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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/ii/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 / ii / Solution by

Codex  0 2026-10-31

First derive

A

by double-negation elimination, then derive

\neg B

by negation introduction. The logical falsity elimination rule is used only inside the attempt to derive

A

1  ¬(A → B)  premise
 2  | ¬A     assumption
 3  | | A    assumption
 4  | | ⊥    ¬-elim 3,2
 5  | | B    ⊥-elim 4
 6  | A → B  →-intro 3–5
 7  | ⊥      ¬-elim 6,1
 8  ¬¬A      ¬-intro 2–7
 9  A        ¬¬-elim 8
10  | B      assumption
11  | | A    assumption
12  | | B    reiteration 10
13  | A → B  →-intro 11–12
14  | ⊥      ¬-elim 13,1
15  ¬B       ¬-intro 10–14
16  A ∧ ¬B   ∧-intro 9,15

Reiteration at line 12 merely uses an assumption that remains in scope. The final $A \land \neg B$ follows by conjunction introduction. This explicitly supplies the classical step needed to obtain

A

rather than just

\neg\neg A

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