International audienceWe discuss several constructions of orthonormal wavelet bases on the interval, and we introduce a new construction that avoids some of the disadvantages of earlier constructions

Starting from an exact correspondence between linear approximations and non-idempotent intersection types, we develop a general framework for building systems of intersection types characterizing normalization properties. We show how this construction, which uses in a fundamental way Melliès and Zeilberger's łtype systems as functorsž viewpoint, allows us to recover equivalent versions of every well known intersection type system (including Coppo and Dezani's original system, as well as its non-idempotent variants independently introduced by Gardner and de Carvalho). We also show how new systems of intersection types may be built almost automatically in this way.

No abstract

We provide a type-theoretical characterization of weakly-normalizing terms in an infinitary lambda-calculus. We adapt for this purpose the standard quantitative (with non-idempotent intersections) type assignment system of the lambda-calculus to our infinite calculus.Our work provides a new answer to Klop's HHN-problem, namely, finding out if there is a type system characterizing the hereditary head-normalizing (HHN) lambda-terms. Tatsuta showed that HHN could not be characterized by a finite type system. We prove that an infinitary type system endowed with a validity condition called approximability can achieve it.

No abstract

Infinite types and formulas are known to have really curious and unsound behaviors. For instance, they allow to type Ω, the auto-autoapplication and they thus do not ensure any form of normalization/productivity. Moreover, in most infinitary frameworks, it is not difficult to define a type R that can be assigned to every λ-term. However, these observations do not say much about what coinductive (i.e. infinitary) type grammars are able to provide: it is for instance very difficult to know what types (besides R) can be assigned to a term in this setting. We begin with a discussion on the expressivity of different forms of infinite types. Using the resource-awareness of sequential intersection types (system S) and tracking, we then prove that infinite types are able to characterize the order (arity) of every λ-terms and that, in the infinitary extension of the relational model, every term has a "meaning" i.e. a non-empty denotation. From the technical point of view, we must deal with the total lack of productivity guarantee for typable terms: we do so by importing methods inspired by first order model theory.

Whereas proof assistants based on Higher-Order Logic benefit from external solvers' automation, those based on Type Theory resist automation and thus require more expertise. Indeed, the latter use a more expressive logic which is further away from first-order logic, the logic of most automatic theorem provers. In this article, we develop a methodology to transform a subset of Coq goals into first-order statements that can be automatically discharged by automatic provers. The general idea is to write modular, pairwise independent transformations and combine them. Each of these eliminates a specific aspect of Coq logic towards first-order logic. As a proof of concept, we apply this methodology to a set of simple but crucial transformations which extend the local context with proven first-order assertions that make Coq definitions and algebraic types explicit. They allow users of Coq to solve non-trivial goals automatically. This methodology paves the way towards the definition and combination of more complex transformations, making Coq more accessible. * This work is funded by a Nomadic Labs-Inria collaboration.

In the context of interactive theorem provers based on a dependent type theory, automation tactics (dedicated decision procedures, call of automated solvers, ...) are often limited to goals which are exactly in some expected logical fragment. This very often prevents users from applying these tactics in other contexts, even similar ones.This paper discusses the design and the implementation of pre-processing operations for automating formal proofs in the Coq proof assistant. It presents the implementation of a wide variety of predictible, atomic goal transformations, which can be composed in various ways to target different backends. A gallery of examples illustrates how it helps to expand significantly the power of automation engines.