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Funcons-beta : Functions.cbs | PLAIN | PDF
[ Datatype functions Funcon function Funcon apply Funcon supply Funcon compose Funcon uncurry Funcon curry Funcon partial-apply ] \begin{align*}
[ \
\KEY{Datatype} \quad & \NAMEREF{functions} \\
\KEY{Funcon} \quad & \NAMEREF{function} \\
\KEY{Funcon} \quad & \NAMEREF{apply} \\
\KEY{Funcon} \quad & \NAMEREF{supply} \\
\KEY{Funcon} \quad & \NAMEREF{compose} \\
\KEY{Funcon} \quad & \NAMEREF{uncurry} \\
\KEY{Funcon} \quad & \NAMEREF{curry} \\
\KEY{Funcon} \quad & \NAMEREF{partial-apply}
\ ]
\end{align*} [ Datatype Funcon Funcon Funcon Funcon Funcon Funcon Funcon functions function apply supply compose uncurry curry partial-apply ] Meta-variables T , T ′ , T 1 , T 2 < : values \begin{align*}
\KEY{Meta-variables} \quad
& \VAR{T}, \VAR{T}', \VAR{T}\SUB{1}, \VAR{T}\SUB{2} <: \NAMEHYPER{../..}{Value-Types}{values}
\end{align*} Meta-variables T , T ′ , T 1 , T 2 <: values Datatype functions ( T , T ′ ) : : = function ( A : abstractions ( T ⇒ T ′ ) ) \begin{align*}
\KEY{Datatype} \quad
\NAMEDECL{functions}(
\VAR{T} , \VAR{T}' )
\ ::= \ & \NAMEDECL{function}(
\VAR{A} : \NAMEHYPER{../.}{Generic}{abstractions}
( \VAR{T} \TO \VAR{T}' ))
\end{align*} Datatype functions ( T , T ′ ) ::= function ( A : abstractions ( T ⇒ T ′ )) functions ( T , T ′ ) \SHADE{\NAMEREF{functions}
( \VAR{T},
\VAR{T}' )} functions ( T , T ′ ) T \SHADE{\VAR{T}} T T ′ \SHADE{\VAR{T}'} T ′ function ( abstraction ( X ) ) \SHADE{\NAMEREF{function}
( \NAMEHYPER{../.}{Generic}{abstraction}
( \VAR{X} ) )} function ( abstraction ( X )) function ( closure ( X ) ) \SHADE{\NAMEREF{function}
( \NAMEHYPER{../.}{Generic}{closure}
( \VAR{X} ) )} function ( closure ( X ))
Funcon apply ( _ : functions ( T , T ′ ) , _ : T ) : ⇒ T ′ \begin{align*}
\KEY{Funcon} \quad
& \NAMEDECL{apply}(
\_ : \NAMEREF{functions}
( \VAR{T},
\VAR{T}' ), \_ : \VAR{T})
: \TO \VAR{T}'
\end{align*} Funcon apply ( _ : functions ( T , T ′ ) , _ : T ) : ⇒ T ′ apply ( F , V ) \SHADE{\NAMEREF{apply}
( \VAR{F},
\VAR{V} )} apply ( F , V ) F \SHADE{\VAR{F}} F V \SHADE{\VAR{V}} V
Rule apply ( function ( abstraction ( X ) ) , V : T ) ⇝ give ( V , X ) \begin{align*}
\KEY{Rule} \quad
& \NAMEREF{apply}
( \NAMEREF{function}
( \NAMEHYPER{../.}{Generic}{abstraction}
( \VAR{X} ) ),
\VAR{V} : \VAR{T} ) \leadsto
\NAMEHYPER{../../../Computations/Normal}{Giving}{give}
( \VAR{V},
\VAR{X} )
\end{align*} Rule apply ( function ( abstraction ( X )) , V : T ) ⇝ give ( V , X ) Funcon supply ( _ : functions ( T , T ′ ) , _ : T ) : ⇒ thunks ( T ′ ) \begin{align*}
\KEY{Funcon} \quad
& \NAMEDECL{supply}(
\_ : \NAMEREF{functions}
( \VAR{T},
\VAR{T}' ), \_ : \VAR{T})
: \TO \NAMEHYPER{../.}{Thunks}{thunks}
