Processing math: 96%

Sequence Notation

Notation	Equivalent SML with `SEQUENCE` library
$S[i]$	`nth` $S$ $i$
$\|S\|$	`length` $S$
$\langle \; \rangle$	`empty ()`
$\langle v \rangle$	`singleton` $v$
$\langle i, \ldots, j \rangle$	`tabulate (fn k =>` $i$ `+ k) (` $j$ `-` $i$ `+ 1)`
$S \langle i, \ldots, j \rangle$	`subseq` $S$ `(` $i$ `,` $j$ `-` $i$ `+ 1)`
$\langle e : p \in S \rangle$	`map (fn` $p$ `=>` $e$ `)` $S$
$\langle e : 0 \leq i < n \rangle$	`tabulate (fn` $i$ `=>` $e$ `)` $n$
$\langle p \in S \;\|\; e \rangle$	`filter (fn` $p$ `=>` $e$ `)` $S$
$\langle e_1 : p \in S \;\|\; e_2\rangle$	`map (fn` $p$ `=>` $e_1$ `) (filter (fn` $p$ `=>` $e_2$ `)` $S$ `)`
$\langle e : p_1 \in S_1,\; p_2 \in S_2 \rangle$	`flatten (map (fn` $p_1$ `=> map (fn` $p_2$ `=>` $e$ `)` $S_2$ `)` $S_1$ `)`
$\langle e : 0 \leq i < n,\; 0 \leq j < i \rangle$	`flatten (tabulate (fn` $i$ `=> tabulate (fn` $j$ `=>` $e$ `)` $i$ `)` $n$ `)`
$\sum_{p \in S} e$	`reduce add 0 (map (fn` $p$ `=>` $e$ `)` $S$ `)`
$\sum_{i = k}^n e$	`reduce add 0 (map (fn` $i$ `=>` $e$ `)` $\langle k, \ldots, n\rangle$ `)`

The meaning of add and 0 in the reduce will depend on the type (e.g. Int.+ and 0 for integers or Real.+ and 0.0 for reals). We can also replace $\sum$ with other operations such as $\min$ and $\max$ with the implied semantics.