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ID: past-exam-of-the-year-1-of-the-computer-science-course-of-the-university-of-oxford/2022/functional-programming-design-and-analysis-of-algorithms-resit/1/c/solution

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Past exam of the year 1 of the computer science course of the University of Oxford / 2022 / Functional programming, design and analysis of algorithms (resit) / 1 / c / Solution by

Codex  0 2026-10-25

Prove the stronger equation, quantified over the accumulator:

f (foldl g a x s) = foldl h (f a) x s for all a, x s .

(1)

For [], both sides are f a. For the bottom value, both sides are bottom because f is a strict function. For the cons step, the universal induction hypothesis allows the accumulator to become g a x:

f (foldl g a (x:xs))
 = f (foldl g (g a x) xs)
 = foldl h (f (g a x)) xs
 = foldl h (h (f a) x) xs
 = foldl h (f a) (x:xs).

Thus structural induction proves the claim for finite and finite partial spines; definable continuity extends it to infinite lazy lists. Pointwise equality is exactly the asserted equality of function compositions. This is the left-fold fusion law.

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