( \VAR{T}' )
\end{align*} Funcon supply ( _ : functions ( T , T ′ ) , _ : T ) : ⇒ thunks ( T ′ ) supply ( F , V ) \SHADE{\NAMEREF{supply}
( \VAR{F},
\VAR{V} )} supply ( F , V )
Rule supply ( function ( abstraction ( X ) ) , V : T ) ⇝ thunk ( abstraction ( give ( V , X ) ) ) \begin{align*}
\KEY{Rule} \quad
& \NAMEREF{supply}
( \NAMEREF{function}
( \NAMEHYPER{../.}{Generic}{abstraction}
( \VAR{X} ) ),
\VAR{V} : \VAR{T} ) \leadsto
\NAMEHYPER{../.}{Thunks}{thunk}
( \NAMEHYPER{../.}{Generic}{abstraction}
( \NAMEHYPER{../../../Computations/Normal}{Giving}{give}
( \VAR{V},
\VAR{X} ) ) )
\end{align*} Rule supply ( function ( abstraction ( X )) , V : T ) ⇝ thunk ( abstraction ( give ( V , X ))) Funcon compose ( _ : functions ( T 2 , T ′ ) , _ : functions ( T 1 , T 2 ) ) : ⇒ functions ( T 1 , T ′ ) \begin{align*}
\KEY{Funcon} \quad
& \NAMEDECL{compose}(
\_ : \NAMEREF{functions}
( \VAR{T}\SUB{2},
\VAR{T}' ), \_ : \NAMEREF{functions}
( \VAR{T}\SUB{1},
\VAR{T}\SUB{2} ))
: \TO \NAMEREF{functions}
( \VAR{T}\SUB{1},
\VAR{T}' )
\end{align*} Funcon compose ( _ : functions ( T 2 , T ′ ) , _ : functions ( T 1 , T 2 )) : ⇒ functions ( T 1 , T ′ ) compose ( F 2 , F 1 ) \SHADE{\NAMEREF{compose}
( \VAR{F}\SUB{2},
\VAR{F}\SUB{1} )} compose ( F 2 , F 1 ) F 1 \SHADE{\VAR{F}\SUB{1}} F 1 F 2 \SHADE{\VAR{F}\SUB{2}} F 2 F 1 \SHADE{\VAR{F}\SUB{1}} F 1
Rule compose ( function ( abstraction ( Y ) ) , function ( abstraction ( X ) ) ) ⇝ function ( abstraction ( give ( X , Y ) ) ) \begin{align*}
\KEY{Rule} \quad
& \NAMEREF{compose}
( \NAMEREF{function}
( \NAMEHYPER{../.}{Generic}{abstraction}
( \VAR{Y} ) ),
\NAMEREF{function}
( \NAMEHYPER{../.}{Generic}{abstraction}
( \VAR{X} ) ) ) \leadsto \\&\quad
\NAMEREF{function}
( \NAMEHYPER{../.}{Generic}{abstraction}
( \NAMEHYPER{../../../Computations/Normal}{Giving}{give}
( \VAR{X},
\VAR{Y} ) ) )
\end{align*} Rule compose ( function ( abstraction ( Y )) , function ( abstraction ( X ))) ⇝ function ( abstraction ( give ( X , Y ))) Funcon uncurry ( F : functions ( T 1 , functions ( T 2 , T ′ ) ) ) : ⇒ functions ( tuples ( T 1 , T 2 ) , T ′ ) ⇝ function ( abstraction ( apply ( apply ( F , checked index ( 1 , tuple-elements given ) ) , checked index ( 2 , tuple-elements given ) ) ) ) \begin{align*}
\KEY{Funcon} \quad
& \NAMEDECL{uncurry}(
\VAR{F} : \NAMEREF{functions}
( \VAR{T}\SUB{1},
\NAMEREF{functions}
( \VAR{T}\SUB{2},
\VAR{T}' ) )) \\&\quad
: \TO \NAMEREF{functions}
( \NAMEHYPER{../../Composite}{Tuples}{tuples}
( \VAR{T}\SUB{1},
\VAR{T}\SUB{2} ),
\VAR{T}' ) \\&\quad
\leadsto \NAMEREF{function}
( \\&\quad\quad\quad\quad \NAMEHYPER{../.}{Generic}{abstraction}
( \\&\quad\quad\quad\quad\quad \NAMEREF{apply}
( \\&\quad\quad\quad\quad\quad\quad \NAMEREF{apply}
( \VAR{F},
