Definitional Functoriality for Dependent (Sub)Types

Théo Laurent (1) , Meven Lennon-Bertrand (2) , Kenji Maillard (3, 4)

1 PROSECCO - Programming securely with cryptography
2 Computer Laboratory [Cambridge]
3 GALLINETTE - Gallinette : vers une nouvelle génération d'assistant à la preuve
4 LS2N - Laboratoire des Sciences du Numérique de Nantes

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 : Type -> Type, equipped with a mapF : (A -> B) -> FA -> FB, 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. This is the extended version of "Definitional functoriality for Dependent (Sub)Types" (ESOP'24).

Mots clés: Subtyping; Dependent type system; Bidirectional typing; Logical relations

Domaines: Informatique [cs]

Théo Laurent, Meven Lennon-Bertrand, Kenji Maillard. Definitional Functoriality for Dependent (Sub)Types. 2023. ⟨hal-04160858v3⟩