# The Relationship Between the Proof Complexities of Linear Proofs in Quantified Sequent Calculus and Substitution Frege Systems

## DOI:

https://doi.org/10.51408/1963-0099## Keywords:

Sequent systems, Frege systems, Proof size, Number of proof lines, Exponential speed-up## Abstract

It has formerly been proved that there is an exponential speed-up in the number of lines of the quantified propositional sequent calculus over substitution Frege systems when considering proofs as trees. This paper shows that a linear proof of any quantifierfree tautology in quantified propositional sequent calculus can be transformed into a linear proof of the same tautology in a substitution Frege systems with no more than polynomially increasing proof lines and size.

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*Mathematical Problems of Computer Science*,

*59*, 27–34. https://doi.org/10.51408/1963-0099

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Copyright (c) 2023 Hakob A. Tamazyan

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