How many times do we need and assumption ?

In this article we present a class of formulas φn, n ∈ Nat, that need at least 2 assumptions to be proved in a normal proof in Natural Deduction for purely implicational minimal propositional logic. In purely implicational classical propositional logic, with Peirce’s rule, each φn is proved with only one assumption in Natural Deduction in a normal proof. Hence, the formulas φn have exponentially sized proofs in cut-free Sequent Calculus and Tableaux. In fact 2 is the lower-bound for normal proofs in ND, cut-free Sequent proofs and Tableaux. We discuss the consequences of the existence of this class of formulas for designing automatic proof-procedures based on these deductive systems.