Extending the Extensional Lambda Calculus with Surjective Pairing is Conservative

We answer Klop and de Vrijer's question whether adding surjective-pairing axioms to the extensional lambda calculus yields a conservative extension. The answer is positive. As a byproduct we obtain a "syntactic" proof that the extensional lambda calculus with surjective pairing is consistent.Comment: To appear in Logical Methods in Computer Scienc

Cyclic Datatypes modulo Bisimulation based on Second-Order Algebraic Theories

Cyclic data structures, such as cyclic lists, in functional programming are tricky to handle because of their cyclicity. This paper presents an investigation of categorical, algebraic, and computational foundations of cyclic datatypes. Our framework of cyclic datatypes is based on second-order algebraic theories of Fiore et al., which give a uniform setting for syntax, types, and computation rules for describing and reasoning about cyclic datatypes. We extract the "fold" computation rules from the categorical semantics based on iteration categories of Bloom and Esik. Thereby, the rules are correct by construction. We prove strong normalisation using the General Schema criterion for second-order computation rules. Rather than the fixed point law, we particularly choose Bekic law for computation, which is a key to obtaining strong normalisation. We also prove the property of "Church-Rosser modulo bisimulation" for the computation rules. Combining these results, we have a remarkable decidability result of the equational theory of cyclic data and fold.Comment: 38 page

Dialectica Categories for the Lambek Calculus

We revisit the old work of de Paiva on the models of the Lambek Calculus in dialectica models making sure that the syntactic details that were sketchy on the first version got completed and verified. We extend the Lambek Calculus with a \kappa modality, inspired by Yetter's work, which makes the calculus commutative. Then we add the of-course modality !, as Girard did, to re-introduce weakening and contraction for all formulas and get back the full power of intuitionistic and classical logic. We also present the categorical semantics, proved sound and complete. Finally we show the traditional properties of type systems, like subject reduction, the Church-Rosser theorem and normalization for the calculi of extended modalities, which we did not have before

Pure Type System conversion is always typable

International audiencePure Type Systems are usually described in two different ways, one that uses an external notion of computation like beta-reduction, and one that relies on a typed judgment of equality, directly in the typing system. For a long time, the question was open to know whether both presentations described the same theory. A first step toward this equivalence has been made by Adams for a particular class of \emph{Pure Type Systems} (PTS) called functional. Then, his result has been relaxed to all semi-full PTS in previous work. In this paper, we finally give a positive answer to the general issue, and prove that equivalence holds for any Pure Type System.Les Systèmes de Types Purs (PTS) sont habituellement présentés de deux manières différentes, une qui utilise une notion de calcul indépendante du typage, comme la béta-reduction, et une qui défini un jugement d'égalité typée au sein du système de types. La question de savoir si ces deux présentations représentaient la même théorie est restée ouverte pendant de nombreuses années. Une première réponse partielle à cette question a été apportée par Adams pour une classe particulière de PTS dit "fonctionnels". Nous avons récement étendu ce résultat à tous les PTS "semi-complets" . Dans cet article, nous pouvons finalement donner une réponse positive à la question dans toute sa généralité: l'équivalence entre les deux présentations est prouvée correcte pour n'importe quel Système de Types Purs

On the strength of proof-irrelevant type theories

We present a type theory with some proof-irrelevance built into the conversion rule. We argue that this feature is useful when type theory is used as the logical formalism underlying a theorem prover. We also show a close relation with the subset types of the theory of PVS. We show that in these theories, because of the additional extentionality, the axiom of choice implies the decidability of equality, that is, almost classical logic. Finally we describe a simple set-theoretic semantics.Comment: 20 pages, Logical Methods in Computer Science, Long version of IJCAR 2006 pape

Combining Algebra and Higher-Order Types

We study the higher-order rewrite/equational proof systems obtained by adding the simply typed lambda calculus to algebraic rewrite/equational proof systems. We show that if a many-sorted algebraic rewrite system has the Church-Rosser property, then the corresponding higher-order rewrite system which adds simply typed ß-reduction has the Church-Rosser property too. This result is relevant to parallel implementations of functional programming languages. We also show that provability in the higher-order equational proof system obtained by adding the simply typed ß and η axioms to some many-sorted algebraic proof system is effectively reducible to provability in that algebraic proof system. This effective reduction also establishes transformations between higher-order and algebraic equational proofs, transformations which can be useful in automated deduction

Encoding Proofs in Dedukti: the case of Coq proofs

International audienceA main ambition of the Inria project Dedukti is to serve as a common language for representing and type checking proof objects originating from other proof systems. Encoding these proof objects makes heavy use of the rewriting capabilities of LambdaPiModulo, the formal system on which Dedukti is based. So far, the proofs generated by two automatic proofsystems, Zenon and iProver, have been encoded, and can therefore be read and checked by Dedukti. But Dedukti goes far beyond this so-called hammering technique of sending goals to automated provers. Proofs from HOL and Matita can be encoded as well. Some Coq’s proofs can be encoded already, when they do not use universe polymorphism. Our ambition here is to close this remaining gap. To this end, we describe a rewrite-based encoding in LambdaPiModulo of the Calculus of Constructions with a cumulative hierarchy of predicative universes above Prop, which is confluent on open terms