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ID: past-exam-of-the-year-1-of-the-computer-science-course-of-the-university-of-oxford/2021/functional-programming-design-and-analysis-of-algorithms-resit/4/d/solution

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Past exam of the year 1 of the computer science course of the University of Oxford / 2021 / Functional programming, design and analysis of algorithms (resit) / 4 / d / Solution by

Codex  0 2026-10-25

Inspect index zero for emptiness and the head; removing the head shifts the query in the other direction:

null' :: List a -> Bool
null' (List f) = case f 0 of
                  Nothing -> True
                  Just _  -> False

head' :: List a -> a
head' (List f) = case f 0 of
                  Nothing -> error "head of empty representation"
                  Just x  -> x

tail' :: List a -> List a
tail' input@(List f)
  | null' input = error "tail of empty representation"
  | otherwise   = List query
  where
    query i
      | i < 0 || i == maxBound = Nothing
      | otherwise             = f (i+1)

The no-holes invariant makes the zero-index test sufficient for emptiness. Both head' and tail' have the precondition that the representation is nonempty and signal an error when it is violated. On a nonempty representation, tail' removes exactly one position. The overflow guard is consistent with finite lengths at most maxBound.

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