FTtype

FTtype is a theoretical construct in programming language theory used to classify function types by their arity. In this framework, a type FTtype[n](A1, ..., An, R) denotes the type of functions that take n arguments with types A1 through An and return a result of type R. The arity index n provides a uniform way to distinguish between uncurried multi-argument functions and the traditional curried function form, enabling more explicit type-level reasoning about application.

Formal semantics around FTtype treat application as consuming a fixed number of inputs to produce a result. If f has type FTtype[n](A1, ..., An, R) and arguments a1 through an have types A1 through An, then applying f to these arguments yields a value of type R. In languages with currying, FTtype can be related to a sequence of single-argument functions via eta-conversion or explicit uncurry transformations, allowing both representations to be treated within a single type system.

FTtype has appeared in academic discussions of polyadic or multi-argument function types, arity polymorphism, and type-level programming. It is not part of a single standardized syntax, and different authors use varied notations to express the same concept. Its primary value is to provide a precise, arity-indexed abstraction for function types, aiding formal proofs, type-directed optimizations, and the design of experimental languages that need explicit handling of function arity.

Example use cases include formalizing libraries that expose high-arity combinators, reasoning about composition of uncurried functions, and comparing uncurried versus curried representations in type-safe code bases. See also function type, arity, and polyadic types.