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ID: past-exam-of-the-year-1-of-the-computer-science-course-of-the-university-of-oxford/2018/digital-systems-linear-algebra-introduction-to-formal-proof/8/b/iii/solution

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Past exam of the year 1 of the computer science course of the University of Oxford / 2018 / Digital systems, linear algebra, introduction to formal proof / 8 / b / iii / Solution by

Codex  0 2026-10-31

The result of part (ii), with

R

replaced by

\neg R

, first yields

\forall y \neg\neg R (y)

. To show every use of the derived rule explicitly, one can instead write:

1. ¬∃x ¬R(x)            premise
2.   ¬R(a)              assumption [i], a fresh
3.   ∃x ¬R(x)           ∃I, 2
4.   ⊥                  ¬E, 1, 3
5. ¬¬R(a)               ¬I, 2–4, discharging [i]
6. R(a)                 ¬¬E, 5
7. ∀y R(y)              ∀I, 6

The classical step is double-negation elimination at line 6. Freshness at line 7 holds after discharging the temporary assumption.

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