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A type theory with definitional proof-irrelevance

Abstract : Definitional equality, a.k.a 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 thesis, 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 improvingthe so-called singleton elimination of Coq) by using techniques coming from recent development on dependent pattern matching without UIP. We show the decidability of type checking using logical relations. This extension of type theory has been implemented both in Coq and Agda.
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Submitted on : Wednesday, May 26, 2021 - 10:59:23 AM
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  • HAL Id : tel-03236271, version 1

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Gaëtan Gilbert. A type theory with definitional proof-irrelevance. Logic in Computer Science [cs.LO]. Ecole nationale supérieure Mines-Télécom Atlantique, 2019. English. ⟨NNT : 2019IMTA0169⟩. ⟨tel-03236271⟩

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