\NAMEHYPER{../../../Computations/Abnormal}{Failing}{checked} \
\NAMEHYPER{../../Composite}{Sequences}{index}
( 1,
\NAMEHYPER{../../Composite}{Tuples}{tuple-elements} \
\NAMEHYPER{../../../Computations/Normal}{Giving}{given} ) ), \\&\quad\quad\quad\quad\quad\quad
\NAMEHYPER{../../../Computations/Abnormal}{Failing}{checked} \
\NAMEHYPER{../../Composite}{Sequences}{index}
( 2,
\NAMEHYPER{../../Composite}{Tuples}{tuple-elements} \
\NAMEHYPER{../../../Computations/Normal}{Giving}{given} ) ) ) )
\end{align*} Funcon uncurry ( F : functions ( T 1 , functions ( T 2 , T ′ ))) : ⇒ functions ( tuples ( T 1 , T 2 ) , T ′ ) ⇝ function ( abstraction ( apply ( apply ( F , checked index ( 1 , tuple-elements given )) , checked index ( 2 , tuple-elements given )))) uncurry ( F ) \SHADE{\NAMEREF{uncurry}
( \VAR{F} )} uncurry ( F ) F \SHADE{\VAR{F}} F
Funcon curry ( F : functions ( tuples ( T 1 , T 2 ) , T ′ ) ) : ⇒ functions ( T 1 , functions ( T 2 , T ′ ) ) ⇝ function ( abstraction ( partial-apply ( F , given ) ) ) \begin{align*}
\KEY{Funcon} \quad
& \NAMEDECL{curry}(
\VAR{F} : \NAMEREF{functions}
( \NAMEHYPER{../../Composite}{Tuples}{tuples}
( \VAR{T}\SUB{1},
\VAR{T}\SUB{2} ),
\VAR{T}' ))
: \TO \NAMEREF{functions}
( \VAR{T}\SUB{1},
\NAMEREF{functions}
( \VAR{T}\SUB{2},
\VAR{T}' ) ) \\&\quad
\leadsto \NAMEREF{function}
( \NAMEHYPER{../.}{Generic}{abstraction}
( \NAMEREF{partial-apply}
( \VAR{F},
\NAMEHYPER{../../../Computations/Normal}{Giving}{given} ) ) )
\end{align*} Funcon curry ( F : functions ( tuples ( T 1 , T 2 ) , T ′ )) : ⇒ functions ( T 1 , functions ( T 2 , T ′ )) ⇝ function ( abstraction ( partial-apply ( F , given ))) curry ( F ) \SHADE{\NAMEREF{curry}
( \VAR{F} )} curry ( F ) F \SHADE{\VAR{F}} F
Funcon partial-apply ( F : functions ( tuples ( T 1 , T 2 ) , T ′ ) , V : T 1 ) : ⇒ functions ( T 2 , T ′ ) ⇝ function ( abstraction ( apply ( F , tuple ( V , given ) ) ) ) \begin{align*}
\KEY{Funcon} \quad
& \NAMEDECL{partial-apply}(
\VAR{F} : \NAMEREF{functions}
( \NAMEHYPER{../../Composite}{Tuples}{tuples}
( \VAR{T}\SUB{1},
\VAR{T}\SUB{2} ),
\VAR{T}' ), \VAR{V} : \VAR{T}\SUB{1})
: \TO \NAMEREF{functions}
( \VAR{T}\SUB{2},
\VAR{T}' ) \\&\quad
\leadsto \NAMEREF{function}
( \NAMEHYPER{../.}{Generic}{abstraction}
( \NAMEREF{apply}
( \VAR{F},
\NAMEHYPER{../../Composite}{Tuples}{tuple}
( \VAR{V},
\NAMEHYPER{../../../Computations/Normal}{Giving}{given} ) ) ) )
\end{align*} Funcon partial-apply ( F : functions ( tuples ( T 1 , T 2 ) , T ′ ) , V : T 1 ) : ⇒ functions ( T 2 , T ′ ) ⇝ function ( abstraction ( apply ( F , tuple ( V , given )))) partial-apply ( F , V ) \SHADE{\NAMEREF{partial-apply}
( \VAR{F},
\VAR{V} )} partial-apply ( F , V ) F \SHADE{\VAR{F}} F
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