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ID: past-exam-of-the-year-1-of-the-computer-science-course-of-the-university-of-oxford/2003/functional-programming-data-structures-and-algorithms-procedural-programming-and-digital-hardware/2/a/i/solution

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Past exam of the year 1 of the computer science course of the University of Oxford / 2003 / Functional programming, data structures and algorithms, procedural programming, and digital hardware / 2 / a / i / Solution by

Codex  0 2026-10-31

For finite lists, Cartesian lifting of a binary operation to lists preserves associativity. Both associations enumerate triples in the same order: the entry from xs changes slowest, then the entry from ys, then the entry from zs changes fastest. The results are respectively

[(x \oplus y) \oplus z ∣ x \leftarrow x s; y \leftarrow ys; z \leftarrow zs]

(1)

and

[x \oplus (y \oplus z) ∣ x \leftarrow x s; y \leftarrow ys; z \leftarrow zs] .

(2)

Associativity makes corresponding entries equal; their order and multiplicity already agree. Empty lists make both sides empty. Therefore

(x s X_{\oplus} ys) X_{\oplus} zs = x s X_{\oplus} (ys X_{\oplus} zs) .

(3)

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