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

Codex  0 2026-10-31

Abbreviate

H = (\neg P) \to Q

and

K = (\neg P) \to \neg Q

. Use the primitive natural deduction rules, including double-negation elimination:

1  H             premise
 2  | H          assumption
 3  | | K        assumption
 4  | | | ¬P     assumption
 5  | | | Q      →-elim 2,4
 6  | | | ¬Q     →-elim 3,4
 7  | | | ⊥      ¬-elim 5,6
 8  | | ¬¬P      ¬-intro 4–7
 9  | | P        ¬¬-elim 8
10  | K → P      →-intro 3–9
11  H → (K → P)  →-intro 2–10

The outer premise is redundant: the proof uses the locally assumed

H

, so the conclusion is already a theorem. Keeping the redundant premise is allowed. The successive discharges give $((\neg P) \to Q) \to (((\neg P) \to \neg Q) \to P)$ with exactly the requested assumption scopes.

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