n\ {\isaliteral{3D}{\isacharequal}}\ plus{\isaliteral{5F}{\isacharunderscore}}nat\ m\ {\isaliteral{28}{\isacharparenleft}}Suc\ n{\isaliteral{29}{\isacharparenright}}{\isaliteral{3B}{\isacharsemicolon}}\isanewline -plus{\isaliteral{5F}{\isacharunderscore}}nat\ Zero{\isaliteral{5F}{\isacharunderscore}}nat\ n\ {\isaliteral{3D}{\isacharequal}}\ n{\isaliteral{3B}{\isacharsemicolon}}\isanewline -\isanewline -nat\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ Integer\ {\isaliteral{2D}{\isacharminus}}{\isaliteral{3E}{\isachargreater}}\ Nat{\isaliteral{3B}{\isacharsemicolon}}\isanewline -nat\ k\ {\isaliteral{3D}{\isacharequal}}\isanewline -\ \ {\isaliteral{28}{\isacharparenleft}}if\ k\ {\isaliteral{3C}{\isacharless}}{\isaliteral{3D}{\isacharequal}}\ {\isadigit{0}}\ then\ Zero{\isaliteral{5F}{\isacharunderscore}}nat\isanewline -\ \ \ \ else\ let\ {\isaliteral{7B}{\isacharbraceleft}}\isanewline -\ \ \ \ \ \ \ \ \ \ \ {\isaliteral{28}{\isacharparenleft}}l{\isaliteral{2C}{\isacharcomma}}\ j{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ divmod{\isaliteral{5F}{\isacharunderscore}}int\ k\ {\isadigit{2}}{\isaliteral{3B}{\isacharsemicolon}}\isanewline -\ \ \ \ \ \ \ \ \ \ \ n\ {\isaliteral{3D}{\isacharequal}}\ nat\ l{\isaliteral{3B}{\isacharsemicolon}}\isanewline -\ \ \ \ \ \ \ \ \ \ \ la\ {\isaliteral{3D}{\isacharequal}}\ plus{\isaliteral{5F}{\isacharunderscore}}nat\ n\ n{\isaliteral{3B}{\isacharsemicolon}}\isanewline -\ \ \ \ \ \ \ \ \ {\isaliteral{7D}{\isacharbraceright}}\ in\ {\isaliteral{28}{\isacharparenleft}}if\ j\ {\isaliteral{3D}{\isacharequal}}{\isaliteral{3D}{\isacharequal}}\ {\isadigit{0}}\ then\ la\ else\ Suc\ la{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{3B}{\isacharsemicolon}}\isanewline -\isanewline -class\ Semigroup\ a\ where\ {\isaliteral{7B}{\isacharbraceleft}}\isanewline -\ \ mult\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ a\ {\isaliteral{2D}{\isacharminus}}{\isaliteral{3E}{\isachargreater}}\ a\ {\isaliteral{2D}{\isacharminus}}{\isaliteral{3E}{\isachargreater}}\ a{\isaliteral{3B}{\isacharsemicolon}}\isanewline -{\isaliteral{7D}{\isacharbraceright}}{\isaliteral{3B}{\isacharsemicolon}}\isanewline -\isanewline -class\ {\isaliteral{28}{\isacharparenleft}}Semigroup\ a{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}{\isaliteral{3E}{\isachargreater}}\ Monoidl\ a\ where\ {\isaliteral{7B}{\isacharbraceleft}}\isanewline -\ \ neutral\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ a{\isaliteral{3B}{\isacharsemicolon}}\isanewline -{\isaliteral{7D}{\isacharbraceright}}{\isaliteral{3B}{\isacharsemicolon}}\isanewline -\isanewline -class\ {\isaliteral{28}{\isacharparenleft}}Monoidl\ a{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}{\isaliteral{3E}{\isachargreater}}\ Monoid\ a\ where\ {\isaliteral{7B}{\isacharbraceleft}}\isanewline -{\isaliteral{7D}{\isacharbraceright}}{\isaliteral{3B}{\isacharsemicolon}}\isanewline -\isanewline -class\ {\isaliteral{28}{\isacharparenleft}}Monoid\ a{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}{\isaliteral{3E}{\isachargreater}}\ Group\ a\ where\ {\isaliteral{7B}{\isacharbraceleft}}\isanewline -\ \ inverse\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ a\ {\isaliteral{2D}{\isacharminus}}{\isaliteral{3E}{\isachargreater}}\ a{\isaliteral{3B}{\isacharsemicolon}}\isanewline -{\isaliteral{7D}{\isacharbraceright}}{\isaliteral{3B}{\isacharsemicolon}}\isanewline -\isanewline -neutral{\isaliteral{5F}{\isacharunderscore}}int\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ Integer{\isaliteral{3B}{\isacharsemicolon}}\isanewline -neutral{\isaliteral{5F}{\isacharunderscore}}int\ {\isaliteral{3D}{\isacharequal}}\ {\isadigit{0}}{\isaliteral{3B}{\isacharsemicolon}}\isanewline -\isanewline -inverse{\isaliteral{5F}{\isacharunderscore}}int\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ Integer\ {\isaliteral{2D}{\isacharminus}}{\isaliteral{3E}{\isachargreater}}\ Integer{\isaliteral{3B}{\isacharsemicolon}}\isanewline -inverse{\isaliteral{5F}{\isacharunderscore}}int\ i\ {\isaliteral{3D}{\isacharequal}}\ negate\ i{\isaliteral{3B}{\isacharsemicolon}}\isanewline -\isanewline -mult{\isaliteral{5F}{\isacharunderscore}}int\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ Integer\ {\isaliteral{2D}{\isacharminus}}{\isaliteral{3E}{\isachargreater}}\ Integer\ {\isaliteral{2D}{\isacharminus}}{\isaliteral{3E}{\isachargreater}}\ Integer{\isaliteral{3B}{\isacharsemicolon}}\isanewline -mult{\isaliteral{5F}{\isacharunderscore}}int\ i\ j\ {\isaliteral{3D}{\isacharequal}}\ i\ {\isaliteral{2B}{\isacharplus}}\ j{\isaliteral{3B}{\isacharsemicolon}}\isanewline -\isanewline -instance\ Semigroup\ Integer\ where\ {\isaliteral{7B}{\isacharbraceleft}}\isanewline -\ \ mult\ {\isaliteral{3D}{\isacharequal}}\ mult{\isaliteral{5F}{\isacharunderscore}}int{\isaliteral{3B}{\isacharsemicolon}}\isanewline -{\isaliteral{7D}{\isacharbraceright}}{\isaliteral{3B}{\isacharsemicolon}}\isanewline -\isanewline -instance\ Monoidl\ Integer\ where\ {\isaliteral{7B}{\isacharbraceleft}}\isanewline -\ \ neutral\ {\isaliteral{3D}{\isacharequal}}\ neutral{\isaliteral{5F}{\isacharunderscore}}int{\isaliteral{3B}{\isacharsemicolon}}\isanewline -{\isaliteral{7D}{\isacharbraceright}}{\isaliteral{3B}{\isacharsemicolon}}\isanewline -\isanewline -instance\ Monoid\ Integer\ where\ {\isaliteral{7B}{\isacharbraceleft}}\isanewline -{\isaliteral{7D}{\isacharbraceright}}{\isaliteral{3B}{\isacharsemicolon}}\isanewline -\isanewline -instance\ Group\ Integer\ where\ {\isaliteral{7B}{\isacharbraceleft}}\isanewline -\ \ inverse\ {\isaliteral{3D}{\isacharequal}}\ inverse{\isaliteral{5F}{\isacharunderscore}}int{\isaliteral{3B}{\isacharsemicolon}}\isanewline -{\isaliteral{7D}{\isacharbraceright}}{\isaliteral{3B}{\isacharsemicolon}}\isanewline -\isanewline -pow{\isaliteral{5F}{\isacharunderscore}}nat\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ forall\ a{\isaliteral{2E}{\isachardot}}\ {\isaliteral{28}{\isacharparenleft}}Monoid\ a{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}{\isaliteral{3E}{\isachargreater}}\ Nat\ {\isaliteral{2D}{\isacharminus}}{\isaliteral{3E}{\isachargreater}}\ a\ {\isaliteral{2D}{\isacharminus}}{\isaliteral{3E}{\isachargreater}}\ a{\isaliteral{3B}{\isacharsemicolon}}\isanewline -pow{\isaliteral{5F}{\isacharunderscore}}nat\ Zero{\isaliteral{5F}{\isacharunderscore}}nat\ x\ {\isaliteral{3D}{\isacharequal}}\ neutral{\isaliteral{3B}{\isacharsemicolon}}\isanewline -pow{\isaliteral{5F}{\isacharunderscore}}nat\ {\isaliteral{28}{\isacharparenleft}}Suc\ n{\isaliteral{29}{\isacharparenright}}\ x\ {\isaliteral{3D}{\isacharequal}}\ mult\ x\ {\isaliteral{28}{\isacharparenleft}}pow{\isaliteral{5F}{\isacharunderscore}}nat\ n\ x{\isaliteral{29}{\isacharparenright}}{\isaliteral{3B}{\isacharsemicolon}}\isanewline -\isanewline -pow{\isaliteral{5F}{\isacharunderscore}}int\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ forall\ a{\isaliteral{2E}{\isachardot}}\ {\isaliteral{28}{\isacharparenleft}}Group\ a{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}{\isaliteral{3E}{\isachargreater}}\ Integer\ {\isaliteral{2D}{\isacharminus}}{\isaliteral{3E}{\isachargreater}}\ a\ {\isaliteral{2D}{\isacharminus}}{\isaliteral{3E}{\isachargreater}}\ a{\isaliteral{3B}{\isacharsemicolon}}\isanewline -pow{\isaliteral{5F}{\isacharunderscore}}int\ k\ x\ {\isaliteral{3D}{\isacharequal}}\isanewline -\ \ {\isaliteral{28}{\isacharparenleft}}if\ {\isadigit{0}}\ {\isaliteral{3C}{\isacharless}}{\isaliteral{3D}{\isacharequal}}\ k\ then\ pow{\isaliteral{5F}{\isacharunderscore}}nat\ {\isaliteral{28}{\isacharparenleft}}nat\ k{\isaliteral{29}{\isacharparenright}}\ x\isanewline -\ \ \ \ else\ inverse\ {\isaliteral{28}{\isacharparenleft}}pow{\isaliteral{5F}{\isacharunderscore}}nat\ {\isaliteral{28}{\isacharparenleft}}nat\ {\isaliteral{28}{\isacharparenleft}}negate\ k{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}\ x{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{3B}{\isacharsemicolon}}\isanewline -\isanewline -example\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ Integer{\isaliteral{3B}{\isacharsemicolon}}\isanewline -example\ {\isaliteral{3D}{\isacharequal}}\ pow{\isaliteral{5F}{\isacharunderscore}}int\ {\isadigit{1}}{\isadigit{0}}\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{2D}{\isacharminus}}{\isadigit{2}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{3B}{\isacharsemicolon}}\isanewline -\isanewline -{\isaliteral{7D}{\isacharbraceright}}\isanewline% -\end{isamarkuptext}% -\isamarkuptrue% -% -\endisatagquotetypewriter -{\isafoldquotetypewriter}% -% -\isadelimquotetypewriter -% -\endisadelimquotetypewriter -% -\begin{isamarkuptext}% -\noindent The code in SML has explicit dictionary passing:% -\end{isamarkuptext}% -\isamarkuptrue% -% -\isadelimquotetypewriter -% -\endisadelimquotetypewriter -% -\isatagquotetypewriter -% -\begin{isamarkuptext}% -structure\ Example\ {\isaliteral{3A}{\isacharcolon}}\ sig\isanewline -\ \ datatype\ nat\ {\isaliteral{3D}{\isacharequal}}\ Zero{\isaliteral{5F}{\isacharunderscore}}nat\ {\isaliteral{7C}{\isacharbar}}\ Suc\ of\ nat\isanewline -\ \ datatype\ num\ {\isaliteral{3D}{\isacharequal}}\ One\ {\isaliteral{7C}{\isacharbar}}\ Bit{\isadigit{0}}\ of\ num\ {\isaliteral{7C}{\isacharbar}}\ Bit{\isadigit{1}}\ of\ num\isanewline -\ \ val\ apsnd\ {\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{27}{\isacharprime}}a\ {\isaliteral{2D}{\isacharminus}}{\isaliteral{3E}{\isachargreater}}\ {\isaliteral{27}{\isacharprime}}b{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{2D}{\isacharminus}}{\isaliteral{3E}{\isachargreater}}\ {\isaliteral{27}{\isacharprime}}c\ {\isaliteral{2A}{\isacharasterisk}}\ {\isaliteral{27}{\isacharprime}}a\ {\isaliteral{2D}{\isacharminus}}{\isaliteral{3E}{\isachargreater}}\ {\isaliteral{27}{\isacharprime}}c\ {\isaliteral{2A}{\isacharasterisk}}\ {\isaliteral{27}{\isacharprime}}b\isanewline -\ \ val\ sgn{\isaliteral{5F}{\isacharunderscore}}int\ {\isaliteral{3A}{\isacharcolon}}\ IntInf{\isaliteral{2E}{\isachardot}}int\ {\isaliteral{2D}{\isacharminus}}{\isaliteral{3E}{\isachargreater}}\ IntInf{\isaliteral{2E}{\isachardot}}int\isanewline -\ \ val\ abs{\isaliteral{5F}{\isacharunderscore}}int\ {\isaliteral{3A}{\isacharcolon}}\ IntInf{\isaliteral{2E}{\isachardot}}int\ {\isaliteral{2D}{\isacharminus}}{\isaliteral{3E}{\isachargreater}}\ IntInf{\isaliteral{2E}{\isachardot}}int\isanewline -\ \ val\ divmod{\isaliteral{5F}{\isacharunderscore}}int\ {\isaliteral{3A}{\isacharcolon}}\ IntInf{\isaliteral{2E}{\isachardot}}int\ {\isaliteral{2D}{\isacharminus}}{\isaliteral{3E}{\isachargreater}}\ IntInf{\isaliteral{2E}{\isachardot}}int\ {\isaliteral{2D}{\isacharminus}}{\isaliteral{3E}{\isachargreater}}\ IntInf{\isaliteral{2E}{\isachardot}}int\ {\isaliteral{2A}{\isacharasterisk}}\ IntInf{\isaliteral{2E}{\isachardot}}int\isanewline -\ \ val\ plus{\isaliteral{5F}{\isacharunderscore}}nat\ {\isaliteral{3A}{\isacharcolon}}\ nat\ {\isaliteral{2D}{\isacharminus}}{\isaliteral{3E}{\isachargreater}}\ nat\ {\isaliteral{2D}{\isacharminus}}{\isaliteral{3E}{\isachargreater}}\ nat\isanewline -\ \ val\ nat\ {\isaliteral{3A}{\isacharcolon}}\ IntInf{\isaliteral{2E}{\isachardot}}int\ {\isaliteral{2D}{\isacharminus}}{\isaliteral{3E}{\isachargreater}}\ nat\isanewline -\ \ type\ {\isaliteral{27}{\isacharprime}}a\ semigroup\isanewline -\ \ val\ mult\ {\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{27}{\isacharprime}}a\ semigroup\ {\isaliteral{2D}{\isacharminus}}{\isaliteral{3E}{\isachargreater}}\ {\isaliteral{27}{\isacharprime}}a\ {\isaliteral{2D}{\isacharminus}}{\isaliteral{3E}{\isachargreater}}\ {\isaliteral{27}{\isacharprime}}a\ {\isaliteral{2D}{\isacharminus}}{\isaliteral{3E}{\isachargreater}}\ {\isaliteral{27}{\isacharprime}}a\isanewline -\ \ type\ {\isaliteral{27}{\isacharprime}}a\ monoidl\isanewline -\ \ val\ semigroup{\isaliteral{5F}{\isacharunderscore}}monoidl\ {\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{27}{\isacharprime}}a\ monoidl\ {\isaliteral{2D}{\isacharminus}}{\isaliteral{3E}{\isachargreater}}\ {\isaliteral{27}{\isacharprime}}a\ semigroup\isanewline -\ \ val\ neutral\ {\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{27}{\isacharprime}}a\ monoidl\ {\isaliteral{2D}{\isacharminus}}{\isaliteral{3E}{\isachargreater}}\ {\isaliteral{27}{\isacharprime}}a\isanewline -\ \ type\ {\isaliteral{27}{\isacharprime}}a\ monoid\isanewline -\ \ val\ monoidl{\isaliteral{5F}{\isacharunderscore}}monoid\ {\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{27}{\isacharprime}}a\ monoid\ {\isaliteral{2D}{\isacharminus}}{\isaliteral{3E}{\isachargreater}}\ {\isaliteral{27}{\isacharprime}}a\ monoidl\isanewline -\ \ type\ {\isaliteral{27}{\isacharprime}}a\ group\isanewline -\ \ val\ monoid{\isaliteral{5F}{\isacharunderscore}}group\ {\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{27}{\isacharprime}}a\ group\ {\isaliteral{2D}{\isacharminus}}{\isaliteral{3E}{\isachargreater}}\ {\isaliteral{27}{\isacharprime}}a\ monoid\isanewline -\ \ val\ inverse\ {\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{27}{\isacharprime}}a\ group\ {\isaliteral{2D}{\isacharminus}}{\isaliteral{3E}{\isachargreater}}\ {\isaliteral{27}{\isacharprime}}a\ {\isaliteral{2D}{\isacharminus}}{\isaliteral{3E}{\isachargreater}}\ {\isaliteral{27}{\isacharprime}}a\isanewline -\ \ val\ neutral{\isaliteral{5F}{\isacharunderscore}}int\ {\isaliteral{3A}{\isacharcolon}}\ IntInf{\isaliteral{2E}{\isachardot}}int\isanewline -\ \ val\ inverse{\isaliteral{5F}{\isacharunderscore}}int\ {\isaliteral{3A}{\isacharcolon}}\ IntInf{\isaliteral{2E}{\isachardot}}int\ {\isaliteral{2D}{\isacharminus}}{\isaliteral{3E}{\isachargreater}}\ IntInf{\isaliteral{2E}{\isachardot}}int\isanewline -\ \ val\ mult{\isaliteral{5F}{\isacharunderscore}}int\ {\isaliteral{3A}{\isacharcolon}}\ IntInf{\isaliteral{2E}{\isachardot}}int\ {\isaliteral{2D}{\isacharminus}}{\isaliteral{3E}{\isachargreater}}\ IntInf{\isaliteral{2E}{\isachardot}}int\ {\isaliteral{2D}{\isacharminus}}{\isaliteral{3E}{\isachargreater}}\ IntInf{\isaliteral{2E}{\isachardot}}int\isanewline -\ \ val\ semigroup{\isaliteral{5F}{\isacharunderscore}}int\ {\isaliteral{3A}{\isacharcolon}}\ IntInf{\isaliteral{2E}{\isachardot}}int\ semigroup\isanewline -\ \ val\ monoidl{\isaliteral{5F}{\isacharunderscore}}int\ {\isaliteral{3A}{\isacharcolon}}\ IntInf{\isaliteral{2E}{\isachardot}}int\ monoidl\isanewline -\ \ val\ monoid{\isaliteral{5F}{\isacharunderscore}}int\ {\isaliteral{3A}{\isacharcolon}}\ IntInf{\isaliteral{2E}{\isachardot}}int\ monoid\isanewline -\ \ val\ group{\isaliteral{5F}{\isacharunderscore}}int\ {\isaliteral{3A}{\isacharcolon}}\ IntInf{\isaliteral{2E}{\isachardot}}int\ group\isanewline -\ \ val\ pow{\isaliteral{5F}{\isacharunderscore}}nat\ {\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{27}{\isacharprime}}a\ monoid\ {\isaliteral{2D}{\isacharminus}}{\isaliteral{3E}{\isachargreater}}\ nat\ {\isaliteral{2D}{\isacharminus}}{\isaliteral{3E}{\isachargreater}}\ {\isaliteral{27}{\isacharprime}}a\ {\isaliteral{2D}{\isacharminus}}{\isaliteral{3E}{\isachargreater}}\ {\isaliteral{27}{\isacharprime}}a\isanewline -\ \ val\ pow{\isaliteral{5F}{\isacharunderscore}}int\ {\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{27}{\isacharprime}}a\ group\ {\isaliteral{2D}{\isacharminus}}{\isaliteral{3E}{\isachargreater}}\ IntInf{\isaliteral{2E}{\isachardot}}int\ {\isaliteral{2D}{\isacharminus}}{\isaliteral{3E}{\isachargreater}}\ {\isaliteral{27}{\isacharprime}}a\ {\isaliteral{2D}{\isacharminus}}{\isaliteral{3E}{\isachargreater}}\ {\isaliteral{27}{\isacharprime}}a\isanewline -\ \ val\ example\ {\isaliteral{3A}{\isacharcolon}}\ IntInf{\isaliteral{2E}{\isachardot}}int\isanewline -end\ {\isaliteral{3D}{\isacharequal}}\ struct\isanewline -\isanewline -datatype\ nat\ {\isaliteral{3D}{\isacharequal}}\ Zero{\isaliteral{5F}{\isacharunderscore}}nat\ {\isaliteral{7C}{\isacharbar}}\ Suc\ of\ nat{\isaliteral{3B}{\isacharsemicolon}}\isanewline -\isanewline -datatype\ num\ {\isaliteral{3D}{\isacharequal}}\ One\ {\isaliteral{7C}{\isacharbar}}\ Bit{\isadigit{0}}\ of\ num\ {\isaliteral{7C}{\isacharbar}}\ Bit{\isadigit{1}}\ of\ num{\isaliteral{3B}{\isacharsemicolon}}\isanewline -\isanewline -fun\ apsnd\ f\ {\isaliteral{28}{\isacharparenleft}}x{\isaliteral{2C}{\isacharcomma}}\ y{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{28}{\isacharparenleft}}x{\isaliteral{2C}{\isacharcomma}}\ f\ y{\isaliteral{29}{\isacharparenright}}{\isaliteral{3B}{\isacharsemicolon}}\isanewline -\isanewline -fun\ sgn{\isaliteral{5F}{\isacharunderscore}}int\ i\ {\isaliteral{3D}{\isacharequal}}\isanewline -\ \ {\isaliteral{28}{\isacharparenleft}}if\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{28}{\isacharparenleft}}i\ {\isaliteral{3A}{\isacharcolon}}\ IntInf{\isaliteral{2E}{\isachardot}}int{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ {\isadigit{0}}{\isaliteral{29}{\isacharparenright}}\ then\ {\isadigit{0}}\isanewline -\ \ \ \ else\ {\isaliteral{28}{\isacharparenleft}}if\ IntInf{\isaliteral{2E}{\isachardot}}{\isaliteral{3C}{\isacharless}}\ {\isaliteral{28}{\isacharparenleft}}{\isadigit{0}}{\isaliteral{2C}{\isacharcomma}}\ i{\isaliteral{29}{\isacharparenright}}\ then\ {\isaliteral{28}{\isacharparenleft}}{\isadigit{1}}\ {\isaliteral{3A}{\isacharcolon}}\ IntInf{\isaliteral{2E}{\isachardot}}int{\isaliteral{29}{\isacharparenright}}\isanewline -\ \ \ \ \ \ \ \ \ \ \ else\ IntInf{\isaliteral{2E}{\isachardot}}{\isaliteral{7E}{\isachartilde}}\ {\isaliteral{28}{\isacharparenleft}}{\isadigit{1}}\ {\isaliteral{3A}{\isacharcolon}}\ IntInf{\isaliteral{2E}{\isachardot}}int{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{3B}{\isacharsemicolon}}\isanewline -\isanewline -fun\ abs{\isaliteral{5F}{\isacharunderscore}}int\ i\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{28}{\isacharparenleft}}if\ IntInf{\isaliteral{2E}{\isachardot}}{\isaliteral{3C}{\isacharless}}\ {\isaliteral{28}{\isacharparenleft}}i{\isaliteral{2C}{\isacharcomma}}\ {\isadigit{0}}{\isaliteral{29}{\isacharparenright}}\ then\ IntInf{\isaliteral{2E}{\isachardot}}{\isaliteral{7E}{\isachartilde}}\ i\ else\ i{\isaliteral{29}{\isacharparenright}}{\isaliteral{3B}{\isacharsemicolon}}\isanewline -\isanewline -fun\ divmod{\isaliteral{5F}{\isacharunderscore}}int\ k\ l\ {\isaliteral{3D}{\isacharequal}}\isanewline -\ \ {\isaliteral{28}{\isacharparenleft}}if\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{28}{\isacharparenleft}}k\ {\isaliteral{3A}{\isacharcolon}}\ IntInf{\isaliteral{2E}{\isachardot}}int{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ {\isadigit{0}}{\isaliteral{29}{\isacharparenright}}\ then\ {\isaliteral{28}{\isacharparenleft}}{\isadigit{0}}{\isaliteral{2C}{\isacharcomma}}\ {\isadigit{0}}{\isaliteral{29}{\isacharparenright}}\isanewline -\ \ \ \ else\ {\isaliteral{28}{\isacharparenleft}}if\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{28}{\isacharparenleft}}l\ {\isaliteral{3A}{\isacharcolon}}\ IntInf{\isaliteral{2E}{\isachardot}}int{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ {\isadigit{0}}{\isaliteral{29}{\isacharparenright}}\ then\ {\isaliteral{28}{\isacharparenleft}}{\isadigit{0}}{\isaliteral{2C}{\isacharcomma}}\ k{\isaliteral{29}{\isacharparenright}}\isanewline -\ \ \ \ \ \ \ \ \ \ \ else\ apsnd\ {\isaliteral{28}{\isacharparenleft}}fn\ a\ {\isaliteral{3D}{\isacharequal}}{\isaliteral{3E}{\isachargreater}}\ IntInf{\isaliteral{2E}{\isachardot}}{\isaliteral{2A}{\isacharasterisk}}\ {\isaliteral{28}{\isacharparenleft}}sgn{\isaliteral{5F}{\isacharunderscore}}int\ l{\isaliteral{2C}{\isacharcomma}}\ a{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}\isanewline -\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isaliteral{28}{\isacharparenleft}}if\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{28}{\isacharparenleft}}{\isaliteral{28}{\isacharparenleft}}sgn{\isaliteral{5F}{\isacharunderscore}}int\ k{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3A}{\isacharcolon}}\ IntInf{\isaliteral{2E}{\isachardot}}int{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{28}{\isacharparenleft}}sgn{\isaliteral{5F}{\isacharunderscore}}int\ l{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}\isanewline -\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ then\ IntInf{\isaliteral{2E}{\isachardot}}divMod\ {\isaliteral{28}{\isacharparenleft}}IntInf{\isaliteral{2E}{\isachardot}}abs\ k{\isaliteral{2C}{\isacharcomma}}\ IntInf{\isaliteral{2E}{\isachardot}}abs\ l{\isaliteral{29}{\isacharparenright}}\isanewline -\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ else\ let\isanewline -\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ val\ {\isaliteral{28}{\isacharparenleft}}r{\isaliteral{2C}{\isacharcomma}}\ s{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\isanewline -\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ IntInf{\isaliteral{2E}{\isachardot}}divMod\ {\isaliteral{28}{\isacharparenleft}}IntInf{\isaliteral{2E}{\isachardot}}abs\ k{\isaliteral{2C}{\isacharcomma}}\ IntInf{\isaliteral{2E}{\isachardot}}abs\ l{\isaliteral{29}{\isacharparenright}}{\isaliteral{3B}{\isacharsemicolon}}\isanewline -\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ in\isanewline -\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isaliteral{28}{\isacharparenleft}}if\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{28}{\isacharparenleft}}s\ {\isaliteral{3A}{\isacharcolon}}\ IntInf{\isaliteral{2E}{\isachardot}}int{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ {\isadigit{0}}{\isaliteral{29}{\isacharparenright}}\ then\ {\isaliteral{28}{\isacharparenleft}}IntInf{\isaliteral{2E}{\isachardot}}{\isaliteral{7E}{\isachartilde}}\ r{\isaliteral{2C}{\isacharcomma}}\ {\isadigit{0}}{\isaliteral{29}{\isacharparenright}}\isanewline -\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ else\ {\isaliteral{28}{\isacharparenleft}}IntInf{\isaliteral{2E}{\isachardot}}{\isaliteral{2D}{\isacharminus}}\ {\isaliteral{28}{\isacharparenleft}}IntInf{\isaliteral{2E}{\isachardot}}{\isaliteral{7E}{\isachartilde}}\isanewline -\ \ \ \ \ \ \ \ \ \ r{\isaliteral{2C}{\isacharcomma}}\ {\isaliteral{28}{\isacharparenleft}}{\isadigit{1}}\ {\isaliteral{3A}{\isacharcolon}}\ IntInf{\isaliteral{2E}{\isachardot}}int{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{2C}{\isacharcomma}}\isanewline -\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ IntInf{\isaliteral{2E}{\isachardot}}{\isaliteral{2D}{\isacharminus}}\ {\isaliteral{28}{\isacharparenleft}}abs{\isaliteral{5F}{\isacharunderscore}}int\ l{\isaliteral{2C}{\isacharcomma}}\ s{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}\isanewline -\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ end{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{3B}{\isacharsemicolon}}\isanewline -\isanewline -fun\ plus{\isaliteral{5F}{\isacharunderscore}}nat\ {\isaliteral{28}{\isacharparenleft}}Suc\ m{\isaliteral{29}{\isacharparenright}}\ n\ {\isaliteral{3D}{\isacharequal}}\ plus{\isaliteral{5F}{\isacharunderscore}}nat\ m\ {\isaliteral{28}{\isacharparenleft}}Suc\ n{\isaliteral{29}{\isacharparenright}}\isanewline -\ \ {\isaliteral{7C}{\isacharbar}}\ plus{\isaliteral{5F}{\isacharunderscore}}nat\ Zero{\isaliteral{5F}{\isacharunderscore}}nat\ n\ {\isaliteral{3D}{\isacharequal}}\ n{\isaliteral{3B}{\isacharsemicolon}}\isanewline -\isanewline -fun\ nat\ k\ {\isaliteral{3D}{\isacharequal}}\isanewline -\ \ {\isaliteral{28}{\isacharparenleft}}if\ IntInf{\isaliteral{2E}{\isachardot}}{\isaliteral{3C}{\isacharless}}{\isaliteral{3D}{\isacharequal}}\ {\isaliteral{28}{\isacharparenleft}}k{\isaliteral{2C}{\isacharcomma}}\ {\isadigit{0}}{\isaliteral{29}{\isacharparenright}}\ then\ Zero{\isaliteral{5F}{\isacharunderscore}}nat\isanewline -\ \ \ \ else\ let\isanewline -\ \ \ \ \ \ \ \ \ \ \ val\ {\isaliteral{28}{\isacharparenleft}}l{\isaliteral{2C}{\isacharcomma}}\ j{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ divmod{\isaliteral{5F}{\isacharunderscore}}int\ k\ {\isaliteral{28}{\isacharparenleft}}{\isadigit{2}}\ {\isaliteral{3A}{\isacharcolon}}\ IntInf{\isaliteral{2E}{\isachardot}}int{\isaliteral{29}{\isacharparenright}}{\isaliteral{3B}{\isacharsemicolon}}\isanewline -\ \ \ \ \ \ \ \ \ \ \ val\ n\ {\isaliteral{3D}{\isacharequal}}\ nat\ l{\isaliteral{3B}{\isacharsemicolon}}\isanewline -\ \ \ \ \ \ \ \ \ \ \ val\ la\ {\isaliteral{3D}{\isacharequal}}\ plus{\isaliteral{5F}{\isacharunderscore}}nat\ n\ n{\isaliteral{3B}{\isacharsemicolon}}\isanewline -\ \ \ \ \ \ \ \ \ in\isanewline -\ \ \ \ \ \ \ \ \ \ \ {\isaliteral{28}{\isacharparenleft}}if\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{28}{\isacharparenleft}}j\ {\isaliteral{3A}{\isacharcolon}}\ IntInf{\isaliteral{2E}{\isachardot}}int{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ {\isadigit{0}}{\isaliteral{29}{\isacharparenright}}\ then\ la\ else\ Suc\ la{\isaliteral{29}{\isacharparenright}}\isanewline -\ \ \ \ \ \ \ \ \ end{\isaliteral{29}{\isacharparenright}}{\isaliteral{3B}{\isacharsemicolon}}\isanewline -\isanewline -type\ {\isaliteral{27}{\isacharprime}}a\ semigroup\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{7B}{\isacharbraceleft}}mult\ {\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{27}{\isacharprime}}a\ {\isaliteral{2D}{\isacharminus}}{\isaliteral{3E}{\isachargreater}}\ {\isaliteral{27}{\isacharprime}}a\ {\isaliteral{2D}{\isacharminus}}{\isaliteral{3E}{\isachargreater}}\ {\isaliteral{27}{\isacharprime}}a{\isaliteral{7D}{\isacharbraceright}}{\isaliteral{3B}{\isacharsemicolon}}\isanewline -val\ mult\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{23}{\isacharhash}}mult\ {\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{27}{\isacharprime}}a\ semigroup\ {\isaliteral{2D}{\isacharminus}}{\isaliteral{3E}{\isachargreater}}\ {\isaliteral{27}{\isacharprime}}a\ {\isaliteral{2D}{\isacharminus}}{\isaliteral{3E}{\isachargreater}}\ {\isaliteral{27}{\isacharprime}}a\ {\isaliteral{2D}{\isacharminus}}{\isaliteral{3E}{\isachargreater}}\ {\isaliteral{27}{\isacharprime}}a{\isaliteral{3B}{\isacharsemicolon}}\isanewline -\isanewline -type\ {\isaliteral{27}{\isacharprime}}a\ monoidl\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{7B}{\isacharbraceleft}}semigroup{\isaliteral{5F}{\isacharunderscore}}monoidl\ {\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{27}{\isacharprime}}a\ semigroup{\isaliteral{2C}{\isacharcomma}}\ neutral\ {\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{27}{\isacharprime}}a{\isaliteral{7D}{\isacharbraceright}}{\isaliteral{3B}{\isacharsemicolon}}\isanewline -val\ semigroup{\isaliteral{5F}{\isacharunderscore}}monoidl\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{23}{\isacharhash}}semigroup{\isaliteral{5F}{\isacharunderscore}}monoidl\ {\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{27}{\isacharprime}}a\ monoidl\ {\isaliteral{2D}{\isacharminus}}{\isaliteral{3E}{\isachargreater}}\ {\isaliteral{27}{\isacharprime}}a\ semigroup{\isaliteral{3B}{\isacharsemicolon}}\isanewline -val\ neutral\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{23}{\isacharhash}}neutral\ {\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{27}{\isacharprime}}a\ monoidl\ {\isaliteral{2D}{\isacharminus}}{\isaliteral{3E}{\isachargreater}}\ {\isaliteral{27}{\isacharprime}}a{\isaliteral{3B}{\isacharsemicolon}}\isanewline -\isanewline -type\ {\isaliteral{27}{\isacharprime}}a\ monoid\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{7B}{\isacharbraceleft}}monoidl{\isaliteral{5F}{\isacharunderscore}}monoid\ {\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{27}{\isacharprime}}a\ monoidl{\isaliteral{7D}{\isacharbraceright}}{\isaliteral{3B}{\isacharsemicolon}}\isanewline -val\ monoidl{\isaliteral{5F}{\isacharunderscore}}monoid\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{23}{\isacharhash}}monoidl{\isaliteral{5F}{\isacharunderscore}}monoid\ {\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{27}{\isacharprime}}a\ monoid\ {\isaliteral{2D}{\isacharminus}}{\isaliteral{3E}{\isachargreater}}\ {\isaliteral{27}{\isacharprime}}a\ monoidl{\isaliteral{3B}{\isacharsemicolon}}\isanewline -\isanewline -type\ {\isaliteral{27}{\isacharprime}}a\ group\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{7B}{\isacharbraceleft}}monoid{\isaliteral{5F}{\isacharunderscore}}group\ {\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{27}{\isacharprime}}a\ monoid{\isaliteral{2C}{\isacharcomma}}\ inverse\ {\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{27}{\isacharprime}}a\ {\isaliteral{2D}{\isacharminus}}{\isaliteral{3E}{\isachargreater}}\ {\isaliteral{27}{\isacharprime}}a{\isaliteral{7D}{\isacharbraceright}}{\isaliteral{3B}{\isacharsemicolon}}\isanewline -val\ monoid{\isaliteral{5F}{\isacharunderscore}}group\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{23}{\isacharhash}}monoid{\isaliteral{5F}{\isacharunderscore}}group\ {\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{27}{\isacharprime}}a\ group\ {\isaliteral{2D}{\isacharminus}}{\isaliteral{3E}{\isachargreater}}\ {\isaliteral{27}{\isacharprime}}a\ monoid{\isaliteral{3B}{\isacharsemicolon}}\isanewline -val\ inverse\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{23}{\isacharhash}}inverse\ {\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{27}{\isacharprime}}a\ group\ {\isaliteral{2D}{\isacharminus}}{\isaliteral{3E}{\isachargreater}}\ {\isaliteral{27}{\isacharprime}}a\ {\isaliteral{2D}{\isacharminus}}{\isaliteral{3E}{\isachargreater}}\ {\isaliteral{27}{\isacharprime}}a{\isaliteral{3B}{\isacharsemicolon}}\isanewline -\isanewline -val\ neutral{\isaliteral{5F}{\isacharunderscore}}int\ {\isaliteral{3A}{\isacharcolon}}\ IntInf{\isaliteral{2E}{\isachardot}}int\ {\isaliteral{3D}{\isacharequal}}\ {\isadigit{0}}{\isaliteral{3B}{\isacharsemicolon}}\isanewline -\isanewline -fun\ inverse{\isaliteral{5F}{\isacharunderscore}}int\ i\ {\isaliteral{3D}{\isacharequal}}\ IntInf{\isaliteral{2E}{\isachardot}}{\isaliteral{7E}{\isachartilde}}\ i{\isaliteral{3B}{\isacharsemicolon}}\isanewline -\isanewline -fun\ mult{\isaliteral{5F}{\isacharunderscore}}int\ i\ j\ {\isaliteral{3D}{\isacharequal}}\ IntInf{\isaliteral{2E}{\isachardot}}{\isaliteral{2B}{\isacharplus}}\ {\isaliteral{28}{\isacharparenleft}}i{\isaliteral{2C}{\isacharcomma}}\ j{\isaliteral{29}{\isacharparenright}}{\isaliteral{3B}{\isacharsemicolon}}\isanewline -\isanewline -val\ semigroup{\isaliteral{5F}{\isacharunderscore}}int\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{7B}{\isacharbraceleft}}mult\ {\isaliteral{3D}{\isacharequal}}\ mult{\isaliteral{5F}{\isacharunderscore}}int{\isaliteral{7D}{\isacharbraceright}}\ {\isaliteral{3A}{\isacharcolon}}\ IntInf{\isaliteral{2E}{\isachardot}}int\ semigroup{\isaliteral{3B}{\isacharsemicolon}}\isanewline -\isanewline -val\ monoidl{\isaliteral{5F}{\isacharunderscore}}int\ {\isaliteral{3D}{\isacharequal}}\isanewline -\ \ {\isaliteral{7B}{\isacharbraceleft}}semigroup{\isaliteral{5F}{\isacharunderscore}}monoidl\ {\isaliteral{3D}{\isacharequal}}\ semigroup{\isaliteral{5F}{\isacharunderscore}}int{\isaliteral{2C}{\isacharcomma}}\ neutral\ {\isaliteral{3D}{\isacharequal}}\ neutral{\isaliteral{5F}{\isacharunderscore}}int{\isaliteral{7D}{\isacharbraceright}}\ {\isaliteral{3A}{\isacharcolon}}\isanewline -\ \ IntInf{\isaliteral{2E}{\isachardot}}int\ monoidl{\isaliteral{3B}{\isacharsemicolon}}\isanewline -\isanewline -val\ monoid{\isaliteral{5F}{\isacharunderscore}}int\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{7B}{\isacharbraceleft}}monoidl{\isaliteral{5F}{\isacharunderscore}}monoid\ {\isaliteral{3D}{\isacharequal}}\ monoidl{\isaliteral{5F}{\isacharunderscore}}int{\isaliteral{7D}{\isacharbraceright}}\ {\isaliteral{3A}{\isacharcolon}}\ IntInf{\isaliteral{2E}{\isachardot}}int\ monoid{\isaliteral{3B}{\isacharsemicolon}}\isanewline -\isanewline -val\ group{\isaliteral{5F}{\isacharunderscore}}int\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{7B}{\isacharbraceleft}}monoid{\isaliteral{5F}{\isacharunderscore}}group\ {\isaliteral{3D}{\isacharequal}}\ monoid{\isaliteral{5F}{\isacharunderscore}}int{\isaliteral{2C}{\isacharcomma}}\ inverse\ {\isaliteral{3D}{\isacharequal}}\ inverse{\isaliteral{5F}{\isacharunderscore}}int{\isaliteral{7D}{\isacharbraceright}}\ {\isaliteral{3A}{\isacharcolon}}\isanewline -\ \ IntInf{\isaliteral{2E}{\isachardot}}int\ group{\isaliteral{3B}{\isacharsemicolon}}\isanewline -\isanewline -fun\ pow{\isaliteral{5F}{\isacharunderscore}}nat\ A{\isaliteral{5F}{\isacharunderscore}}\ Zero{\isaliteral{5F}{\isacharunderscore}}nat\ x\ {\isaliteral{3D}{\isacharequal}}\ neutral\ {\isaliteral{28}{\isacharparenleft}}monoidl{\isaliteral{5F}{\isacharunderscore}}monoid\ A{\isaliteral{5F}{\isacharunderscore}}{\isaliteral{29}{\isacharparenright}}\isanewline -\ \ {\isaliteral{7C}{\isacharbar}}\ pow{\isaliteral{5F}{\isacharunderscore}}nat\ A{\isaliteral{5F}{\isacharunderscore}}\ {\isaliteral{28}{\isacharparenleft}}Suc\ n{\isaliteral{29}{\isacharparenright}}\ x\ {\isaliteral{3D}{\isacharequal}}\isanewline -\ \ \ \ mult\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{28}{\isacharparenleft}}semigroup{\isaliteral{5F}{\isacharunderscore}}monoidl\ o\ monoidl{\isaliteral{5F}{\isacharunderscore}}monoid{\isaliteral{29}{\isacharparenright}}\ A{\isaliteral{5F}{\isacharunderscore}}{\isaliteral{29}{\isacharparenright}}\ x\ {\isaliteral{28}{\isacharparenleft}}pow{\isaliteral{5F}{\isacharunderscore}}nat\ A{\isaliteral{5F}{\isacharunderscore}}\ n\ x{\isaliteral{29}{\isacharparenright}}{\isaliteral{3B}{\isacharsemicolon}}\isanewline -\isanewline -fun\ pow{\isaliteral{5F}{\isacharunderscore}}int\ A{\isaliteral{5F}{\isacharunderscore}}\ k\ x\ {\isaliteral{3D}{\isacharequal}}\isanewline -\ \ {\isaliteral{28}{\isacharparenleft}}if\ IntInf{\isaliteral{2E}{\isachardot}}{\isaliteral{3C}{\isacharless}}{\isaliteral{3D}{\isacharequal}}\ {\isaliteral{28}{\isacharparenleft}}{\isadigit{0}}{\isaliteral{2C}{\isacharcomma}}\ k{\isaliteral{29}{\isacharparenright}}\ then\ pow{\isaliteral{5F}{\isacharunderscore}}nat\ {\isaliteral{28}{\isacharparenleft}}monoid{\isaliteral{5F}{\isacharunderscore}}group\ A{\isaliteral{5F}{\isacharunderscore}}{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{28}{\isacharparenleft}}nat\ k{\isaliteral{29}{\isacharparenright}}\ x\isanewline -\ \ \ \ else\ inverse\ A{\isaliteral{5F}{\isacharunderscore}}\ {\isaliteral{28}{\isacharparenleft}}pow{\isaliteral{5F}{\isacharunderscore}}nat\ {\isaliteral{28}{\isacharparenleft}}monoid{\isaliteral{5F}{\isacharunderscore}}group\ A{\isaliteral{5F}{\isacharunderscore}}{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{28}{\isacharparenleft}}nat\ {\isaliteral{28}{\isacharparenleft}}IntInf{\isaliteral{2E}{\isachardot}}{\isaliteral{7E}{\isachartilde}}\ k{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}\ x{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{3B}{\isacharsemicolon}}\isanewline -\isanewline -val\ example\ {\isaliteral{3A}{\isacharcolon}}\ IntInf{\isaliteral{2E}{\isachardot}}int\ {\isaliteral{3D}{\isacharequal}}\isanewline -\ \ pow{\isaliteral{5F}{\isacharunderscore}}int\ group{\isaliteral{5F}{\isacharunderscore}}int\ {\isaliteral{28}{\isacharparenleft}}{\isadigit{1}}{\isadigit{0}}\ {\isaliteral{3A}{\isacharcolon}}\ IntInf{\isaliteral{2E}{\isachardot}}int{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{7E}{\isachartilde}}{\isadigit{2}}\ {\isaliteral{3A}{\isacharcolon}}\ IntInf{\isaliteral{2E}{\isachardot}}int{\isaliteral{29}{\isacharparenright}}{\isaliteral{3B}{\isacharsemicolon}}\isanewline -\isanewline -end{\isaliteral{3B}{\isacharsemicolon}}\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{2A}{\isacharasterisk}}struct\ Example{\isaliteral{2A}{\isacharasterisk}}{\isaliteral{29}{\isacharparenright}}\isanewline% -\end{isamarkuptext}% -\isamarkuptrue% -% -\endisatagquotetypewriter -{\isafoldquotetypewriter}% -% -\isadelimquotetypewriter -% -\endisadelimquotetypewriter -% -\begin{isamarkuptext}% -\noindent In Scala, implicts are used as dictionaries:% -\end{isamarkuptext}% -\isamarkuptrue% -% -\isadeliminvisible -% -\endisadeliminvisible -% -\isataginvisible -% -\endisataginvisible -{\isafoldinvisible}% -% -\isadeliminvisible -% -\endisadeliminvisible -% -\isadelimquotetypewriter -% -\endisadelimquotetypewriter -% -\isatagquotetypewriter -% -\begin{isamarkuptext}% -object\ Example\ {\isaliteral{7B}{\isacharbraceleft}}\isanewline -\isanewline -abstract\ sealed\ class\ nat\isanewline -final\ case\ object\ Zero{\isaliteral{5F}{\isacharunderscore}}nat\ extends\ nat\isanewline -final\ case\ class\ Suc{\isaliteral{28}{\isacharparenleft}}a{\isaliteral{3A}{\isacharcolon}}\ nat{\isaliteral{29}{\isacharparenright}}\ extends\ nat\isanewline -\isanewline -abstract\ sealed\ class\ num\isanewline -final\ case\ object\ One\ extends\ num\isanewline -final\ case\ class\ Bit{\isadigit{0}}{\isaliteral{28}{\isacharparenleft}}a{\isaliteral{3A}{\isacharcolon}}\ num{\isaliteral{29}{\isacharparenright}}\ extends\ num\isanewline -final\ case\ class\ Bit{\isadigit{1}}{\isaliteral{28}{\isacharparenleft}}a{\isaliteral{3A}{\isacharcolon}}\ num{\isaliteral{29}{\isacharparenright}}\ extends\ num\isanewline -\isanewline -def\ apsnd{\isaliteral{5B}{\isacharbrackleft}}A{\isaliteral{2C}{\isacharcomma}}\ B{\isaliteral{2C}{\isacharcomma}}\ C{\isaliteral{5D}{\isacharbrackright}}{\isaliteral{28}{\isacharparenleft}}f{\isaliteral{3A}{\isacharcolon}}\ A\ {\isaliteral{3D}{\isacharequal}}{\isaliteral{3E}{\isachargreater}}\ B{\isaliteral{2C}{\isacharcomma}}\ x{\isadigit{1}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{28}{\isacharparenleft}}C{\isaliteral{2C}{\isacharcomma}}\ A{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{28}{\isacharparenleft}}C{\isaliteral{2C}{\isacharcomma}}\ B{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{28}{\isacharparenleft}}f{\isaliteral{2C}{\isacharcomma}}\ x{\isadigit{1}}{\isaliteral{29}{\isacharparenright}}\ match\ {\isaliteral{7B}{\isacharbraceleft}}\isanewline -\ \ case\ {\isaliteral{28}{\isacharparenleft}}f{\isaliteral{2C}{\isacharcomma}}\ {\isaliteral{28}{\isacharparenleft}}x{\isaliteral{2C}{\isacharcomma}}\ y{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}{\isaliteral{3E}{\isachargreater}}\ {\isaliteral{28}{\isacharparenleft}}x{\isaliteral{2C}{\isacharcomma}}\ f{\isaliteral{28}{\isacharparenleft}}y{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}\isanewline -{\isaliteral{7D}{\isacharbraceright}}\isanewline -\isanewline -def\ sgn{\isaliteral{5F}{\isacharunderscore}}int{\isaliteral{28}{\isacharparenleft}}i{\isaliteral{3A}{\isacharcolon}}\ BigInt{\isaliteral{29}{\isacharparenright}}{\isaliteral{3A}{\isacharcolon}}\ BigInt\ {\isaliteral{3D}{\isacharequal}}\isanewline -\ \ {\isaliteral{28}{\isacharparenleft}}if\ {\isaliteral{28}{\isacharparenleft}}i\ {\isaliteral{3D}{\isacharequal}}{\isaliteral{3D}{\isacharequal}}\ BigInt{\isaliteral{28}{\isacharparenleft}}{\isadigit{0}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}\ BigInt{\isaliteral{28}{\isacharparenleft}}{\isadigit{0}}{\isaliteral{29}{\isacharparenright}}\isanewline -\ \ \ \ else\ {\isaliteral{28}{\isacharparenleft}}if\ {\isaliteral{28}{\isacharparenleft}}BigInt{\isaliteral{28}{\isacharparenleft}}{\isadigit{0}}{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3C}{\isacharless}}\ i{\isaliteral{29}{\isacharparenright}}\ BigInt{\isaliteral{28}{\isacharparenleft}}{\isadigit{1}}{\isaliteral{29}{\isacharparenright}}\ else\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{2D}{\isacharminus}}\ BigInt{\isaliteral{28}{\isacharparenleft}}{\isadigit{1}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}\isanewline -\isanewline -def\ abs{\isaliteral{5F}{\isacharunderscore}}int{\isaliteral{28}{\isacharparenleft}}i{\isaliteral{3A}{\isacharcolon}}\ BigInt{\isaliteral{29}{\isacharparenright}}{\isaliteral{3A}{\isacharcolon}}\ BigInt\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{28}{\isacharparenleft}}if\ {\isaliteral{28}{\isacharparenleft}}i\ {\isaliteral{3C}{\isacharless}}\ BigInt{\isaliteral{28}{\isacharparenleft}}{\isadigit{0}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{2D}{\isacharminus}}\ i{\isaliteral{29}{\isacharparenright}}\ else\ i{\isaliteral{29}{\isacharparenright}}\isanewline -\isanewline -def\ divmod{\isaliteral{5F}{\isacharunderscore}}int{\isaliteral{28}{\isacharparenleft}}k{\isaliteral{3A}{\isacharcolon}}\ BigInt{\isaliteral{2C}{\isacharcomma}}\ l{\isaliteral{3A}{\isacharcolon}}\ BigInt{\isaliteral{29}{\isacharparenright}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{28}{\isacharparenleft}}BigInt{\isaliteral{2C}{\isacharcomma}}\ BigInt{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\isanewline -\ \ {\isaliteral{28}{\isacharparenleft}}if\ {\isaliteral{28}{\isacharparenleft}}k\ {\isaliteral{3D}{\isacharequal}}{\isaliteral{3D}{\isacharequal}}\ BigInt{\isaliteral{28}{\isacharparenleft}}{\isadigit{0}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{28}{\isacharparenleft}}BigInt{\isaliteral{28}{\isacharparenleft}}{\isadigit{0}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{2C}{\isacharcomma}}\ BigInt{\isaliteral{28}{\isacharparenleft}}{\isadigit{0}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}\isanewline -\ \ \ \ else\ {\isaliteral{28}{\isacharparenleft}}if\ {\isaliteral{28}{\isacharparenleft}}l\ {\isaliteral{3D}{\isacharequal}}{\isaliteral{3D}{\isacharequal}}\ BigInt{\isaliteral{28}{\isacharparenleft}}{\isadigit{0}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{28}{\isacharparenleft}}BigInt{\isaliteral{28}{\isacharparenleft}}{\isadigit{0}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{2C}{\isacharcomma}}\ k{\isaliteral{29}{\isacharparenright}}\isanewline -\ \ \ \ \ \ \ \ \ \ \ else\ apsnd{\isaliteral{5B}{\isacharbrackleft}}BigInt{\isaliteral{2C}{\isacharcomma}}\ BigInt{\isaliteral{2C}{\isacharcomma}}\isanewline -\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ BigInt{\isaliteral{5D}{\isacharbrackright}}{\isaliteral{28}{\isacharparenleft}}{\isaliteral{28}{\isacharparenleft}}a{\isaliteral{3A}{\isacharcolon}}\ BigInt{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}{\isaliteral{3E}{\isachargreater}}\ sgn{\isaliteral{5F}{\isacharunderscore}}int{\isaliteral{28}{\isacharparenleft}}l{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{2A}{\isacharasterisk}}\ a{\isaliteral{2C}{\isacharcomma}}\isanewline -\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isaliteral{28}{\isacharparenleft}}if\ {\isaliteral{28}{\isacharparenleft}}sgn{\isaliteral{5F}{\isacharunderscore}}int{\isaliteral{28}{\isacharparenleft}}k{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}{\isaliteral{3D}{\isacharequal}}\ sgn{\isaliteral{5F}{\isacharunderscore}}int{\isaliteral{28}{\isacharparenleft}}l{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}\isanewline -\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{28}{\isacharparenleft}}k{\isaliteral{3A}{\isacharcolon}}\ BigInt{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}{\isaliteral{3E}{\isachargreater}}\ {\isaliteral{28}{\isacharparenleft}}l{\isaliteral{3A}{\isacharcolon}}\ BigInt{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}{\isaliteral{3E}{\isachargreater}}\isanewline -\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ if\ {\isaliteral{28}{\isacharparenleft}}l\ {\isaliteral{3D}{\isacharequal}}{\isaliteral{3D}{\isacharequal}}\ {\isadigit{0}}{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{28}{\isacharparenleft}}BigInt{\isaliteral{28}{\isacharparenleft}}{\isadigit{0}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{2C}{\isacharcomma}}\ k{\isaliteral{29}{\isacharparenright}}\ else\isanewline -\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isaliteral{28}{\isacharparenleft}}k{\isaliteral{2E}{\isachardot}}abs\ {\isaliteral{2F}{\isacharslash}}{\isaliteral{25}{\isacharpercent}}\ l{\isaliteral{2E}{\isachardot}}abs{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{2E}{\isachardot}}apply{\isaliteral{28}{\isacharparenleft}}k{\isaliteral{29}{\isacharparenright}}{\isaliteral{2E}{\isachardot}}apply{\isaliteral{28}{\isacharparenleft}}l{\isaliteral{29}{\isacharparenright}}\isanewline -\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ else\ {\isaliteral{7B}{\isacharbraceleft}}\isanewline -\ \ \ \ val\ {\isaliteral{28}{\isacharparenleft}}r{\isaliteral{2C}{\isacharcomma}}\ s{\isaliteral{29}{\isacharparenright}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{28}{\isacharparenleft}}BigInt{\isaliteral{2C}{\isacharcomma}}\ BigInt{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\isanewline -\ \ \ \ \ \ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{28}{\isacharparenleft}}k{\isaliteral{3A}{\isacharcolon}}\ BigInt{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}{\isaliteral{3E}{\isachargreater}}\ {\isaliteral{28}{\isacharparenleft}}l{\isaliteral{3A}{\isacharcolon}}\ BigInt{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}{\isaliteral{3E}{\isachargreater}}\ if\ {\isaliteral{28}{\isacharparenleft}}l\ {\isaliteral{3D}{\isacharequal}}{\isaliteral{3D}{\isacharequal}}\ {\isadigit{0}}{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{28}{\isacharparenleft}}BigInt{\isaliteral{28}{\isacharparenleft}}{\isadigit{0}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{2C}{\isacharcomma}}\ k{\isaliteral{29}{\isacharparenright}}\ else\isanewline -\ \ \ \ \ \ \ \ {\isaliteral{28}{\isacharparenleft}}k{\isaliteral{2E}{\isachardot}}abs\ {\isaliteral{2F}{\isacharslash}}{\isaliteral{25}{\isacharpercent}}\ l{\isaliteral{2E}{\isachardot}}abs{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{2E}{\isachardot}}apply{\isaliteral{28}{\isacharparenleft}}k{\isaliteral{29}{\isacharparenright}}{\isaliteral{2E}{\isachardot}}apply{\isaliteral{28}{\isacharparenleft}}l{\isaliteral{29}{\isacharparenright}}{\isaliteral{3B}{\isacharsemicolon}}\isanewline -\ \ \ \ {\isaliteral{28}{\isacharparenleft}}if\ {\isaliteral{28}{\isacharparenleft}}s\ {\isaliteral{3D}{\isacharequal}}{\isaliteral{3D}{\isacharequal}}\ BigInt{\isaliteral{28}{\isacharparenleft}}{\isadigit{0}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{28}{\isacharparenleft}}{\isaliteral{2D}{\isacharminus}}\ r{\isaliteral{29}{\isacharparenright}}{\isaliteral{2C}{\isacharcomma}}\ BigInt{\isaliteral{28}{\isacharparenleft}}{\isadigit{0}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}\isanewline -\ \ \ \ \ \ else\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{28}{\isacharparenleft}}{\isaliteral{2D}{\isacharminus}}\ r{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{2D}{\isacharminus}}\ BigInt{\isaliteral{28}{\isacharparenleft}}{\isadigit{1}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{2C}{\isacharcomma}}\ abs{\isaliteral{5F}{\isacharunderscore}}int{\isaliteral{28}{\isacharparenleft}}l{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{2D}{\isacharminus}}\ s{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}\isanewline -\ \ {\isaliteral{7D}{\isacharbraceright}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}\isanewline -\isanewline -def\ plus{\isaliteral{5F}{\isacharunderscore}}nat{\isaliteral{28}{\isacharparenleft}}x{\isadigit{0}}{\isaliteral{3A}{\isacharcolon}}\ nat{\isaliteral{2C}{\isacharcomma}}\ n{\isaliteral{3A}{\isacharcolon}}\ nat{\isaliteral{29}{\isacharparenright}}{\isaliteral{3A}{\isacharcolon}}\ nat\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{28}{\isacharparenleft}}x{\isadigit{0}}{\isaliteral{2C}{\isacharcomma}}\ n{\isaliteral{29}{\isacharparenright}}\ match\ {\isaliteral{7B}{\isacharbraceleft}}\isanewline -\ \ case\ {\isaliteral{28}{\isacharparenleft}}Suc{\isaliteral{28}{\isacharparenleft}}m{\isaliteral{29}{\isacharparenright}}{\isaliteral{2C}{\isacharcomma}}\ n{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}{\isaliteral{3E}{\isachargreater}}\ plus{\isaliteral{5F}{\isacharunderscore}}nat{\isaliteral{28}{\isacharparenleft}}m{\isaliteral{2C}{\isacharcomma}}\ Suc{\isaliteral{28}{\isacharparenleft}}n{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}\isanewline -\ \ case\ {\isaliteral{28}{\isacharparenleft}}Zero{\isaliteral{5F}{\isacharunderscore}}nat{\isaliteral{2C}{\isacharcomma}}\ n{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}{\isaliteral{3E}{\isachargreater}}\ n\isanewline -{\isaliteral{7D}{\isacharbraceright}}\isanewline -\isanewline -def\ nat{\isaliteral{28}{\isacharparenleft}}k{\isaliteral{3A}{\isacharcolon}}\ BigInt{\isaliteral{29}{\isacharparenright}}{\isaliteral{3A}{\isacharcolon}}\ nat\ {\isaliteral{3D}{\isacharequal}}\isanewline -\ \ {\isaliteral{28}{\isacharparenleft}}if\ {\isaliteral{28}{\isacharparenleft}}k\ {\isaliteral{3C}{\isacharless}}{\isaliteral{3D}{\isacharequal}}\ BigInt{\isaliteral{28}{\isacharparenleft}}{\isadigit{0}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}\ Zero{\isaliteral{5F}{\isacharunderscore}}nat\isanewline -\ \ \ \ else\ {\isaliteral{7B}{\isacharbraceleft}}\isanewline -\ \ \ \ \ \ \ \ \ \ \ val\ {\isaliteral{28}{\isacharparenleft}}l{\isaliteral{2C}{\isacharcomma}}\ j{\isaliteral{29}{\isacharparenright}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{28}{\isacharparenleft}}BigInt{\isaliteral{2C}{\isacharcomma}}\ BigInt{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ divmod{\isaliteral{5F}{\isacharunderscore}}int{\isaliteral{28}{\isacharparenleft}}k{\isaliteral{2C}{\isacharcomma}}\ BigInt{\isaliteral{28}{\isacharparenleft}}{\isadigit{2}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}\isanewline -\ \ \ \ \ \ \ \ \ \ \ val\ n{\isaliteral{3A}{\isacharcolon}}\ nat\ {\isaliteral{3D}{\isacharequal}}\ nat{\isaliteral{28}{\isacharparenleft}}l{\isaliteral{29}{\isacharparenright}}\isanewline -\ \ \ \ \ \ \ \ \ \ \ val\ la{\isaliteral{3A}{\isacharcolon}}\ nat\ {\isaliteral{3D}{\isacharequal}}\ plus{\isaliteral{5F}{\isacharunderscore}}nat{\isaliteral{28}{\isacharparenleft}}n{\isaliteral{2C}{\isacharcomma}}\ n{\isaliteral{29}{\isacharparenright}}{\isaliteral{3B}{\isacharsemicolon}}\isanewline -\ \ \ \ \ \ \ \ \ \ \ {\isaliteral{28}{\isacharparenleft}}if\ {\isaliteral{28}{\isacharparenleft}}j\ {\isaliteral{3D}{\isacharequal}}{\isaliteral{3D}{\isacharequal}}\ BigInt{\isaliteral{28}{\isacharparenleft}}{\isadigit{0}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}\ la\ else\ Suc{\isaliteral{28}{\isacharparenleft}}la{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}\isanewline -\ \ \ \ \ \ \ \ \ {\isaliteral{7D}{\isacharbraceright}}{\isaliteral{29}{\isacharparenright}}\isanewline -\isanewline -trait\ semigroup{\isaliteral{5B}{\isacharbrackleft}}A{\isaliteral{5D}{\isacharbrackright}}\ {\isaliteral{7B}{\isacharbraceleft}}\isanewline -\ \ val\ {\isaliteral{60}{\isacharbackquote}}Example{\isaliteral{2E}{\isachardot}}mult{\isaliteral{60}{\isacharbackquote}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{28}{\isacharparenleft}}A{\isaliteral{2C}{\isacharcomma}}\ A{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}{\isaliteral{3E}{\isachargreater}}\ A\isanewline -{\isaliteral{7D}{\isacharbraceright}}\isanewline -def\ mult{\isaliteral{5B}{\isacharbrackleft}}A{\isaliteral{5D}{\isacharbrackright}}{\isaliteral{28}{\isacharparenleft}}a{\isaliteral{3A}{\isacharcolon}}\ A{\isaliteral{2C}{\isacharcomma}}\ b{\isaliteral{3A}{\isacharcolon}}\ A{\isaliteral{29}{\isacharparenright}}{\isaliteral{28}{\isacharparenleft}}implicit\ A{\isaliteral{3A}{\isacharcolon}}\ semigroup{\isaliteral{5B}{\isacharbrackleft}}A{\isaliteral{5D}{\isacharbrackright}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{3A}{\isacharcolon}}\ A\ {\isaliteral{3D}{\isacharequal}}\isanewline -\ \ A{\isaliteral{2E}{\isachardot}}{\isaliteral{60}{\isacharbackquote}}Example{\isaliteral{2E}{\isachardot}}mult{\isaliteral{60}{\isacharbackquote}}{\isaliteral{28}{\isacharparenleft}}a{\isaliteral{2C}{\isacharcomma}}\ b{\isaliteral{29}{\isacharparenright}}\isanewline -\isanewline -trait\ monoidl{\isaliteral{5B}{\isacharbrackleft}}A{\isaliteral{5D}{\isacharbrackright}}\ extends\ semigroup{\isaliteral{5B}{\isacharbrackleft}}A{\isaliteral{5D}{\isacharbrackright}}\ {\isaliteral{7B}{\isacharbraceleft}}\isanewline -\ \ val\ {\isaliteral{60}{\isacharbackquote}}Example{\isaliteral{2E}{\isachardot}}neutral{\isaliteral{60}{\isacharbackquote}}{\isaliteral{3A}{\isacharcolon}}\ A\isanewline -{\isaliteral{7D}{\isacharbraceright}}\isanewline -def\ neutral{\isaliteral{5B}{\isacharbrackleft}}A{\isaliteral{5D}{\isacharbrackright}}{\isaliteral{28}{\isacharparenleft}}implicit\ A{\isaliteral{3A}{\isacharcolon}}\ monoidl{\isaliteral{5B}{\isacharbrackleft}}A{\isaliteral{5D}{\isacharbrackright}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{3A}{\isacharcolon}}\ A\ {\isaliteral{3D}{\isacharequal}}\ A{\isaliteral{2E}{\isachardot}}{\isaliteral{60}{\isacharbackquote}}Example{\isaliteral{2E}{\isachardot}}neutral{\isaliteral{60}{\isacharbackquote}}\isanewline -\isanewline -trait\ monoid{\isaliteral{5B}{\isacharbrackleft}}A{\isaliteral{5D}{\isacharbrackright}}\ extends\ monoidl{\isaliteral{5B}{\isacharbrackleft}}A{\isaliteral{5D}{\isacharbrackright}}\ {\isaliteral{7B}{\isacharbraceleft}}\isanewline -{\isaliteral{7D}{\isacharbraceright}}\isanewline -\isanewline -trait\ group{\isaliteral{5B}{\isacharbrackleft}}A{\isaliteral{5D}{\isacharbrackright}}\ extends\ monoid{\isaliteral{5B}{\isacharbrackleft}}A{\isaliteral{5D}{\isacharbrackright}}\ {\isaliteral{7B}{\isacharbraceleft}}\isanewline -\ \ val\ {\isaliteral{60}{\isacharbackquote}}Example{\isaliteral{2E}{\isachardot}}inverse{\isaliteral{60}{\isacharbackquote}}{\isaliteral{3A}{\isacharcolon}}\ A\ {\isaliteral{3D}{\isacharequal}}{\isaliteral{3E}{\isachargreater}}\ A\isanewline -{\isaliteral{7D}{\isacharbraceright}}\isanewline -def\ inverse{\isaliteral{5B}{\isacharbrackleft}}A{\isaliteral{5D}{\isacharbrackright}}{\isaliteral{28}{\isacharparenleft}}a{\isaliteral{3A}{\isacharcolon}}\ A{\isaliteral{29}{\isacharparenright}}{\isaliteral{28}{\isacharparenleft}}implicit\ A{\isaliteral{3A}{\isacharcolon}}\ group{\isaliteral{5B}{\isacharbrackleft}}A{\isaliteral{5D}{\isacharbrackright}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{3A}{\isacharcolon}}\ A\ {\isaliteral{3D}{\isacharequal}}\ A{\isaliteral{2E}{\isachardot}}{\isaliteral{60}{\isacharbackquote}}Example{\isaliteral{2E}{\isachardot}}inverse{\isaliteral{60}{\isacharbackquote}}{\isaliteral{28}{\isacharparenleft}}a{\isaliteral{29}{\isacharparenright}}\isanewline -\isanewline -def\ neutral{\isaliteral{5F}{\isacharunderscore}}int{\isaliteral{3A}{\isacharcolon}}\ BigInt\ {\isaliteral{3D}{\isacharequal}}\ BigInt{\isaliteral{28}{\isacharparenleft}}{\isadigit{0}}{\isaliteral{29}{\isacharparenright}}\isanewline -\isanewline -def\ inverse{\isaliteral{5F}{\isacharunderscore}}int{\isaliteral{28}{\isacharparenleft}}i{\isaliteral{3A}{\isacharcolon}}\ BigInt{\isaliteral{29}{\isacharparenright}}{\isaliteral{3A}{\isacharcolon}}\ BigInt\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{2D}{\isacharminus}}\ i{\isaliteral{29}{\isacharparenright}}\isanewline -\isanewline -def\ mult{\isaliteral{5F}{\isacharunderscore}}int{\isaliteral{28}{\isacharparenleft}}i{\isaliteral{3A}{\isacharcolon}}\ BigInt{\isaliteral{2C}{\isacharcomma}}\ j{\isaliteral{3A}{\isacharcolon}}\ BigInt{\isaliteral{29}{\isacharparenright}}{\isaliteral{3A}{\isacharcolon}}\ BigInt\ {\isaliteral{3D}{\isacharequal}}\ i\ {\isaliteral{2B}{\isacharplus}}\ j\isanewline -\isanewline -implicit\ def\ semigroup{\isaliteral{5F}{\isacharunderscore}}int{\isaliteral{3A}{\isacharcolon}}\ semigroup{\isaliteral{5B}{\isacharbrackleft}}BigInt{\isaliteral{5D}{\isacharbrackright}}\ {\isaliteral{3D}{\isacharequal}}\ new\ semigroup{\isaliteral{5B}{\isacharbrackleft}}BigInt{\isaliteral{5D}{\isacharbrackright}}\ {\isaliteral{7B}{\isacharbraceleft}}\isanewline -\ \ val\ {\isaliteral{60}{\isacharbackquote}}Example{\isaliteral{2E}{\isachardot}}mult{\isaliteral{60}{\isacharbackquote}}\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{28}{\isacharparenleft}}a{\isaliteral{3A}{\isacharcolon}}\ BigInt{\isaliteral{2C}{\isacharcomma}}\ b{\isaliteral{3A}{\isacharcolon}}\ BigInt{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}{\isaliteral{3E}{\isachargreater}}\ mult{\isaliteral{5F}{\isacharunderscore}}int{\isaliteral{28}{\isacharparenleft}}a{\isaliteral{2C}{\isacharcomma}}\ b{\isaliteral{29}{\isacharparenright}}\isanewline -{\isaliteral{7D}{\isacharbraceright}}\isanewline -\isanewline -implicit\ def\ monoidl{\isaliteral{5F}{\isacharunderscore}}int{\isaliteral{3A}{\isacharcolon}}\ monoidl{\isaliteral{5B}{\isacharbrackleft}}BigInt{\isaliteral{5D}{\isacharbrackright}}\ {\isaliteral{3D}{\isacharequal}}\ new\ monoidl{\isaliteral{5B}{\isacharbrackleft}}BigInt{\isaliteral{5D}{\isacharbrackright}}\ {\isaliteral{7B}{\isacharbraceleft}}\isanewline -\ \ val\ {\isaliteral{60}{\isacharbackquote}}Example{\isaliteral{2E}{\isachardot}}neutral{\isaliteral{60}{\isacharbackquote}}\ {\isaliteral{3D}{\isacharequal}}\ neutral{\isaliteral{5F}{\isacharunderscore}}int\isanewline -\ \ val\ {\isaliteral{60}{\isacharbackquote}}Example{\isaliteral{2E}{\isachardot}}mult{\isaliteral{60}{\isacharbackquote}}\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{28}{\isacharparenleft}}a{\isaliteral{3A}{\isacharcolon}}\ BigInt{\isaliteral{2C}{\isacharcomma}}\ b{\isaliteral{3A}{\isacharcolon}}\ BigInt{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}{\isaliteral{3E}{\isachargreater}}\ mult{\isaliteral{5F}{\isacharunderscore}}int{\isaliteral{28}{\isacharparenleft}}a{\isaliteral{2C}{\isacharcomma}}\ b{\isaliteral{29}{\isacharparenright}}\isanewline -{\isaliteral{7D}{\isacharbraceright}}\isanewline -\isanewline -implicit\ def\ monoid{\isaliteral{5F}{\isacharunderscore}}int{\isaliteral{3A}{\isacharcolon}}\ monoid{\isaliteral{5B}{\isacharbrackleft}}BigInt{\isaliteral{5D}{\isacharbrackright}}\ {\isaliteral{3D}{\isacharequal}}\ new\ monoid{\isaliteral{5B}{\isacharbrackleft}}BigInt{\isaliteral{5D}{\isacharbrackright}}\ {\isaliteral{7B}{\isacharbraceleft}}\isanewline -\ \ val\ {\isaliteral{60}{\isacharbackquote}}Example{\isaliteral{2E}{\isachardot}}neutral{\isaliteral{60}{\isacharbackquote}}\ {\isaliteral{3D}{\isacharequal}}\ neutral{\isaliteral{5F}{\isacharunderscore}}int\isanewline -\ \ val\ {\isaliteral{60}{\isacharbackquote}}Example{\isaliteral{2E}{\isachardot}}mult{\isaliteral{60}{\isacharbackquote}}\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{28}{\isacharparenleft}}a{\isaliteral{3A}{\isacharcolon}}\ BigInt{\isaliteral{2C}{\isacharcomma}}\ b{\isaliteral{3A}{\isacharcolon}}\ BigInt{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}{\isaliteral{3E}{\isachargreater}}\ mult{\isaliteral{5F}{\isacharunderscore}}int{\isaliteral{28}{\isacharparenleft}}a{\isaliteral{2C}{\isacharcomma}}\ b{\isaliteral{29}{\isacharparenright}}\isanewline -{\isaliteral{7D}{\isacharbraceright}}\isanewline -\isanewline -implicit\ def\ group{\isaliteral{5F}{\isacharunderscore}}int{\isaliteral{3A}{\isacharcolon}}\ group{\isaliteral{5B}{\isacharbrackleft}}BigInt{\isaliteral{5D}{\isacharbrackright}}\ {\isaliteral{3D}{\isacharequal}}\ new\ group{\isaliteral{5B}{\isacharbrackleft}}BigInt{\isaliteral{5D}{\isacharbrackright}}\ {\isaliteral{7B}{\isacharbraceleft}}\isanewline -\ \ val\ {\isaliteral{60}{\isacharbackquote}}Example{\isaliteral{2E}{\isachardot}}inverse{\isaliteral{60}{\isacharbackquote}}\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{28}{\isacharparenleft}}a{\isaliteral{3A}{\isacharcolon}}\ BigInt{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}{\isaliteral{3E}{\isachargreater}}\ inverse{\isaliteral{5F}{\isacharunderscore}}int{\isaliteral{28}{\isacharparenleft}}a{\isaliteral{29}{\isacharparenright}}\isanewline -\ \ val\ {\isaliteral{60}{\isacharbackquote}}Example{\isaliteral{2E}{\isachardot}}neutral{\isaliteral{60}{\isacharbackquote}}\ {\isaliteral{3D}{\isacharequal}}\ neutral{\isaliteral{5F}{\isacharunderscore}}int\isanewline -\ \ val\ {\isaliteral{60}{\isacharbackquote}}Example{\isaliteral{2E}{\isachardot}}mult{\isaliteral{60}{\isacharbackquote}}\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{28}{\isacharparenleft}}a{\isaliteral{3A}{\isacharcolon}}\ BigInt{\isaliteral{2C}{\isacharcomma}}\ b{\isaliteral{3A}{\isacharcolon}}\ BigInt{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}{\isaliteral{3E}{\isachargreater}}\ mult{\isaliteral{5F}{\isacharunderscore}}int{\isaliteral{28}{\isacharparenleft}}a{\isaliteral{2C}{\isacharcomma}}\ b{\isaliteral{29}{\isacharparenright}}\isanewline -{\isaliteral{7D}{\isacharbraceright}}\isanewline -\isanewline -def\ pow{\isaliteral{5F}{\isacharunderscore}}nat{\isaliteral{5B}{\isacharbrackleft}}A\ {\isaliteral{3A}{\isacharcolon}}\ monoid{\isaliteral{5D}{\isacharbrackright}}{\isaliteral{28}{\isacharparenleft}}xa{\isadigit{0}}{\isaliteral{3A}{\isacharcolon}}\ nat{\isaliteral{2C}{\isacharcomma}}\ x{\isaliteral{3A}{\isacharcolon}}\ A{\isaliteral{29}{\isacharparenright}}{\isaliteral{3A}{\isacharcolon}}\ A\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{28}{\isacharparenleft}}xa{\isadigit{0}}{\isaliteral{2C}{\isacharcomma}}\ x{\isaliteral{29}{\isacharparenright}}\ match\ {\isaliteral{7B}{\isacharbraceleft}}\isanewline -\ \ case\ {\isaliteral{28}{\isacharparenleft}}Zero{\isaliteral{5F}{\isacharunderscore}}nat{\isaliteral{2C}{\isacharcomma}}\ x{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}{\isaliteral{3E}{\isachargreater}}\ neutral{\isaliteral{5B}{\isacharbrackleft}}A{\isaliteral{5D}{\isacharbrackright}}\isanewline -\ \ case\ {\isaliteral{28}{\isacharparenleft}}Suc{\isaliteral{28}{\isacharparenleft}}n{\isaliteral{29}{\isacharparenright}}{\isaliteral{2C}{\isacharcomma}}\ x{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}{\isaliteral{3E}{\isachargreater}}\ mult{\isaliteral{5B}{\isacharbrackleft}}A{\isaliteral{5D}{\isacharbrackright}}{\isaliteral{28}{\isacharparenleft}}x{\isaliteral{2C}{\isacharcomma}}\ pow{\isaliteral{5F}{\isacharunderscore}}nat{\isaliteral{5B}{\isacharbrackleft}}A{\isaliteral{5D}{\isacharbrackright}}{\isaliteral{28}{\isacharparenleft}}n{\isaliteral{2C}{\isacharcomma}}\ x{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}\isanewline -{\isaliteral{7D}{\isacharbraceright}}\isanewline -\isanewline -def\ pow{\isaliteral{5F}{\isacharunderscore}}int{\isaliteral{5B}{\isacharbrackleft}}A\ {\isaliteral{3A}{\isacharcolon}}\ group{\isaliteral{5D}{\isacharbrackright}}{\isaliteral{28}{\isacharparenleft}}k{\isaliteral{3A}{\isacharcolon}}\ BigInt{\isaliteral{2C}{\isacharcomma}}\ x{\isaliteral{3A}{\isacharcolon}}\ A{\isaliteral{29}{\isacharparenright}}{\isaliteral{3A}{\isacharcolon}}\ A\ {\isaliteral{3D}{\isacharequal}}\isanewline -\ \ {\isaliteral{28}{\isacharparenleft}}if\ {\isaliteral{28}{\isacharparenleft}}BigInt{\isaliteral{28}{\isacharparenleft}}{\isadigit{0}}{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3C}{\isacharless}}{\isaliteral{3D}{\isacharequal}}\ k{\isaliteral{29}{\isacharparenright}}\ pow{\isaliteral{5F}{\isacharunderscore}}nat{\isaliteral{5B}{\isacharbrackleft}}A{\isaliteral{5D}{\isacharbrackright}}{\isaliteral{28}{\isacharparenleft}}nat{\isaliteral{28}{\isacharparenleft}}k{\isaliteral{29}{\isacharparenright}}{\isaliteral{2C}{\isacharcomma}}\ x{\isaliteral{29}{\isacharparenright}}\isanewline -\ \ \ \ else\ inverse{\isaliteral{5B}{\isacharbrackleft}}A{\isaliteral{5D}{\isacharbrackright}}{\isaliteral{28}{\isacharparenleft}}pow{\isaliteral{5F}{\isacharunderscore}}nat{\isaliteral{5B}{\isacharbrackleft}}A{\isaliteral{5D}{\isacharbrackright}}{\isaliteral{28}{\isacharparenleft}}nat{\isaliteral{28}{\isacharparenleft}}{\isaliteral{28}{\isacharparenleft}}{\isaliteral{2D}{\isacharminus}}\ k{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{2C}{\isacharcomma}}\ x{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}\isanewline -\isanewline -def\ example{\isaliteral{3A}{\isacharcolon}}\ BigInt\ {\isaliteral{3D}{\isacharequal}}\ pow{\isaliteral{5F}{\isacharunderscore}}int{\isaliteral{5B}{\isacharbrackleft}}BigInt{\isaliteral{5D}{\isacharbrackright}}{\isaliteral{28}{\isacharparenleft}}BigInt{\isaliteral{28}{\isacharparenleft}}{\isadigit{1}}{\isadigit{0}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{2C}{\isacharcomma}}\ BigInt{\isaliteral{28}{\isacharparenleft}}{\isaliteral{2D}{\isacharminus}}\ {\isadigit{2}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}\isanewline -\isanewline -{\isaliteral{7D}{\isacharbraceright}}\ {\isaliteral{2F}{\isacharslash}}{\isaliteral{2A}{\isacharasterisk}}\ object\ Example\ {\isaliteral{2A}{\isacharasterisk}}{\isaliteral{2F}{\isacharslash}}\isanewline% -\end{isamarkuptext}% -\isamarkuptrue% -% -\endisatagquotetypewriter -{\isafoldquotetypewriter}% -% -\isadelimquotetypewriter -% -\endisadelimquotetypewriter -% -\isamarkupsubsection{Inspecting the type class universe% -} -\isamarkuptrue% -% -\begin{isamarkuptext}% -To facilitate orientation in complex subclass structures, two - diagnostics commands are provided: - - \begin{description} - - \item[\hyperlink{command.print-classes}{\mbox{\isa{\isacommand{print{\isaliteral{5F}{\isacharunderscore}}classes}}}}] print a list of all classes - together with associated operations etc. - - \item[\hyperlink{command.class-deps}{\mbox{\isa{\isacommand{class{\isaliteral{5F}{\isacharunderscore}}deps}}}}] visualizes the subclass relation - between all classes as a Hasse diagram. - - \end{description}% -\end{isamarkuptext}% -\isamarkuptrue% -% -\isadelimtheory -% -\endisadelimtheory -% -\isatagtheory -\isacommand{end}\isamarkupfalse% -% -\endisatagtheory -{\isafoldtheory}% -% -\isadelimtheory -% -\endisadelimtheory -\isanewline -\end{isabellebody}% -%%% Local Variables: -%%% mode: latex -%%% TeX-master: "root" -%%% End: