nate a maximum formula. Any derivations containing maximum formulas are not in normal form and does not represent a direct proof. Such derivations have a detour reasoning. Therefore, we say that a derivation has a detour reasoning if it has an application of an elimination rule with its maximum formula. On the other hand, it is not easy to characterize a 'detour' in terms of the length of a derivation. There are cases that the length of a normal derivation is longer than the length of an original derivation. It cannot be said that a derivation of a longer length has a detour and a derivation with a shorter length has no detour. Thus, we only use a 'detour reasoning' as defined through a maximum formula. Then, (T2) means that a proper reduction should neither introduce an application of an elimination rule containing a maximum formula nor increase the degree of maximum formulas. To check (T2), it is useful to translate a reduction in natural deduction to a one in sequent calculus. It is often claimed that a natural deduction system is isomorphic to a sequent calculus system if the generalized eliminations are used. (Cf. Tennant (2002), Negri and Von Plato (2001) and Von Plato (2011)). Especially, Negri and Von Plato (2001, Ch.1 and Ch. 8) propose an inductive definition of the translation algorithm between an intuitionistic natural deduction system with generalized elimination rules and an intuitionistic sequent calculus system with independent contexts. Let Γ, ∆, Θ be a finite multiset, i.e. a list with multiplicity but no order, of assumptions in sequent calculus. We use a binary derivation symbol ⇒ and 'Γ ⇒ φ' means that the antecedent Γ derives the succedent φ . A sequent calculus has the following left and right rules for → and structural rules, such as cut-, weakening, and contraction rules. Γ ⇒ φ ψ,∆ ⇒ σ L → φ → ψ,Γ,∆ ⇒ σ φ,Γ ⇒ ψ R → Γ ⇒ φ → ψ 119 Γ ⇒ φ φ,∆ ⇒ Ψ Cut Γ,∆ ⇒ Ψ Γ ⇒ Ψ Wk φ,Γ ⇒ ψ φ,φ,Γ ⇒ Ψ Ctr φ,Γ ⇒ Ψ φ in Cut