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\( % cbs-katex.sty % \newcommand{\STYLE}[2]{\htmlClass{cbs-#1}{#2}} \newcommand{\DECL}[3]{\htmlId{#1:#2}{#3}} \newcommand{\REF}[3]{\href{###1:#2}{#3}} \newcommand{\HYPER}[5]{\href{#1/#2/index.html###3:#4}{#5}} % \SHADE{MATH} can be defined to produce a shaded background to highlight % inline MATH in running text: \newcommand{\SHADE}[1]{#1} % \KEY{TEXT}, \STRING{TEXT}, \ATOM{TEXT}, \LEX{TEXT} can be used in math mode: \newcommand{\KEY}[1]{\textsf{\textit{\STYLE{Key}{#1}}}} \newcommand{\STRING}[1]{\textsf{``\texttt{#1}''}} \newcommand{\ATOM}[1]{\textsf{`\texttt{#1}'}} \newcommand{\LEX}[1]{\textsf{\STYLE{Key}{`}\texttt{#1}\STYLE{Key}{'}}} % The following commands produce ASCII characters that are treated specially by LaTeX: \newcommand{\HASH}{\char`\#} \newcommand{\DOLLAR}{\char`\$} \newcommand{\PERCENT}{\char`\%} \newcommand{\AMPERSAND}{\char`\&} \newcommand{\APOSTROPHE}{\char`\'} \newcommand{\BACKSLASH}{\char`\\} \newcommand{\CARET}{\char`\^} \newcommand{\UNDERSCORE}{\char`\_} \newcommand{\GRAVE}{\char`\`} \newcommand{\LEFTBRACE}{\char`\{} \newcommand{\RIGHTBRACE}{\char`\}} \newcommand{\TILDE}{\textasciitilde} % {\char`\~} % \NAME{name} highlights the name; % \NAMEDECL{name} declares Name.name as the target of links to name; % \NAMEREF{name} links name to the target Name.name in the current file; % \NAMEHYPER{url}{file}{name} links name to Name.name at url/file/file.pdf. % Similarly for \VAR{partvariable}, \SYN{syntaxname}, \SEM{semanticsName}, % and \SECT{sectionnumber} % The kerns in \SUB and \VAR avoid overlaps with primes: \newcommand{\SUB}[1]{_{\kern-2mu\STYLE{PartVariable}{\textsf{#1}}}} % PLAIN \newcommand{\VAR}[1]{\STYLE{PartVariable}{\textsf{\textit{#1}\kern2mu}}} \newcommand{\NAME}[1]{\STYLE{Name}{\textsf{#1}}} \newcommand{\SYN}[1]{\STYLE{SyntaxName}{\textsf{#1}}} \newcommand{\SEM}[1]{\STYLE{SemanticsName}{\textsf{#1}}} \newcommand{\SECT}[1]{\STYLE{SectionNumber}{\textsf{#1}}} % DECL \newcommand{\VARDECL}[1]{\DECL{PartVariable}{#1}{\VAR{#1}}} \newcommand{\NAMEDECL}[1]{\DECL{Name}{#1}{\NAME{#1}}} \newcommand{\SYNDECL}[1]{\DECL{SyntaxName}{#1}{\SYN{#1}}} \newcommand{\SEMDECL}[1]{\DECL{SemanticsName}{#1}{\SEM{#1}}} \newcommand{\SECTDECL}[1]{\DECL{SectionNumber}{#1}{\textsf{#1}}} % REF \newcommand{\VARREF}[1]{\REF{PartVariable}{#1}{\VAR{#1}}} \newcommand{\NAMEREF}[1]{\REF{Name}{#1}{\NAME{#1}}} \newcommand{\SYNREF}[1]{\REF{SyntaxName}{#1}{\SYN{#1}}} \newcommand{\SEMREF}[1]{\REF{SemanticsName}{#1}{\SEM{#1}}} \newcommand{\SECTREF}[1]{\REF{SectionNumber}{#1}{\SECT{#1}}} % HYPER \newcommand{\VARHYPER}[3]{\HYPER{#1}{#2}{PartVariable}{#3}{\VAR{#3}}} \newcommand{\NAMEHYPER}[3]{\HYPER{#1}{#2}{Name}{#3}{\NAME{#3}}} \newcommand{\SYNHYPER}[3]{\HYPER{#1}{#2}{SyntaxName}{#3}{\SYN{#3}}} \newcommand{\SEMHYPER}[3]{\HYPER{#1}{#2}{SemanticsName}{#3}{\SEM{#3}}} \newcommand{\SECTHYPER}[3]{\HYPER{#1}{#2}{SectionNumber}{#3}{\SECT{#3}}} % \LEFTPHRASE MATH \RIGHTPHRASE produces [[ MATH ]] with proper brackets: \newcommand{\LEFTPHRASE}{\llbracket} \newcommand{\RIGHTPHRASE}{\rrbracket} % \LEFTGROUP MATH \RIGHTGROUP produces ( MATH ) where the parentheses are % highlighted the same as keywords: \newcommand{\LEFTGROUP}{\STYLE{Key}{(}} \newcommand{\RIGHTGROUP}{\STYLE{Key}{)}} % MATH\PLUS produces a superscript + % MATH\STAR produces a superscript * % MATH\QUERY produces a superscript ? \newcommand{\PLUS}{{}^{\texttt{+}}} \newcommand{\STAR}{{}^{\texttt{*}}} \newcommand{\QUERY}{{}^{\texttt{?}}} % \RULE{& PREMISE \\ & ...}{& FORMULA ... \\ & ...} produces an inference rule % with separately aligned premises and conclusion % PREMISE % ... % ----------- % FORMULA ... % ... \newcommand{\RULE}[2] {\frac{\begin{aligned}#1\end{aligned}}{\begin{aligned}#2\end{aligned}}} % \AXIOM{& FORMULA ... \\ & ...} produces an aligned formula % % FORMULA ... % ... \newcommand{\AXIOM}[1]{\begin{aligned}#1\end{aligned}} % \TO TYPE produces => TYPE \newcommand{\TO}{\mathop{\Rightarrow}} % TERM \TRANS TERM produces TERM ---> TERM \newcommand{\TRANS}{\longrightarrow} % TERM \xrightarrow{LABEL} TERM puts the label above the long arrow % \)

