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Funcons-beta : Multisets.cbs | PLAIN | PDF


Multisets (bags)

\[\begin{align*} [ \ \KEY{Type} \quad & \NAMEREF{multisets} \\ \KEY{Funcon} \quad & \NAMEREF{multiset} \\ \KEY{Funcon} \quad & \NAMEREF{multiset-elements} \\ \KEY{Funcon} \quad & \NAMEREF{multiset-occurrences} \\ \KEY{Funcon} \quad & \NAMEREF{multiset-insert} \\ \KEY{Funcon} \quad & \NAMEREF{multiset-delete} \\ \KEY{Funcon} \quad & \NAMEREF{is-submultiset} \ ] \end{align*}\] \[\begin{align*} \KEY{Meta-variables} \quad & \VAR{GT} <: \NAMEHYPER{../..}{Value-Types}{ground-values} \end{align*}\] \[\begin{align*} \KEY{Built-in Type} \quad & \NAMEDECL{multisets}( \VAR{GT} ) \end{align*}\]

\(\SHADE{\NAMEREF{multisets} ( \VAR{GT} )}\) is the type of possibly-empty finite multisets of elements of \(\SHADE{\VAR{GT}}\).

\[\begin{align*} \KEY{Built-in Funcon} \quad & \NAMEDECL{multiset}( \_ : ( \VAR{GT} )\STAR) : \TO \NAMEREF{multisets} ( \VAR{GT} ) \end{align*}\]

Note that \(\SHADE{\NAMEREF{multiset} ( \cdots )}\) is not a constructor operation. The order of argument values is ignored, but duplicates are significant, e.g., \(\SHADE{\NAMEREF{multiset} ( 1, 2, 2 )}\) is equivalent to \(\SHADE{\NAMEREF{multiset} ( 2, 1, 2 )}\), but not to \(\SHADE{\NAMEREF{multiset} ( 1, 2 )}\) or \(\SHADE{\NAMEREF{multiset} ( 2, 1 )}\).

\[\begin{align*} \KEY{Built-in Funcon} \quad & \NAMEDECL{multiset-elements}( \_ : \NAMEREF{multisets} ( \VAR{GT} )) : \TO ( \VAR{GT} )\STAR \end{align*}\]

For each multiset \(\SHADE{\VAR{MS}}\), the sequence of values \(\SHADE{\VAR{V}\STAR}\) returned by \(\SHADE{\NAMEREF{multiset-elements} ( \VAR{MS} )}\) contains each element of \(\SHADE{\VAR{MS}}\) the same number of times as \(\SHADE{\VAR{MS}}\) does. The order of the values in \(\SHADE{\VAR{V}\STAR}\) is unspecified, and may vary between multisets.

\[\begin{align*} \KEY{Assert} \quad & \NAMEREF{multiset} ( \NAMEREF{multiset-elements} ( \VAR{S} ) ) == \VAR{S} \end{align*}\] \[\begin{align*} \KEY{Built-in Funcon} \quad & \NAMEDECL{multiset-occurrences}( \_ : \VAR{GT}, \_ : \NAMEREF{multisets} ( \VAR{GT} )) : \TO \NAMEHYPER{../../Primitive}{Integers}{natural-numbers} \end{align*}\]

\(\SHADE{\NAMEREF{multiset-occurrences} ( \VAR{GV}, \VAR{MS} )}\) returns the number of occurrences of \(\SHADE{\VAR{GV}}\) in \(\SHADE{\VAR{MS}}\).

\[\begin{align*} \KEY{Built-in Funcon} \quad & \NAMEDECL{multiset-insert}( \_ : \VAR{GT}, \_ : \NAMEHYPER{../../Primitive}{Integers}{natural-numbers}, \_ : \NAMEREF{multisets} ( \VAR{GT} )) : \TO \NAMEREF{multisets} ( \VAR{GT} ) \end{align*}\]

\(\SHADE{\NAMEREF{multiset-insert} ( \VAR{GV}, \VAR{N}, \VAR{MS} )}\) returns the multiset that differs from \(\SHADE{\VAR{MS}}\) by containing \(\SHADE{\VAR{N}}\) more copies of \(\SHADE{\VAR{GV}}\).

\[\begin{align*} \KEY{Built-in Funcon} \quad & \NAMEDECL{multiset-delete}( \_ : \NAMEREF{multisets} ( \VAR{GT} ), \_ : \VAR{GT}, \_ : \NAMEHYPER{../../Primitive}{Integers}{natural-numbers}) : \TO \NAMEREF{multisets} ( \VAR{GT} ) \end{align*}\]

\(\SHADE{\NAMEREF{multiset-delete} ( \VAR{MS}, \VAR{GV}, \VAR{N} )}\) removes \(\SHADE{\VAR{N}}\) copies of \(\SHADE{\VAR{V}}\) from the multiset \(\SHADE{\VAR{MS}}\), or all copies of \(\SHADE{\VAR{GV}}\) if there are fewer than \(\SHADE{\VAR{N}}\) in \(\SHADE{\VAR{MS}}\).

\[\begin{align*} \KEY{Built-in Funcon} \quad & \NAMEDECL{is-submultiset}( \_ : \NAMEREF{multisets} ( \VAR{GT} ), \_ : \NAMEREF{multisets} ( \VAR{GT} )) : \TO \NAMEHYPER{../../Primitive}{Booleans}{booleans} \end{align*}\]

\(\SHADE{\NAMEREF{is-submultiset} ( \VAR{MS}\SUB{1}, \VAR{MS}\SUB{2} )}\) tests whether every element of \(\SHADE{\VAR{MS}\SUB{1}}\) has equal or fewer occurrences in \(\SHADE{\VAR{MS}\SUB{1}}\) than in \(\SHADE{\VAR{MS}\SUB{2}}\).