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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/ii/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 / ii / Solution by

Codex  0 2026-10-31

Take a fresh arbitrary variable

a

. Assuming

R (a)

would provide precisely the witness forbidden by the premise; discharge that assumption before universal introduction.

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. ∀y ¬R(y)             ∀I, 5

At line 6,

a

is free in no undischarged assumption: line 2 has already been discharged. The change of bound variable name is part of the universal-introduction instance.

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