Every super-polynomial proof in purely implicational minimal logic has a polynomially sized proof in classical implicational propositional logic

In this note we show how any formula δ with a proof of super-polynomial size in M→ has a poly-sized proof in K→ . This fact entails that any propositional classical tautology has short proofs, i.e., NP=CoNP. 1 Every classical tautology has short proofs This article reports the results on the investigation on the existence of short proofs for every Classical Propositional Logic tautology. Firstly, the functional completeness of the fragment {→,⊥} of the propositional language says that any tautology in the full fragment can be expressed in this implicational based fragment. Let t(α) be the translation of the propositional formula α in terms of {→,⊥}. We know that, from any proof Π of α, in the system of Natural Deduction for CPL, there is a proof Π of t(α), such that, p(| Π |) ≤| Π |, where p(x) is a polynomial. Thus, if there is a super-polynomially sized proof of α then, there is a super-polynomially sized proof of t(α). Proving that every formula in {→,⊥} has short (polynomially sized proofs on the length (|| α ||) entails that every formula in the full language has short proofs too. Without generality loss we focus the discussion on the existence of short proofs in the fragment {→,⊥}. We remember that the negation ¬β can be seen as a shorthand for β → ⊥ whenever we find it convenient. The length of a formula α, || α ||, is the number of occurrences of symbols of the alphabet of α in the string α. The size of a proof Π, | Π |, is the number of occurrences of symbols in the string that represents a linearization of Π (viewed as a tree). In what follows we use ⊢Cla α, ⊢Int α and ⊢Min α to denote Classical, Intuitionistic and Minimal validity, respectively. Our proof-theoretical analysis is based on the systems of Natural Deduction for Classical, Intuitionistic and