On type-based termination and dependent pattern matching in the calculus of inductive constructions. (Terminaison basée sur les types et filtrage dépendant pour le calcul des constructions inductives)

Proof assistants based on dependent type theory are progressively used as a tool to develop certified programs. A successful example is the Coq proof assistant, an implementation of a dependent type theory called the Calculus of Inductive Constructions (CIC). Coq is a functional programming language with an expressive type system that allows to specify and prove properties of programs in a higher-order predicate logic. Motivated by the success of Coq and the desire of improving its usability, in this thesis we study some limitations of current implementations of Coq and its underlying theory, CIC. We propose two extension of CIC that partially overcome these limitations and serve as a theoretical basis for future implementations of Coq. First, we study the problem of termination of recursive functions. In Coq, all recursive functions must be terminating, in order to ensure the consistency of the underlying logic. Current techniques for checking termination are based on syntactical criteria and their limitations appear often in practice. We propose an extension of CIC using a type-based mechanism for ensuring termination of recursive functions. Our main contribution is a proof of Strong Normalization and Logical Consistency for this extension. Second, we study pattern-matching definitions in CIC. With dependent types it is possible to write more precise and safer definitions by pattern matching than with traditional functional programming languages such as Haskell and ML. Based on the success of dependentlytyped programming languages such as Epigram and Agda, we develop an extension of CIC with similar features.