Martin-Löf à la Coq

Adjedj, Arthur; Lennon-Bertrand, Meven; Maillard, Kenji; Pédrot, Pierre-Marie; Pujet, Loïc

doi:10.1145/3636501.3636951

Cited by 4 publications

(2 citation statements)

References 10 publications

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“…While the Abel et al proof has been implemented in Agda [1], our mechanization is written in Coq and is based upon a recent work by Adjedj et al [2] that encodes away induction-recursion into standard inductive types through a technique known as small induction recursion [15]. Apart from this, the two formalizations are globally the same.…”

Section: Mechanizationmentioning

confidence: 99%

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“Upon This Quote I Will Build My Church Thesis”

Pédrot

2024

Proceedings of the 39th Annual ACM/IEEE Symposium on Logic in Computer Science

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Section: Mechanizationmentioning

confidence: 99%

“…But this is a consequence of the fundamental lemma, and actually, if we squint at our reducibility proofs long enough, they already contain a conversion algorithm. Also, given that our Coq proof is based on a development that proves decidability of type-checking for MLTT [2], it should be quite simple to prove it for "MLTT".…”

Section: Future Workmentioning

confidence: 99%

“Upon This Quote I Will Build My Church Thesis”

Pédrot

2024

Proceedings of the 39th Annual ACM/IEEE Symposium on Logic in Computer Science

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Observational Equality Meets CIC

Pujet,

Tabareau

2024

Lecture Notes in Computer Science

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Equality is at the heart of dependent type theory, as it plays a fundamental role in specifications and mathematical reasoning. The standard way to handle it in mainstream proof assistants such as Agda, Lean or Coq is based on Martin-Löf’s identity type, which comes straight out of the ’70s—its elegance and simplicity have earned it a long-standing use, despite a major discrepancy with traditional mathematical formulations: it does not satisfy any extensionality principles. Recently, the work on observational equality has regained interest as a new way to encode equality in proof assistants that support a universe of definitionally proof-irrelevant propositions; however it has yet to be integrated in any major proof assistant, because it is not fully compatible with another important feature of type theory: indexed inductive types. In this paper, we propose a systematic integration of indexed inductive types with an observational equality, and show that this integration can only be completely satisfactory if the observational equality satisfies the computational rule of Martin-Löf’s identity type. The second contribution of this paper is a formal proof that this additional computation rule, although not present in previous works on observational equality, can be integrated to the system without compromising the decidability of conversion.

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Definitional Functoriality for Dependent (Sub)Types

Laurent,

Lennon-Bertrand,

Maillard

2024

Lecture Notes in Computer Science

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Dependently typed proof assistant rely crucially on definitional equality, which relates types and terms that are automatically identified in the underlying type theory. This paper extends type theory with definitional functor laws, equations satisfied propositionally by a large class of container-like type constructors $$F : {\text {Type}}\rightarrow {\text {Type}}$$ F : Type → Type , equipped with a $$\textrm{map}_{F} {{\,\mathrm{:}\,}}(A \rightarrow B) \rightarrow F\,A \rightarrow F\,B$$ map F : ( A → B ) → F A → F B , such as lists or trees. Promoting these equations to definitional ones strengthens the theory, enabling slicker proofs and more automation for functorial type constructors. This extension is used to modularly justify a structural form of coercive subtyping, propagating subtyping through type formers in a map-like fashion. We show that the resulting notion of coercive subtyping, thanks to the extra definitional equations, is equivalent to a natural and implicit form of subsumptive subtyping. The key result of decidability of type-checking in a dependent type system with functor laws for lists has been entirely mechanized in Coq.

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Martin-Löf à la Coq

Cited by 4 publications

References 10 publications

“Upon This Quote I Will Build My Church Thesis”

“Upon This Quote I Will Build My Church Thesis”

Observational Equality Meets CIC

Definitional Functoriality for Dependent (Sub)Types

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