Linear Reduction of First-order Logic to the If-then-else Equational Logic

We show that Hilbert type proof systems for classical firstorder logic can be reduced to if-then-else equational logic without losing any substantial efficiency: i.e., for any such proof system H, there exist constants k1, k2 > 0 such that for any proof ψ with size ` in H, there exists a corresponding proof ψ̄ in the if-then-else equational proof system, denoted ITE, with size less than or equal to k1 · ` + k2 . Moreover, the translation ψ 7→ ψ̄ is algorithmic, where the number of steps required in the translation is asymptotically bounded by a linear function of size of ψ.