|Abstract||When working with a ﬁrst-order theory, it is often convenient to use deﬁnitions. That is, if ϕ(x) is a ﬁrst-order formula with the free variables shown, one can introduce a new relation symbol R to abbreviate ϕ, with deﬁning axiom ∀x (R(x) ↔ ϕ(x)). Of course, this deﬁnition can later be eliminated from a proof, simply by replacing every instance of R by ϕ. But suppose the proof involves nested deﬁnitions, with a sequence of relation symbols R0, . . . , Rk abbreviating formulae ϕ0, . . . , ϕk, where each ϕi may have multiple occurrences of R0, . . . , Ri−1. In that case, the naive elimination procedure described above can yield an exponential increase in the length of the proof. In Section 2, I show that if the underlying theory proves that there are at least two elements in the universe, a more careful translation allows one to eliminate the new deﬁnitions with at most a polynomial increase in length. In fact, I will describe an explicit algorithm that can be implemented in polynomial time. The proof is not diﬃcult, but it relies on the assumption that equality is included in the logic. A similar trick has been used by Solovay in simulating iterated deﬁnitions eﬃciently, as discussed in Section 3.2 of [Pudl´ak 1998]. Consequently, the result proved here may be folklore, but to my knowledge it has not appeared in the literature, and it is needed in Section 3. It is also sometimes convenient, in a ﬁrst-order setting, to introduce Skolem func-.|
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