# Linear lower bounds and simulations in frege systems with substitutions

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## Abstract

We investigate the complexity of proofs in Frege (*F*), Substitution Frege (*sF*) and Renaming Frege (*rF*) systems. Starting from a recent work of Urquhart and using Kolmogorov Complexity we give a more general framework to obtain superlogarithmic lower bounds for the number of lines in both tree-like and dag-like *sF*. We show the previous known lower bound, extend it to the tree-like case and, for another class of tautologies, we give new lower bounds that in the dag-like case slightly improve the previous one. Also we show optimality of Urquhart's lower bounds giving optimal proofs. Finally we give the following two simulation results: (1) tree-like *sF p*-simulates dag-like *sF*; (2) Tree-like *F p*-simulates tree-like *rF*.

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