Title:

Normalisation & equivalence in proof theory & type theory

At the heart of the connections between Proof Theory and Type Theory, the CurryHoward correspondence provides proofterms with computational features and equational theories, i.e. notions of normalisation and equivalence. This dissertation contributes to extend its framework in the directions of prooftheoretic formalisms (such as sequent calculus) that are appealing for logical purposes like proofsearch, powerful systems beyond propositional logic such as type theories, and classical (rather than intuitionistic) reasoning. Part I is entitled Proofterms for Intuitionistic Implicational Logic. Its contributions use rewriting techniques on proofterms for natural deduction (Lambdacalculus) and sequent calculus, and investigate normalisation and cutelimination, with callbyname and callbyvalue semantics. In particular, it introduces proofterm calculi for multiplicative natural deduction and for the depthbounded sequent calculus G4. The former gives rise to the calculus Lambdalxr with explicit substitutions, weakenings and contractions that refines the Lambdacalculus and Betareduction, and preserves strong normalisation with a full notion of composition of substitutions. The latter gives a new insight to cutelimination in G4. Part II, entitled Type Theory in Sequent Calculus develops a theory of Pure Type Sequent Calculi (PTSC), which are sequent calculi that are equivalent (with respect to provability and normalisation) to Pure Type Systems but better suited for proofsearch, in connection with proofassistant tactics and proofterm enumeration algorithms. Part III, entitled Towards Classical Logic, presents some approaches to classical type theory. In particular it develops a sequent calculus for a classical version of System F_omega. Beyond such a type theory, the notion of equivalence of classical proofs becomes crucial and, with such a notion based on parallel rewriting in the Calculus of Structures, we compute canonical representatives of equivalent proofs.
