Type Theory and Decision Procedures

Pierre-Yves Strub 1, 2, 3
1 LOGICAL - Logic and computing
UP11 - Université Paris-Sud - Paris 11, Inria Saclay - Ile de France, X - École polytechnique, CNRS - Centre National de la Recherche Scientifique : UMR8623
2 TYPICAL - Types, Logic and computing
LIX - Laboratoire d'informatique de l'École polytechnique [Palaiseau], Inria Saclay - Ile de France
Abstract : The goal of this thesis is to develop a logical system in which formal proofs of mathematical statements can be carried out in a way which is close to mathematical practice.

Our main contribution is the definition and study of a new calculus, the Calculus of Congruent and Inductive Constructions, an extension of the Calculus of Inductive Constructions (CIC) integrating in its computational part the entailment relation - via decision procedures - of a first order theory over equality. A major technical innovation of this work lies in the computational mechanism: goals are sent to the decision procedure together with a set of user hypotheses available from the current context.

Our main results show that this extension of CIC does not compromise its main properties: confluence, strong normalization, consistency and decidability of proof checking are all preserved (as soon as the incorporated theory is itself decidable). As such, our calculus can be seen as a decidable restriction of the Extensional Calculus of Constructions. It can therefore serve as the basis for an extension of the Coq proof assistant.
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Pierre-Yves Strub. Type Theory and Decision Procedures. Formal Languages and Automata Theory [cs.FL]. Ecole Polytechnique X, 2008. English. ⟨NNT : 2008EPXX0054⟩. ⟨tel-00351837⟩

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