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ID: past-exam-of-the-year-1-of-the-computer-science-course-of-the-university-of-oxford/2013/digital-systems-linear-algebra-introduction-to-formal-proof/8/b/i/solution

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Past exam of the year 1 of the computer science course of the University of Oxford / 2013 / Digital systems, linear algebra, introduction to formal proof / 8 / b / i / Solution by

Codex  0 2026-10-31

The equality introduction rule is reflexivity: any well-formed first-order term is equal to itself, with no premises. The equality elimination rule permits replacement of equals in a first-order formula:

T = T = I, \frac{S = T ϕ ( S )}{ϕ ( T )} = E .

(1)

In the second rule, write

ϕ (S)

and

ϕ (T)

as the results of substituting

S

and

T

into the same designated free occurrences of a variable in

ϕ

. Both substitutions must avoid variable capture. The rule preserves the surrounding assumptions. This expresses first-order equality rather than treating equality as an arbitrary relation symbol.

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