Funcons-beta : Throwing.cbs | PLAIN | PDF

Throwing

\[\begin{align*} [ \ \KEY{Datatype} \quad & \NAMEREF{throwing} \\ \KEY{Funcon} \quad & \NAMEREF{thrown} \\ \KEY{Funcon} \quad & \NAMEREF{finalise-throwing} \\ \KEY{Funcon} \quad & \NAMEREF{throw} \\ \KEY{Funcon} \quad & \NAMEREF{handle-thrown} \\ \KEY{Funcon} \quad & \NAMEREF{handle-recursively} \\ \KEY{Funcon} \quad & \NAMEREF{catch-else-throw} \ ] \end{align*}\] \[\begin{align*} \KEY{Meta-variables} \quad & \VAR{R}, \VAR{S}, \VAR{T}, \VAR{T}', \VAR{T}'' <: \NAMEHYPER{../../../Values}{Value-Types}{values} \end{align*}\] \[\begin{align*} \KEY{Datatype} \quad \NAMEDECL{throwing} \ ::= \ & \NAMEDECL{thrown}( \_ : \NAMEHYPER{../../../Values}{Value-Types}{values}) \end{align*}\]

\(\SHADE{\NAMEREF{thrown} ( \VAR{V} )}\) is a reason for abrupt termination.

\[\begin{align*} \KEY{Funcon} \quad & \NAMEDECL{finalise-throwing}( \VAR{X} : \TO \VAR{T}) : \TO \VAR{T} \mid \NAMEHYPER{../../../Values/Primitive}{Null}{null-type} \\&\quad \leadsto \NAMEHYPER{../.}{Abrupting}{finalise-abrupting} ( \VAR{X} ) \end{align*}\]

\(\SHADE{\NAMEREF{finalise-throwing} ( \VAR{X} )}\) handles abrupt termination of \(\SHADE{\VAR{X}}\) due to executing \(\SHADE{\NAMEREF{throw} ( \VAR{V} )}\).

\[\begin{align*} \KEY{Funcon} \quad & \NAMEDECL{throw}( \VAR{V} : \VAR{T}) : \TO \NAMEHYPER{../../../Values}{Value-Types}{empty-type} \\&\quad \leadsto \NAMEHYPER{../.}{Abrupting}{abrupt} ( \NAMEREF{thrown} ( \VAR{V} ) ) \end{align*}\]

\(\SHADE{\NAMEREF{throw} ( \VAR{V} )}\) abruptly terminates all enclosing computations uTil it is handled.

\[\begin{align*} \KEY{Funcon} \quad & \NAMEDECL{handle-thrown}( \_ : \VAR{T}' \TO \VAR{T}, \_ : \VAR{T}'' \TO \VAR{T}) : \VAR{T}' \TO \VAR{T} \end{align*}\]

\(\SHADE{\NAMEREF{handle-thrown} ( \VAR{X}, \VAR{Y} )}\) first evaluates \(\SHADE{\VAR{X}}\). If \(\SHADE{\VAR{X}}\) terminates normally with value \(\SHADE{\VAR{V}}\), then \(\SHADE{\VAR{V}}\) is returned and \(\SHADE{\VAR{Y}}\) is ignored. If \(\SHADE{\VAR{X}}\) terminates abruptly with a thrown eTity having value \(\SHADE{\VAR{V}}\), then \(\SHADE{\VAR{Y}}\) is executed with \(\SHADE{\VAR{V}}\) as \(\SHADE{\NAMEHYPER{../../Normal}{Giving}{given}}\) value.

