A Note on the Complexity of Classical and Intuitionistic Proofs

We show an effective cut-free variant of Glivenko´s theorem extended to formulas with weak quantifiers (those without eigenvariable conditions): “There is an elementary function f such that if φ is a cut-free LK proof of ⊢ A with symbol complexity ≤ c, then there exists a cut-free LJ proof of 1⊢ ⊣⊣A with symbol complexity ≤ f(c)”. This follows from the more general result: “There is an elementary function f such that if φ is a cut-free LK proof of A ⊢ with symbol complexity ≤ c, then there exists a cut-free LJ proof of A ⊢ with symbol complexity ≤ f(c)”. The result is proved using a suitable variant of cut-elimination by resolution (CERES) and subsumption.