On the lambda calculus with constructors

The lambda calculus with constructors was introduced by Arbiser, Miquel and Rios in the early 2000's as an extension of lambda calculus with pattern matching features. It decomposes the pattern matching a la ML into a case-analysis on constant constructors (in the spirit of the case instruction in Pascal), and a commutation rule between case construction and application. This commutation rule between two different kinds of constructions designs a surprising computational behaviour, a priori} not compatible with usual typing intuitions. However the whole calculus was proved confluent, and it enjoys the separation property (a version of Bohm's lemma).In this thesis we propose a polymorphic type system for this calculus, and we develop a realisability model, based on Girard's reducibility candidates. This leads to a strong normalisation result for the typed calculus, and guaranties that the type system prevents match failure. Next we focus on semantics for the untyped calculus. We first define a generic notion of models for the lambda calculus with constructors in Cartesian closed categories. We then establish the syntactic model in the category of PERs, and deduce a completeness result from it.Finally, we consider a translation of the lambda calculus with constructors into the pure lambda lambda calculus relying on continuation passing style techniques. This enables the simulation of the lambda calculus with constructors by a well known calculus, and provides a transformation of every continuation model into a model of the lambda calculus with constructors. Thereby a categorical equation characteristic of these models appears, which enables the construction of non syntactic models in Scott's domains.