This paper starts by setting the ground for a lambda calculus notation that strongly mirrors the two fundamental operations of term construction, namely abstraction and application. In particular, we single out those parts of a term, called items in the paper, that are added during abstraction and application. This item notation proves to be a powerful device for the representation of basic substitution steps, giving rise to different versions of fi-reduction including local and global fi- reduction. In other words substitution, thanks to the new notation, can be easily formalised as an object language notion rather than remaining a meta language one. Such formalisation will have advantages with respect to various areas including functional application and the partial unfolding of definitions. Moreover our substitution is, we believe, the most general to date. This is shown by the fact that our framework can accommodate most of the known reduction strategies, which range from local to...

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.

this paper appears in Types for Proofs and Programs: International Workshop TYPES'93, Nijmegen, May 1993, Selected Papers, LNCS 806. abstraction, compute a type for its body in an extended context; to compute a type for an application, compute types for its left and right components, and check that they match appropriately. Lets use the algorithm to compute a type for a = [x:ø ][x:oe]x. FAILURE: no rule applies because x 2 Dom (x:ø )

The paper develops notation for strings of abstracters in typed lambda calculus, and shows how to treat them more or less as single abstracters. 0 1991 Academic Press. Inc. 1.

We show that, given an ordinary lambda-term and a normal resource lambda-term which appears in the normal form of its Taylor expansion, the unique resource term of the Taylor expansion of the ordinary lambda-term reducing to this normal resource term can be obtained by running a version of the Krivine abstract machine.

This paper is about mechanical checking of formal mathematics. Given some formal system, we want to construct derivations in that system, or check the correctness of putative derivations; our job is not to ascertain truth (that is the job of the designer of our formal system), but only proof. However, we are quite rigid about this: only a derivation in our given formal system will do; nothing else counts as evidence! Thus it is not a collection of judgements (provability), or a consequence relation [Avr91] (derivability) we are interested in, but the derivations themselves; the formal system used to present a logic is important. This viewpoint seems forced on us by our intention to actually do formal mathematics. There is still a question, however, revolving around whether we insist on objects that are immediately recognisable as proofs (direct proofs), or will accept some meta-notations that only compute to proofs (indirect proofs). For example, we informally refer to previously proved results, lemmas and theorems, without actually inserting the texts of their proofs in our argument. Such an argument could be made into a direct proof by replacing all references to previous results by their direct proofs, so it might be accepted as a kind of indirect proof. In fact, even for very simple formal systems, such an indirect proof may compute to a very much bigger direct proof, and if we will only accept a fully expanded direct proof (in a mechanical proof checker for example), we will not be able to do much mathematics. It is well known that this notion of referring to previous results can be internalized in a logic as a cut rule, or Modus Ponens. In a logic containing a cut rule, proofs containing cuts are considered direct proofs, and can be directly accepted by a proof ch...

In this article, we introduce a -notation that is useful for many concepts of the - calculus. The new notation is a simple translation of the classical one. Yet, it provides many nice advantages. First, we show that definitions such as compatibility, the heart of a term and fi-redexes become simpler in item notation. Second, we show that with this item notation, reduction can be generalised in a nice way. We find a relation ; fi which extends ! fi , which is Church Rosser and Strongly Normalising. This reduction relation may be the way to new reduction strategies. In classical notation, it is much harder to present this generalised reduction in a convincing manner. Third, we show that the item notation enables one to represent in a very simple way the canonical type ø (\Gamma; A) of a term A in context \Gamma. This canonical type plays the role of a preference type and can be used to split \Gamma ` A : B in the two parts: \Gamma ` A and ø (\Gamma; A) = B. This means that the questio...

• In 1967, an internationally renowned mathematician called N.G. de Bruijn wanted to do something never done before: use the computer to formally check the correctness of mathematical books. • Such a task needs a good formalisation of mathematics, a good competence in implementation, and extreme attention to all the details so that nothing is left informal. • No earlier formalisation of mathematics (Frege, Russell, Whitehead, Hilbert, Ramsey, etc.) had ever achieved such attention to details. • Implementing extensive formal systems on the computer was never done before. • De Bruijn, an extremely original mathematician, did every step his own way, quickly replacing existing ideas with his own original genious way of thinking, shaping the road ahead for everyone else to follow.

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