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Funcons-beta : Lists.cbs | PLAIN | PDF

Lists

\[\begin{align*} [ \ \KEY{Datatype} \quad & \NAMEREF{lists} \\ \KEY{Funcon} \quad & \NAMEREF{list} \\ \KEY{Funcon} \quad & \NAMEREF{list-elements} \\ \KEY{Funcon} \quad & \NAMEREF{list-nil} \\ \KEY{Alias} \quad & \NAMEREF{nil} \\ \KEY{Funcon} \quad & \NAMEREF{list-cons} \\ \KEY{Alias} \quad & \NAMEREF{cons} \\ \KEY{Funcon} \quad & \NAMEREF{list-head} \\ \KEY{Alias} \quad & \NAMEREF{head} \\ \KEY{Funcon} \quad & \NAMEREF{list-tail} \\ \KEY{Alias} \quad & \NAMEREF{tail} \\ \KEY{Funcon} \quad & \NAMEREF{list-length} \\ \KEY{Funcon} \quad & \NAMEREF{list-append} \ ] \end{align*}\] \[\begin{align*} \KEY{Meta-variables} \quad & \VAR{T} <: \NAMEHYPER{../..}{Value-Types}{values} \end{align*}\] \[\begin{align*} \KEY{Datatype} \quad \NAMEDECL{lists}( \VAR{T} ) \ ::= \ & \NAMEDECL{list}( \_ : ( \VAR{T} )\STAR) \end{align*}\]

\(\SHADE{\NAMEREF{lists} ( \VAR{T} )}\) is the type of possibly-empty finite lists \(\SHADE{[ \VAR{V}\SUB{1}, \cdots, \VAR{V}\SUB{n} ]}\) where \(\SHADE{\VAR{V}\SUB{1} : \VAR{T}}\), …, \(\SHADE{\VAR{V}\SUB{n} : \VAR{T}}\).

N.B. \(\SHADE{[ \VAR{T} ]}\) is always a single list value, and not interpreted as the type \(\SHADE{\NAMEREF{lists} ( \VAR{T} )}\).

The notation \(\SHADE{[ \VAR{V}\SUB{1}, \cdots, \VAR{V}\SUB{n} ]}\) for \(\SHADE{\NAMEREF{list} ( \VAR{V}\SUB{1}, \cdots, \VAR{V}\SUB{n} )}\) is built-in.

