By Herschell E Filipowski

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Suppose that e is a closed term. Then e is a value or e = C[r] for some context C and term r which is either a β-redex or an implementation. A more general lemma is necessary when we extend the language, and then we want to distinguish arbitrary normal forms (like the bad application 3(4)) from proper values. Since Theorem 1 applies to any well-typed term in System E ∗ , it is easy to check: Theorem 2. If Γ C[r] : R in System E ∗ , ε does not appear in Γ , and C[r] → e, then there exists R such that Γ e : R in System E ∗ .

It has been later simplified by Asperti into Light affine logic ([3]) which allows Work partially supported by Action Sp´ecifique CNRS M´ethodes formelles pour la Mobilit´e and ACI S´ecurit´e Informatique CRISS. I. ): FOSSACS 2004, LNCS 2987, pp. 27–41, 2004. c Springer-Verlag Berlin Heidelberg 2004 28 P. Baillot and V. Mogbil for arbitrary erasing. However formulas in this system are quite complicated as there are two modalities, instead of just one in Intuitionistic linear logic, which makes programming delicate (see [2,4]).

2. if x ∈ T V (t) then: W (t[u/x], n) W (t, n) + nW (u, n). Proposition 5. Let t be a term and n > rank(t). ) reduction rule then W (t , n) < W (t, n). σ Proof. ) then there is a context C and a redex r σ such that t = C[r], t = C[r ] and r → r . Soft lambda-Calculus: A Language for Polynomial Time Computation 33 We prove the statement by induction on the context C, for a given n > rank(t). e. x in r1 , . W (u, n) < W (r, n) . The case of a (β) reduction is easy. C1 the fact that n 1 as we have the strict inequality n > rank(t).

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A table of anti-logarithms to seven places and Gauss logarithms by Herschell E Filipowski

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