Cubical Synthetic Homotopy Theory

International audienc

Martin-L\"of \`a la Coq

We present an extensive mechanization of the meta-theory of Martin-L\"of Type Theory (MLTT) in the Coq proof assistant. Our development builds on pre-existing work in Agda to show not only the decidability of conversion, but also the decidability of type checking, using an approach guided by bidirectional type checking. From our proof of decidability, we obtain a certified and executable type checker for a full-fledged version of MLTT with support for $\Pi$, $\Sigma$, $\mathbb{N}$, and identity types, and one universe. Furthermore, our development does not rely on impredicativity, induction-recursion or any axiom beyond MLTT with a schema for indexed inductive types and a handful of predicative universes, narrowing the gap between the object theory and the meta-theory to a mere difference in universes. Finally, we explain our formalization choices, geared towards a modular development relying on Coq's features, e.g. meta-programming facilities provided by tactics and universe polymorphism

Calculer avec des Principes d'Extensionnalité en Théorie des Types

In this thesis, I study several possibilities to extend intuitionistic type theory with extensionality principles, such as function extensionality or Voevodsky's univalence axiom, while preserving the computational properties of the proofs. In the first part, I develop a complete meta-theory for the observational equality of Altenkirch et al. In particular, I obtain a formal proof of normalization, canonicity and decidability of the conversion for an observational type theory with impredicative proof-irrelevant propositions. Then in a second part, I sketch a translation from homotopy type theory to observational type theory based on the model of Coquand et al in cubical sets. Finally, in the last part I explain how to take advantage of the computational properties of cubical type theory to obtain elegant synthetic proofs of classical results from homotopy theory, in particular the construction of the Hopf fibration and the 3x3 lemma for homotopy pushouts.Dans cette thèse, j'étudie plusieurs manières d'étendre la théorie des types intuitionniste avec des principes d'extensionnalité, comme par exemple l'extensionnalité des fonctions ou l'axiome d'univalence de Voevodsky, en portant une attention particulière à la préservation des propriétés calculatoires des preuves. Dans une première partie, je développe une méta-théorie complète pour l'égalité observationnelle de Altenkirch et al. J'obtiens notamment une preuve formelle de normalisation, de canonicité et de décidabilité de la conversion pour une théorie des types observationnelle avec des propositions imprédicatives. Dans une seconde partie, j'esquisse une traduction de la théorie des types homotopique vers la théorie des types observationnelle, en me basant sur le modèle cubique de Coquand et al. Enfin dans une dernière partie, j'explique comment tirer parti des propriétés calculatoires de la théorie des types cubique pour obtenir des preuves synthétiques élégantes de résultats classiques de la théorie de l'homotopie, notamment la construction de la fibration de Hopf et le lemme 3x3 pour les sommes amalgamées homotopiques

Observational Equality: Now For Good

International audienceBuilding on the recent extension of dependent type theory with a universe of definitionally proof-irrelevant types, we introduce TT obs , a new type theory based on the setoidal interpretation of dependent type theory. TT obs equips every type with an identity relation that satisfies function extensionality, propositional extensionality, and definitional uniqueness of identity proofs (UIP). Compared to other existing proposals to enrich dependent type theory with these principles, our theory features a notion of reduction that is normalizing and provides an algorithmic canonicity result, which we formally prove in Agda using the logical relation framework of Abel et al. Our paper thoroughly develops the meta-theoretical properties of TT obs , such as the decidability of the conversion and of the type checking, as well as consistency. We also explain how to extend our theory with quotient types, and we introduce a setoidal version of Swan's Id types that turn it into a proper extension of MLTT with inductive equality

Impredicative Observational Equality

International audienceIn dependent type theory, impredicativity is a powerful logical principle that allows the definition of propositions that quantify over arbitrarily large types, potentially resulting in self-referential propositions. Impredicativity can provide a system with increased logical strength and flexibility, but in counterpart it comes with multiple incompatibility results. In particular, Abel and Coquand showed that adding definitional uniqueness of identity proofs (UIP) to the main proof assistants that support impredicative propositions (Coq and Lean) breaks the normalization procedure, and thus the type-checking algorithm. However, it was not known whether this stems from a fundamental incompatibility between UIP and impredicativity or if a more suitable algorithm could decide type-checking for a type theory that supports both. In this paper, we design a theory that handles both UIP and impredicativity by extending the recently introduced observational type theory TTobs with an impredicative universe of definitionally proof-irrelevant types, as initially proposed in the seminal work on observational equality of Altenkirch et al. We prove decidability of conversion for the resulting system, that we call CCobs , by harnessing proof-irrelevance to avoid computing with impredicative proof terms. Additionally, we prove normalization for CCobs in plain Martin-Löf type theory, thereby showing that adding proof-irrelevant impredicativity does not increase the computational content of the theory

Engineering logical relations for MLTT in Coq

International audienceWe report on a mechanization in the Coq proof assistant of the decidability of conversion and type-checking for Martin-Löf Type Theory (MLTT), extending a previous Agda formalization. Our development proves the decidability not only of conversion, but also of type-checking, using bidirectional derivations that are canonical for typing. Moreover, we wish to narrow the gap between the object theory we formalize (currently MLTT with Π, Σ, N and one universe) and the metatheory used to prove the normalization result, e.g., MLTT, to a mere difference of universe levels. We thus avoid induction-recursion or impredicativity, which are central in previous work. Working in Coq, we also investigate how its features, including universe polymorphism and the metaprogramming facilities provided by tactics, impact the development of the formalization compared to the development style in Agda. The development is freely accessible on GitHub [2]

Martin-Löf à la Coq

A mechanization of the meta-theory of Martin-Löf type theory, in Coq, using logical relations. Also contains an executable and certified type-checker

Martin-Löf à la Coq

Keywords: Dependent type systems, Bidirectional typing, Logical relationsWe present an extensive mechanization of the metatheory of Martin-Löf Type Theory (MLTT) in the Coq proof assistant. Our development builds on pre-existing work in Agda to show not only the decidability of conversion, but also the decidability of type checking, using an approach guided by bidirectional type checking. From our proof of decidability, we obtain a certified and executable type checker for a full-fledged version of MLTT with support for Π, Σ, ℕ, and Id types, and one universe. Our development does not rely on impredicativity, induction-recursion or any axiom beyond MLTT extended with indexed inductive types and a handful of predicative universes, thus narrowing the gap between the object theory and the metatheory to a mere difference in universes. Furthermore, our formalization choices are geared towards a modular development that relies on Coq's features, e.g. universe polymorphism and metaprogramming with tactics