Homework #5 Solution Sketches

1)

fun fibhelp (1, a1, a2) = a1
|   fibhelp (i, a1, a2) =   
       fibhelp (i-1, a1+a2, a1);

fun fib (0) = 0
|   fib (n) = fibhelp (n, 1, 0);

2)

fun remove (nil, i) = nil
|   remove (x::xs, 1) = xs
|   remove (x::xs, i) = x:: remove (xs, i-1);

3)

exception EmptyList;

fun lreduce (F, nil) = raise EmptyList
|   lreduce (F, [a]) = a
|   lreduce (F, x::y::remaining) = lreduce (F, F (x, y)::remaining);

lreduce can be used to sum a list of numbers as follows:

lreduce ((op +) [1,2,3];

4)

a)

(*
The key observation here is that
1-2+3-4 can be written as: 1-(2-(3-4)))
With this observation, we can write the desired function as:
*)

val asum = foldr (op -) 0;

b)

fun xor2 (0, 0) = 0
| xor2 (0, 1) = 1
| xor2 (1, 0) = 1
| xor2 (1, 1) = 0;

fun xor (list) = foldr xor2 0 list;