A higher-order function is a function that take another function as its argument. For instance, the following defined function rtfind is a higher-order one:

fun rtfind (f: int -> int): int = let fun loop ( f: int -> int, n: int ) : int = if f(n) = 0 then n else loop (f, n+1) // end of [loop] in loop (f, 0) end // end of [rtfind]

Higher-order functions can greatly facilitate code reuse, and I now present a simple example to illustrate this point. The following defined functions sum and prod compute the sum and product of the integers ranging from 1 to a given natural number, respectively:

fun sum (n: int): int = if n > 0 then sum (n-1) + n else 0 fun prod (n: int): int = if n > 0 then prod (n-1) * n else 1

// fun ifold (n: int, f: (int, int) -> int, ini: int): int = if n > 0 then f (ifold (n-1, f, ini), n) else ini // fun sum (n: int): int = ifold (n, lam (res, x) => res + x, 0) fun prod (n: int): int = ifold (n, lam (res, x) => res * x, 1) //

fun sqrmodsum (n: int, d: int): int = ifold (n, lam (res, x) => if x mod d = 0 then res + x * x else res, 0) // end of [sqrmodsum]

// fun ifold2 ( n: int, f: (int, int) -<cloref1> int, ini: int ) : int = if n > 0 then f (ifold2 (n-1, f, ini), n) else ini // end of [ifold2] // fun sqrmodsum (n: int, d: int): int = ifold2(n, lam (res, x) => if x mod d = 0 then res + x * x else res, 0) // end of [sqrmodsum] //

As more features of ATS are introduced, higher-order functions will become even more effective in facilitating code reuse.