Eliminating definitions and Skolem functions in first-order logic

From proofs in any classical first-order theory that proves the existence of at least two elements, one can eliminate definitions in polynomial time. From proofs in any classical first-order theory strong enough to code finite functions, including sequential theories, one can also eliminate Skolem functions in polynomial time.
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Pavel Hrubeš (2009). On Lengths of Proofs in Non-Classical Logics. Annals of Pure and Applied Logic 157 (2):194-205.

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