\(\SHADE{\NAMEREF{handle-thrown} ( \VAR{X}, \VAR{Y} )}\) is associative, with \(\SHADE{\NAMEREF{throw} ( \NAMEHYPER{../../Normal}{Giving}{given} )}\) as unit. \(\SHADE{\NAMEREF{handle-thrown} ( \VAR{X}, \NAMEHYPER{../.}{Failing}{else} ( \VAR{Y}, \NAMEREF{throw} ( \NAMEHYPER{../../Normal}{Giving}{given} ) ) )}\) ensures that if \(\SHADE{\VAR{Y}}\) fails, the thrown value is re-thrown.

\[\begin{align*} \KEY{Rule} \quad & \RULE{ & \VAR{X} \xrightarrow{\NAMEHYPER{../.}{Abrupting}{abrupted}( \ )}_{} \VAR{X}' }{ & \NAMEREF{handle-thrown} ( \VAR{X}, \VAR{Y} ) \xrightarrow{\NAMEHYPER{../.}{Abrupting}{abrupted}( \ )}_{} \NAMEREF{handle-thrown} ( \VAR{X}', \VAR{Y} ) } \\ \KEY{Rule} \quad & \RULE{ & \VAR{X} \xrightarrow{\NAMEHYPER{../.}{Abrupting}{abrupted}( \NAMEREF{thrown} ( \VAR{V}'' : \NAMEHYPER{../../../Values}{Value-Types}{values} ) )}_{} \VAR{X}' }{ & \NAMEREF{handle-thrown} ( \VAR{X}, \VAR{Y} ) \xrightarrow{\NAMEHYPER{../.}{Abrupting}{abrupted}( \ )}_{} \NAMEHYPER{../../Normal}{Giving}{give} ( \VAR{V}'', \VAR{Y} ) } \\ \KEY{Rule} \quad & \RULE{ & \VAR{X} \xrightarrow{\NAMEHYPER{../.}{Abrupting}{abrupted}( \VAR{V}' : \mathop{\sim} \NAMEREF{throwing} )}_{} \VAR{X}' }{ & \NAMEREF{handle-thrown} ( \VAR{X}, \VAR{Y} ) \xrightarrow{\NAMEHYPER{../.}{Abrupting}{abrupted}( \VAR{V}' )}_{} \NAMEREF{handle-thrown} ( \VAR{X}', \VAR{Y} ) } \\ \KEY{Rule} \quad & \NAMEREF{handle-thrown} ( \VAR{V} : \VAR{T}, \VAR{Y} ) \leadsto \VAR{V} \end{align*}\] \[\begin{align*} \KEY{Funcon} \quad & \NAMEDECL{handle-recursively}( \VAR{X} : \VAR{S} \TO \VAR{T}, \VAR{Y} : \VAR{R} \TO \VAR{T}) : \VAR{S} \TO \VAR{T} \\&\quad \leadsto \NAMEREF{handle-thrown} ( \VAR{X}, \NAMEHYPER{../.}{Failing}{else} ( \NAMEREF{handle-recursively} ( \VAR{Y}, \VAR{Y} ), \NAMEREF{throw} ( \NAMEHYPER{../../Normal}{Giving}{given} ) ) ) \end{align*}\]

\(\SHADE{\NAMEREF{handle-recursively} ( \VAR{X}, \VAR{Y} )}\) behaves similarly to \(\SHADE{\NAMEREF{handle-thrown} ( \VAR{X}, \VAR{Y} )}\), except that another copy of the handler attempts to handle any values thrown by \(\SHADE{\VAR{Y}}\). Thus, many thrown values may get handled by the same handler.

\[\begin{align*} \KEY{Funcon} \quad & \NAMEDECL{catch-else-throw}( \VAR{P} : \NAMEHYPER{../../../Values}{Value-Types}{values}, \VAR{Y} : \TO \VAR{T}) : \TO \VAR{T} \\&\quad \leadsto \NAMEHYPER{../.}{Failing}{else} ( \NAMEHYPER{../../../Values/Abstraction}{Patterns}{case-match} ( \VAR{P}, \VAR{Y} ), \NAMEREF{throw} ( \NAMEHYPER{../../Normal}{Giving}{given} ) ) \end{align*}\]

\(\SHADE{\NAMEREF{handle-thrown} ( \VAR{X}, \NAMEREF{catch-else-throw} ( \VAR{P}, \VAR{Y} ) )}\) handles those values thrown by \(\SHADE{\VAR{X}}\) that match pattern \(\SHADE{\VAR{P}}\). Other thrown values are re-thrown.

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