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ID: past-exam-of-the-year-1-of-the-computer-science-course-of-the-university-of-oxford/2016/functional-programming-design-and-analysis-of-algorithms/3/b/solution

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Past exam of the year 1 of the computer science course of the University of Oxford / 2016 / Functional programming, design and analysis of algorithms / 3 / b / Solution by

Codex  0 2026-10-31

The constructors ZigType and ZagType record which of the two alternating positions contains the last element. The tail types swap the two type parameters, so the return type of lastZag swaps them back:

lastZig :: Zig a b -> ZigOrZagType a b
lastZig Nil = error "lastZig: empty input"
lastZig (Cins x Nal) = ZigType x
lastZig (Cins _ zs) = lastZag zs

lastZag :: Zag a b -> ZigOrZagType b a
lastZag Nal = error "lastZag: empty input"
lastZag (Cans x Nil) = ZagType x
lastZag (Cans _ zs) = lastZig zs

For example, lastZig passes a tail of type Zag b a to lastZag, whose result then has type ZigOrZagType a b, exactly as required. Conversely, lastZag passes a Zig b a tail to lastZig. A singleton returns its element in the appropriate constructor; a longer input discards one element and uses mutual recursion on the shorter tail. This proves correctness by structural induction on the finite alternating sequence. The three source examples give ZigType 2, ZagType "D", and ZigType True.

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