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ID: past-exam-of-the-year-1-of-the-computer-science-course-of-the-university-of-oxford/2016/digital-systems-linear-algebra-introduction-to-formal-proof-resit/8/c/solution

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Past exam of the year 1 of the computer science course of the University of Oxford / 2016 / Digital systems, linear algebra, introduction to formal proof (resit) / 8 / c / Solution by

Codex  0 2026-10-31

The proposed proof is invalid: its existential-elimination step exports

Q (x_{0})

while the witness name

x_{0}

still occurs free in that conclusion. Introducing an existential quantifier later does not retrospectively repair that scope error.

Introduce the existential quantifier before discharging the witness box:

1  ∀x (P(x) → Q(x))   premise
2  ∃x P(x)            premise
3  | P(a)             assumption, a fresh
4  | P(a) → Q(a)      ∀-elim 1
5  | Q(a)             →-elim 4,3
6  | ∃x Q(x)          ∃-intro 5
7  ∃x Q(x)            ∃-elim 2,3–6

The conclusion of existential elimination now has no free occurrence of

a

, and the name occurs in neither original premise, satisfying the eigenvariable condition.

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