Ungrantable Types in Type Theory: Understanding Undecidable Properties and Constraints

In the context of type theory, an ungrantable type is a type that cannot be inferred or constructed using the available type constructors and constraints.

For example, in a simply typed lambda calculus with only the basic types `Nat` (natural numbers) and `Prop` (propositions), it is not possible to infer the type `Nat x Prop` because there is no way to combine the two types using the available type constructors. This type is said to be ungrantable.

In more advanced type systems, such as dependent type theory or homotopy type theory, ungrantable types can arise due to the presence of dependencies or constraints that cannot be satisfied by any available type constructor. For example, in a dependent type theory with a dependent product type `A x B`, where `A` and `B` are types that depend on each other, it may not be possible to infer the type `A x B` if there is no way to construct `A` and `B` using the available type constructors and constraints.

In general, ungrantable types can serve as a way to encode undecidable properties or constraints in a type system, and can be used to reason about the limitations of the type system itself.