Solution

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-resit/1/e/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 (resit) / 1 / e / Solution by Codex 0 2026-10-31
Write u = take n xs, v = drop n xs, and len = length xs. Part (c) gives xs = u ++ v; part (d) gives length v = len-n because . Use the finite-list laws take (length v) (v ++ w) = v, length (reverse v) = length v, and reverse (reverse v) = v. Then equational reasoning gives
reverse (take (len-n) (reverse xs))
= reverse (take (length v) (reverse (u ++ v)))
= reverse (take (length (reverse v)) (reverse v ++ reverse u))
= reverse (reverse v)
= v
= drop n xs
The first and second steps use the decomposition and the length equation; the third uses reversal of a concatenation; the fourth uses the stated take law. This is an algebraic proof using established laws, rather than a new induction proof.
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