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ID: past-exam-of-the-year-1-of-the-computer-science-course-of-the-university-of-oxford/2017/digital-systems-linear-algebra-introduction-to-formal-proof/8/b/solution

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Past exam of the year 1 of the computer science course of the University of Oxford / 2017 / Digital systems, linear algebra, introduction to formal proof / 8 / b / Solution by

Codex  0 2026-10-31

The two proofs differ in whether existential elimination is outside or inside negation introduction. For the first proof, choose fresh

a

, derive the negation inside the witness subproof, and then discharge the witness:

1234567 \exists x \neg P (x) \neg P (a) \forall y P (y) P (a) ⊥ \neg\forall y P (y) \neg\forall y P (y) premise fresh witness assumption assumption \forall E 3 \neg E 4, 2 \neg I 3 - 5 \exists E 1, 2 - 6

(1)

The conclusion at line 6 contains no free

a

. For the second proof, open the universal assumption first, eliminate the existential to derive falsity, and then discharge the universal assumption:

1234567 \exists x \neg P (x) \forall y P (y) \neg P (a) P (a) ⊥ ⊥ \neg\forall y P (y) premise assumption fresh witness assumption \forall E 2 \neg E 4, 3 \exists E 1, 3 - 5 \neg I 2 - 6

(2)

At line 6 of the second proof, neither the exported conclusion

⊥

nor the other open assumptions contains

a

free, satisfying the eigenvariable condition. Neither proof needs RAA or double-negation elimination.

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