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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 : Failing.cbs | PLAIN | PDF

Failing

\[\begin{align*} [ \ \KEY{Datatype} \quad & \NAMEREF{failing} \\ \KEY{Funcon} \quad & \NAMEREF{failed} \\ \KEY{Funcon} \quad & \NAMEREF{finalise-failing} \\ \KEY{Funcon} \quad & \NAMEREF{fail} \\ \KEY{Funcon} \quad & \NAMEREF{else} \\ \KEY{Funcon} \quad & \NAMEREF{else-choice} \\ \KEY{Funcon} \quad & \NAMEREF{checked} \\ \KEY{Funcon} \quad & \NAMEREF{check-true} \ ] \end{align*}\] \[\begin{align*} \KEY{Meta-variables} \quad & \VAR{T} <: \NAMEHYPER{../../../Values}{Value-Types}{values} \end{align*}\] \[\begin{align*} \KEY{Datatype} \quad \NAMEDECL{failing} \ ::= \ & \NAMEDECL{failed} \end{align*}\]

\(\SHADE{\NAMEREF{failed}}\) is a reason for abrupt termination.

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

\(\SHADE{\NAMEREF{finalise-failing} ( \VAR{X} )}\) handles abrupt termination of \(\SHADE{\VAR{X}}\) due to executing \(\SHADE{\NAMEREF{fail}}\).

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

\(\SHADE{\NAMEREF{fail}}\) abruptly terminates all enclosing computations until it is handled.

\[\begin{align*} \KEY{Funcon} \quad & \NAMEDECL{else}( \_ : \TO \VAR{T}, \_ : ( \TO \VAR{T} )\PLUS) : \TO \VAR{T} \end{align*}\]

\(\SHADE{\NAMEREF{else} ( \VAR{X}\SUB{1}, \VAR{X}\SUB{2}, \cdots )}\) executes the arguments in turn until either some \(\SHADE{\VAR{Xi}}\) does not fail, or all arguments \(\SHADE{\VAR{Xi}}\) have been executed. The last argument executed determines the result. \(\SHADE{\NAMEREF{else} ( \VAR{X}, \VAR{Y} )}\) is associative, with unit \(\SHADE{\NAMEREF{fail}}\).

\[\begin{align*} \KEY{Rule} \quad & \RULE{ & \VAR{X} \xrightarrow{\NAMEHYPER{../.}{Abrupting}{abrupted}( \ )}_{} \VAR{X}' }{ & \NAMEREF{else} ( \VAR{X}, \VAR{Y} ) \xrightarrow{\NAMEHYPER{../.}{Abrupting}{abrupted}( \ )}_{} \NAMEREF{else} ( \VAR{X}', \VAR{Y} ) } \\ \KEY{Rule} \quad & \RULE{ & \VAR{X} \xrightarrow{\NAMEHYPER{../.}{Abrupting}{abrupted}( \NAMEREF{failed} )}_{} \_ }{ & \NAMEREF{else} ( \VAR{X}, \VAR{Y} ) \xrightarrow{\NAMEHYPER{../.}{Abrupting}{abrupted}( \ )}_{} \VAR{Y} } \\ \KEY{Rule} \quad & \RULE{ & \VAR{X} \xrightarrow{\NAMEHYPER{../.}{Abrupting}{abrupted}( \VAR{V} : \mathop{\sim} \NAMEREF{failing} )}_{} \VAR{X}' }{ & \NAMEREF{else} ( \VAR{X}, \VAR{Y} ) \xrightarrow{\NAMEHYPER{../.}{Abrupting}{abrupted}( \VAR{V} )}_{} \NAMEREF{else} ( \VAR{X}', \VAR{Y} ) } \\ \KEY{Rule} \quad & \NAMEREF{else} ( \VAR{V} : \VAR{T}, \VAR{Y} ) \leadsto \VAR{V} \\ \KEY{Rule} \quad & \NAMEREF{else} ( \VAR{X}, \VAR{Y}, \VAR{Z}\PLUS ) \leadsto \NAMEREF{else} ( \VAR{X}, \NAMEREF{else} ( \VAR{Y}, \VAR{Z}\PLUS ) ) \end{align*}\] \[\begin{align*} \KEY{Funcon} \quad & \NAMEDECL{else-choice}( \_ : ( \TO \VAR{T} )\PLUS) : \TO \VAR{T} \end{align*}\]