\[\begin{align*} \KEY{Assert} \quad & [ \VAR{V}\STAR : \NAMEHYPER{../..}{Value-Types}{values}\STAR ] == \NAMEREF{list} ( \VAR{V}\STAR ) \end{align*}\] \[\begin{align*} \KEY{Funcon} \quad & \NAMEDECL{list-elements}( \_ : \NAMEREF{lists} ( \VAR{T} )) : \TO ( \VAR{T} )\STAR \\ \KEY{Rule} \quad & \NAMEREF{list-elements} ( \NAMEREF{list} ( \VAR{V}\STAR : \NAMEHYPER{../..}{Value-Types}{values}\STAR ) ) \leadsto \VAR{V}\STAR \end{align*}\] \[\begin{align*} \KEY{Funcon} \quad & \NAMEDECL{list-nil} : \TO \NAMEREF{lists} ( \_ ) \\&\quad \leadsto [ \ ] \\ \KEY{Alias} \quad & \NAMEDECL{nil} = \NAMEREF{list-nil} \end{align*}\] \[\begin{align*} \KEY{Funcon} \quad & \NAMEDECL{list-cons}( \_ : \VAR{T}, \_ : \NAMEREF{lists} ( \VAR{T} )) : \TO \NAMEREF{lists} ( \VAR{T} ) \\ \KEY{Alias} \quad & \NAMEDECL{cons} = \NAMEREF{list-cons} \end{align*}\] \[\begin{align*} \KEY{Rule} \quad & \NAMEREF{list-cons} ( \VAR{V} : \NAMEHYPER{../..}{Value-Types}{values}, [ \VAR{V}\STAR : \NAMEHYPER{../..}{Value-Types}{values}\STAR ] ) \leadsto [ \VAR{V}, \VAR{V}\STAR ] \end{align*}\] \[\begin{align*} \KEY{Funcon} \quad & \NAMEDECL{list-head}( \_ : \NAMEREF{lists} ( \VAR{T} )) : \TO ( \VAR{T} )\QUERY \\ \KEY{Alias} \quad & \NAMEDECL{head} = \NAMEREF{list-head} \end{align*}\] \[\begin{align*} \KEY{Rule} \quad & \NAMEREF{list-head} \ [ \VAR{V} : \NAMEHYPER{../..}{Value-Types}{values}, \_\STAR : \NAMEHYPER{../..}{Value-Types}{values}\STAR ] \leadsto \VAR{V} \\ \KEY{Rule} \quad & \NAMEREF{list-head} \ [ \ ] \leadsto ( \ ) \end{align*}\] \[\begin{align*} \KEY{Funcon} \quad & \NAMEDECL{list-tail}( \_ : \NAMEREF{lists} ( \VAR{T} )) : \TO ( \NAMEREF{lists} ( \VAR{T} ) )\QUERY \\ \KEY{Alias} \quad & \NAMEDECL{tail} = \NAMEREF{list-tail} \end{align*}\] \[\begin{align*} \KEY{Rule} \quad & \NAMEREF{list-tail} \ [ \_ : \NAMEHYPER{../..}{Value-Types}{values}, \VAR{V}\STAR : \NAMEHYPER{../..}{Value-Types}{values}\STAR ] \leadsto [ \VAR{V}\STAR ] \\ \KEY{Rule} \quad & \NAMEREF{list-tail} \ [ \ ] \leadsto ( \ ) \end{align*}\] \[\begin{align*} \KEY{Funcon} \quad & \NAMEDECL{list-length}( \_ : \NAMEREF{lists} ( \VAR{T} )) : \TO \NAMEHYPER{../../Primitive}{Integers}{natural-numbers} \\ \KEY{Rule} \quad & \NAMEREF{list-length} \ [ \VAR{V}\STAR : \NAMEHYPER{../..}{Value-Types}{values}\STAR ] \leadsto \NAMEHYPER{../.}{Sequences}{length} ( \VAR{V}\STAR ) \end{align*}\] \[\begin{align*} \KEY{Funcon} \quad & \NAMEDECL{list-append}( \_ : ( \NAMEREF{lists} ( \VAR{T} ) )\STAR) : \TO \NAMEREF{lists} ( \VAR{T} ) \\ \KEY{Rule} \quad & \NAMEREF{list-append} ( [ \VAR{V}\SUB{1}\STAR : \NAMEHYPER{../..}{Value-Types}{values}\STAR ], [ \VAR{V}\SUB{2}\STAR : \NAMEHYPER{../..}{Value-Types}{values}\STAR ] ) \leadsto [ \VAR{V}\SUB{1}\STAR, \VAR{V}\SUB{2}\STAR ] \\ \KEY{Rule} \quad & \NAMEREF{list-append} ( \VAR{L}\SUB{1} : \NAMEREF{lists} ( \_ ), \VAR{L}\SUB{2} : \NAMEREF{lists} ( \_ ), \VAR{L}\SUB{3} : \NAMEREF{lists} ( \_ ), \VAR{L}\STAR : ( \NAMEREF{lists} ( \_ ) )\STAR ) \leadsto \\&\quad \NAMEREF{list-append} ( \VAR{L}\SUB{1}, \NAMEREF{list-append} ( \VAR{L}\SUB{2}, \VAR{L}\SUB{3}, \VAR{L}\STAR ) ) \\ \KEY{Rule} \quad & \NAMEREF{list-append} ( \ ) \leadsto [ \ ] \\ \KEY{Rule} \quad & \NAMEREF{list-append} ( \VAR{L} : \NAMEREF{lists} ( \_ ) ) \leadsto \VAR{L} \end{align*}\]

Datatypes of infinite and possibly-infinite lists can be specified as algebraic datatypes using abstractions.

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