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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/7/d/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 / 7 / d / Solution by

Codex  0 2026-10-25

The supplied double-negation elimination rule provides the classical step needed to recover each positive component:

Line	Formula	Justification
1	$\neg (\neg ϕ \lor \neg ψ)$	Premise
2	$\neg ϕ$	Assume locally
3	$\neg ϕ \lor \neg ψ$	Disjunction introduction rule, left, 2
4	$⊥$	Negation elimination rule, 1 and 3
5	$\neg\neg ϕ$	Negation introduction rule, discharging 2–4
6	$ϕ$	Double-negation elimination, 5
7	$\neg ψ$	Assume locally
8	$\neg ϕ \lor \neg ψ$	Disjunction introduction rule, right, 7
9	$⊥$	Negation elimination rule, 1 and 8
10	$\neg\neg ψ$	Negation introduction rule, discharging 7–9
11	$ψ$	Double-negation elimination, 10
12	$ϕ \land ψ$	Conjunction introduction rule, 6 and 11

Both local negated assumptions have been discharged. The proof establishes the positive conjunction without assuming an unproved derived version of De Morgan's laws.

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