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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/7/e/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 / 7 / e / Solution by

Codex  0 2026-10-25

Use the two derived inference rules whose complete proofs were supplied in (d). The following logical derivation has no nested hypothetical subproofs:

1  P → Q                           premise
2  Q → R                           premise
3  P → (Q → R)                     → creation from 2
4  (P → Q) → (P → R)               → distribution from 3
5  P → R                           →-elim, 1 and 4

The creation instance takes

ϕ = P

and

ψ = (Q \to R)

. The distribution instance takes

(ϕ, ψ, ρ) = (P, Q, R)

. The metarules are justified in (d); this requested flat use does not secretly invoke implication introduction rule in one of its own lines.

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