Skip to Main content Skip to Navigation
New interface
Journal articles

Definitional Proof-Irrelevance without K

Gaëtan Gilbert 1, 2 Jesper Cockx 3 Matthieu Sozeau 4, 5 Nicolas Tabareau 1, 2 
1 LS2N - équipe GALLINETTE - Gallinette : vers une nouvelle génération d'assistant à la preuve
Inria Rennes – Bretagne Atlantique , LS2N - Laboratoire des Sciences du Numérique de Nantes
4 PI.R2 - Design, study and implementation of languages for proofs and programs
UPD7 - Université Paris Diderot - Paris 7, CNRS - Centre National de la Recherche Scientifique, IRIF (UMR_8243) - Institut de Recherche en Informatique Fondamentale, Inria de Paris
Abstract : Definitional equality—or conversion—for a type theory with a decidable type checking is the simplest tool to prove that two objects are the same, letting the system decide just using computation. Therefore, the more things are equal by conversion, the simpler it is to use a language based on type theory. Proof-irrelevance, stating that any two proofs of the same proposition are equal, is a possible way to extend conversion to make a type theory more powerful. However, this new power comes at a price if we integrate it naively, either by making type checking undecidable or by realizing new axioms—such as uniqueness of identity proofs (UIP)—that are incompatible with other extensions, such as univalence. In this paper, taking inspiration from homotopy type theory, we propose a general way to extend a type theory with definitional proof irrelevance, in a way that keeps type checking decidable and is compatible with univalence. We provide a new criterion to decide whether a proposition can be eliminated over a type (correcting and improving the so-called singleton elimination of Coq) by using techniques coming from recent development on dependent pattern matching without UIP. We show the generality of our approach by providing implementations for both Coq and Agda, both of which are planned to be integrated in future versions of those proof assistants.
Document type :
Journal articles
Complete list of metadata

Cited literature [23 references]  Display  Hide  Download
Contributor : Nicolas Tabareau Connect in order to contact the contributor
Submitted on : Tuesday, October 16, 2018 - 1:30:52 PM
Last modification on : Thursday, December 1, 2022 - 11:00:07 AM
Long-term archiving on: : Thursday, January 17, 2019 - 1:51:07 PM


Files produced by the author(s)



Gaëtan Gilbert, Jesper Cockx, Matthieu Sozeau, Nicolas Tabareau. Definitional Proof-Irrelevance without K. Proceedings of the ACM on Programming Languages, 2019, POPL'19, pp.1-28. ⟨10.1145/329031610.1145/3290316⟩. ⟨hal-01859964v2⟩



Record views


Files downloads