 Solution

ID: past-exam-of-the-year-1-of-the-computer-science-course-of-the-university-of-oxford/2017/functional-programming-design-and-analysis-of-algorithms/2/b/solution

Tie the three scaled streams back to the same shared lazy list:

hamming :: [Integer]
hamming = 1 : merge (map (2*) hamming)
                    (merge (map (3*) hamming) (map (5*) hamming))

The resulting values are precisely the Hamming numbers

2^{i} 3^{j} 5^{k}

for nonnegative integers

i, j, k

. Every noninitial value has a smaller predecessor obtained by removing a factor, so the recursive demand can be supplied from earlier cells. This is the productivity argument for a sorted multiplicative closure stream. The merge algorithm removes duplicate representations such as

2 \cdot 3 = 3 \cdot 2

, retaining strict order and every generated value.

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