Normal deduction in the intuitionistic linear logic

A natural deduction system NDIL described here admits normalization and has subformula property. It has standard axioms A ` A, ` 1, standard introduction and elimination rules for &,−◦ (linear implication), ⊕ and quantifiers. The rules for ⊗ are now standard too. Structural rules are (implicit) permutation plus contraction and weakening for m-formulas. The rules for ! use an idea of D. Prawitz. By a m-formula we mean 1, any formula beginning with !, and any expression < Γ >!A, where Γ is a list of formulas and m-formulas, and A is a formula. Derivable objects are sequents Γ ` A where Γ is a multiset of formulas and m-formulas, and A is a formula. The rules for !, weakening and contraction are as follows: <> I Γ `!A < Γ >!A`!A !E Γ `!A Γ ` A !I Γ ` A Γ `!A <> E < Γ >!B, Σ ` A Γ, Σ ` A weak Γ ` A Σ ` B Γ, Σ ` B c B,B, Γ ` D B, Γ ` D provided Γ consists of m-formulas, B is a m-formula and A is 1 or begins with ! . Normal deduction is one without maximal segments. Theorem. Every derivable sequent has a normal deduction. Proof. Apply Prawitz translation to a cut-free sequent derivation