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Normalisation & Equivalence in Proof Theory & Type Theory

Abstract : At the heart of the connections between Proof Theory and Type Theory, the Curry-Howard correspondence provides proof-terms with computational features and equational theories, i.e. notions of normalisation and equivalence. This dissertation contributes to extend its framework in the directions of proof-theoretic formalisms (such as sequent calculus) that are appealing for logical purposes like proof-search, powerful systems beyond propositional logic such as type theories, and classical (rather than intuitionistic) reasoning.
Part I is entitled Proof-terms for Intuitionistic Implicational Logic. Its contributions use rewriting techniques on proof-terms for natural deduction (lambda-calculus) and sequent calculus, and investigate normalisation and cut-elimination, with call-by-name and call-by-value semantics. In particular, it introduces proof-term calculi for multiplicative natural deduction and for the depth-bounded sequent calculus G4. The former gives rise to a calculus with explicit substitutions, weakenings and contractions that refines the lambda-calculus and beta-reduction, and preserves strong normalisation with a full notion of composition of substitutions. The latter gives a new insight to cut-elimination 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 proof-search, in connection with proof-assistant tactics and proof-term 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 Fomega. 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.
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Stéphane Lengrand. Normalisation & Equivalence in Proof Theory & Type Theory. Mathematics [math]. Université Paris-Diderot - Paris VII; University of St Andrews, 2006. English. ⟨tel-00134646⟩

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