We present an extension of the lambda-calculus with differential constructions motivated by a model of linear logic discovered by the first author and presented in [Ehr01]. We state and prove some basic results (confluence, weak normalization in the typed case), and also a theorem relating the usual Taylor series of analysis to the linear head reduction of lambda-calculus.

We study the Taylor expansion of lambda-terms in a non-deterministic or algebraic setting, where terms can be added. The target language is a resource lambda calculus based on a differential lambda-calculus we introduced recently. This operation is not possible in the general untyped case where reduction can produce unbounded coefficients. We endow resource terms with a finiteness structure (in the sense of our earlier work on finiteness spaces) and show that the Taylor expansions of terms typeable in Girard’s system F are finitary by a reducibility method.

We show that the standard method of saturated sets for proving strong normalization of fi-reduction in the simply typed and second order polymorphic lambda calculus incorporates non-structural subtyping systems in a natural way. This shows that strong normalization for non-structural subtyping proved by Wand, O'Keefe and Palsberg [24] via coercion interpretations can be obtained in a straight-forward extension of the standard method. The proof presented here is compared to other proofs of strong normalization for subtyping systems.

We present the titular proof development that has been verified in Isabelle/HOL. As a first, the proof is conducted exclusively by the primitive proof principles of the standard syntax and of the considered reduction relations: the naive way, so to speak. Curiously, the Barendregt Variable Convention takes on a central technical role in the proof. We also show (i) that our presentation of the λ-calculus coincides with Curry’s and Hindley’s when terms are considered equal up to α-equivalence and (ii) that the confluence properties of all considered systems are equivalent.

Abstract. Given a semi-ring with unit which satis es some algebraic conditions, we de ne an exponential functor on the category of sets and relations which allows to de ne a denotational model of di erential linear logic and of the lambda-calculus with resources. We show that, when the semi-ring has an element which is in nite in the sense that it is equal to its successor, this model does not validate the Taylor formula and that it is possible to build, in the associated Kleisli cartesian closed category, a model of the pure lambda-calculus which is not sensible. This is a quantitative analogue of the standard graph model construction in the