\(\SHADE{\NAMEREF{else-choice} ( \VAR{X}, \cdots )}\) executes the arguments in any order until either some \(\SHADE{\VAR{Xi}}\) does not fail, or all arguments \(\SHADE{\VAR{Xi}}\) have been executed. The last argument executed determines the result. \(\SHADE{\NAMEREF{else} ( \VAR{X}, \VAR{Y} )}\) is associative and commutative, with unit \(\SHADE{\NAMEREF{fail}}\).

\[\begin{align*} \KEY{Rule} \quad & \NAMEREF{else-choice} ( \VAR{W}\STAR, \VAR{X}, \VAR{Y}, \VAR{Z}\STAR ) \leadsto \\&\quad \NAMEHYPER{../../Normal}{Flowing}{choice} ( \\&\quad\quad \NAMEREF{else} ( \\&\quad\quad\quad \VAR{X}, \\&\quad\quad\quad \NAMEREF{else-choice} ( \VAR{W}\STAR, \VAR{Y}, \VAR{Z}\STAR ), \\&\quad\quad\quad \NAMEREF{else} ( \VAR{Y}, \NAMEREF{else-choice} ( \VAR{W}\STAR, \VAR{X}, \VAR{Z}\STAR ) ) ) ) \\ \KEY{Rule} \quad & \NAMEREF{else-choice} ( \VAR{X} ) \leadsto \VAR{X} \end{align*}\] \[\begin{align*} \KEY{Funcon} \quad & \NAMEDECL{check-true}( \_ : \NAMEHYPER{../../../Values/Primitive}{Booleans}{booleans}) : \TO \NAMEHYPER{../../../Values/Primitive}{Null}{null-type} \\ \KEY{Alias} \quad & \NAMEDECL{check} = \NAMEREF{check-true} \end{align*}\]

\(\SHADE{\NAMEREF{check-true} ( \VAR{X} )}\) terminates normally if the value computed by \(\SHADE{\VAR{X}}\) is \(\SHADE{\NAMEHYPER{../../../Values/Primitive}{Booleans}{true}}\), and fails if it is \(\SHADE{\NAMEHYPER{../../../Values/Primitive}{Booleans}{false}}\).

\[\begin{align*} \KEY{Rule} \quad & \NAMEREF{check-true} ( \NAMEHYPER{../../../Values/Primitive}{Booleans}{true} ) \leadsto \NAMEHYPER{../../../Values/Primitive}{Null}{null-value} \\ \KEY{Rule} \quad & \NAMEREF{check-true} ( \NAMEHYPER{../../../Values/Primitive}{Booleans}{false} ) \leadsto \NAMEREF{fail} \end{align*}\] \[\begin{align*} \KEY{Funcon} \quad & \NAMEDECL{checked}( \_ : ( \VAR{T} )\QUERY) : \TO \VAR{T} \end{align*}\]

\(\SHADE{\NAMEREF{checked} ( \VAR{X} )}\) fails when \(\SHADE{\VAR{X}}\) gives the empty sequence of values \(\SHADE{( \ )}\), representing that an optional value has not been computed. It otherwise computes the same as \(\SHADE{\VAR{X}}\).

\[\begin{align*} \KEY{Rule} \quad & \NAMEREF{checked} ( \VAR{V} : \VAR{T} ) \leadsto \VAR{V} \\ \KEY{Rule} \quad & \NAMEREF{checked} ( \ ) \leadsto \NAMEREF{fail} \end{align*}\]

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