Labeled types and a new relation between types are added to the lambda calculus of objects as described in [5]. This relation is a trade-off between the possibility of having a restricted form of width subtyping and the features of the delegation-based language itself. The original type inference system allows both specialization of the type of an inherited method to the type of the inheriting object and static detection of errors, such as 'message-not-understood '. The resulting calculus is an extension of the original one. Type soundness follows from the subject reduction property.

We introduce a new framework of algebraic pure type systems in which we consider rewrite rules as lambda terms with patterns and rewrite rule application as abstraction application with built-in matching facilities. This framework, that we call "Pure Pattern Type Systems", is particularly well-suited for the foundations of programming (meta)languages and proof assistants since it provides in a fully unified setting higher-order capabilities and pattern matching ability together with powerful type systems. We prove some standard properties like confluence and subject reduction for the case of a syntactic theory and under a syntactical restriction over the shape of patterns. We also conjecture the strong normalization of typable terms. This work should be seen as a contribution to a formal connection between logics and rewriting, and a step towards new proof engines based on the Curry-Howard isomorphism.

Abstract. The LFP Framework is an extension of the Harper-Honsell-Plotkin's Edinburgh Logical Framework LF with external predicates, hence the name Open Logical Framework. This is accomplished by defining lock type constructors, which are a sort of -modality constructors, releasing their argument under the condition that a possibly external predicate is satisfied on an appropriate typed judgement. Lock types are defined using the standard pattern of constructive type theory, i.e. via introduction, elimination, and equality rules. Using LFP , one can factor out the complexity of encoding specific features of logical systems which would otherwise be awkwardly encoded in LF, e.g. side-conditions in the application of rules in Modal Logics, and sub-structural rules, as in non-commutative Linear Logic. The idea of LFP is that these conditions need only to be specified, while their verification can be delegated to an external proof engine, in the style of the Poincaré Principle or Deduction Modulo. Indeed such paradigms can be adequately formalized in LFP . We investigate and characterize the meta-theoretical properties of the calculus underpinning LFP : strong normalization, confluence, and subject reduction. This latter property holds under the assumption that the predicates are well-behaved, i.e. closed under weakening, permutation, substitution, and reduction in the arguments. Moreover, we provide a canonical presentation of LFP , based on a suitable extension of the notion of βη-long normal form, allowing for smooth formulations of adequacy statements.

International audienceWe investigate a first-order extension of the Theory of Primitive Objects of [5] that supports method extension in presence of object subsumption. Extension is the ability of modifying the behavior of an object by adding new methods (and inheriting the existing ones). Object subsumption allows to use objects with a bigger interface in a context expecting another object with a smaller interface. This extended calculus has a sound type system which allows static detection of run-time errors such as message-not-understood, " width " subtyping and a typed equational theory on objects. Moreover, it can express classes and class-inheritance