Polymorphic type systems such as System F enjoy the parametricity property: polymorphic functions cannot inspect their type argument and will therefore apply the same algorithm to any type they are instantiated on. This idea is formalized mathematically in Reynolds's theory of relational parametricity, which allows the metatheoretical derivation of parametricity theorems about all values of a given type. Although predicative System F embeds into dependent type systems such as Martin-Löf Type Theory (MLTT), parametricity does not carry over as easily. The identity extension lemma, which is crucial if we want to prove theorems involving equality, has only been shown to hold for small types, excluding the universe. We attribute this to the fact that MLTT uses a single type former Π to generalize both the parametric quantifier ∀ and the type former → which is non-parametric in the sense that its elements may use their argument as a value. We equip MLTT with parametric quantifiers ∀ and ∃ alongside the existing Π and Σ, and provide relation type formers for proving parametricity theorems internally. We show internally the existence of initial algebras and final co-algebras of indexed functors both by Church encoding and, for a large class of functors, by using sized types. We prove soundness of our type system by enhancing existing iterated reflexive graph (cubical set) models of dependently typed parametricity by distinguishing between edges that express relatedness of objects (bridges) and edges that express equality (paths). The parametric functions are those that map bridges to paths. We implement an extension to the Agda proof assistant that type-checks proofs in our type system.

We introduce MTT, a dependent type theory which supports multiple modalities. MTT is parametrized by a mode theory which specifies a collection of modes, modalities, and transformations between them. We show that different choices of mode theory allow us to use the same type theory to compute and reason in many modal situations, including guarded recursion, axiomatic cohesion, and parametric quantification. We reproduce examples from prior work in guarded recursion and axiomatic cohesion, thereby demonstrating that MTT constitutes a simple and usable syntax whose instantiations intuitively correspond to previous handcrafted modal type theories. In some cases, instantiating MTT to a particular situation unearths a previously unknown type theory that improves upon prior systems. Finally, we investigate the metatheory of MTT. We prove the consistency of MTT and establish canonicity through an extension of recent type-theoretic gluing techniques. These results hold irrespective of the choice of mode theory, and thus apply to a wide variety of modal situations.

We introduce MTT, a dependent type theory which supports multiple modalities. MTT is parametrized by a mode theory which specifies a collection of modes, modalities, and transformations between them. We show that different choices of mode theory allow us to use the same type theory to compute and reason in many modal situations, including guarded recursion, axiomatic cohesion, and parametric quantification. We reproduce examples from prior work in guarded recursion and axiomatic cohesion-demonstrating that MTT constitutes a simple and usable syntax whose instantiations intuitively correspond to previous handcrafted modal type theories. In some cases, instantiating MTT to a particular situation unearths a previously unknown type theory that improves upon prior systems. Finally, we investigate the metatheory of MTT. We prove the consistency of MTT and establish canonicity through an extension of recent type-theoretic gluing techniques. These results hold irrespective of the choice of mode theory, and thus apply to a wide variety of modal situations.

Dependent type theory allows us to write programs and to prove properties about those programs in the same language. However, some properties do not require much proof, as they are evident from a program's implementation, e.g. if a polymorphic program is not ad hoc but relationally parametric, then we get parametricity theorems for free. If we want to safely shortcut proofs by relying on the evident good behaviour of a program, then we need a type-level guarantee that the program is indeed well-behaved. This can be achieved by annotating function types with a modality describing the behaviour of functions.We consider a dependent type system with modalities for relational parametricity, irrelevance (i.e. type-checking time erasability of an argument) and ad hoc polymorphism. The interplay of three modalities and dependent types raises a number of questions. For example: If a term depends on a variable with a given modality, then how should its type depend on it? Are all modalities always applicable, e.g. should we consider parametric functions from the booleans to the naturals? Do we need any further modalities in order to properly reason about these three?We develop a type system, based on a denotational presheaf model, that answers these questions. The core idea is to equip every type with a number of relations -just equality for small types, but more for larger types -and to describe function behaviour by saying how functions act on those relations. The system has modality-aware equality judgements (ignoring irrelevant parts) and modality-aware proving operators (for proving free theorems) which are even self-applicable. It also supports sized types, some form of intersection and union types, and parametric quantification over algebraic structures. We prove soundness in a denotational presheaf model. CCS Concepts • Theory of computation → Type theory;