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ID: past-exam-of-the-year-1-of-the-computer-science-course-of-the-university-of-oxford/2019/digital-systems-linear-algebra-introduction-to-formal-proof-resit/8/b/solution

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Past exam of the year 1 of the computer science course of the University of Oxford / 2019 / Digital systems, linear algebra, introduction to formal proof resit / 8 / b / Solution by

Codex  0 2026-10-25

The displayed derivation is invalid, although its final conjecture is valid. Its existential-elimination step exports

T (x_{0})

with the local witness name still free, violating the eigenvariable condition. Introducing an existential quantifier afterwards does not repair that earlier invalid rule application.

Move the existential quantifier introduction rule inside the witness scope:

Line	Formula	Justification
1	$\forall x (S (x) \to T (x))$	Premise
2	$\exists x S (x)$	Premise
3	$S (u)$	Assume with fresh $u$
4	$S (u) \to T (u)$	Universal quantifier elimination rule, 1
5	$T (u)$	Implication elimination rule, 3 and 4
6	$\exists x T (x)$	Existential quantifier introduction rule, 5
7	$\exists x T (x)$	Existential quantifier elimination rule, 2 and 3–6

The exported formula contains no free

u

, and neither original premise contains that name. The witness assumption is now discharged legally.

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