System FC with Explicit Kind Equality

System FC, the core language of the Glasgow Haskell Compiler, is an explicitly-typed variant of System F with first-class type equality proofs called coercions. This extensible proof system forms the foundation for type system extensions such as type families (type- level functions) and Generalized Algebraic Datatypes (GADTs). Such features, in conjunction with kind polymorphism and datatype promotion, support expressive compile-time reasoning. However, the core language lacks explicit kind equality proofs. As a result, type-level computation does not have access to kind- level functions or promoted GADTs, the type-level analogues to expression-level features that have been so useful. In this paper, we eliminate such discrepancies by introducing kind equalities to System FC. Our approach is based on dependent type systems with heterogeneous equality and the “Type-in-Type” axiom, yet it preserves the metatheoretic properties of FC. In particular, type checking is simple, decidable and syntax directed. We prove the preservation and progress theorems for the extended language

System FC with Explicit Kind Equality (extended version)

System FC, the core language of the Glasgow Haskell Compiler, is an explicitly-typed variant of System F with first-class type equality proofs called coercions. This extensible proof system forms the foundation for type system extensions such as type families (type- level functions) and Generalized Algebraic Datatypes (GADTs). Such features, in conjunction with kind polymorphism and datatype promotion, support expressive compile-time reasoning. However, the core language lacks explicit kind equality proofs. As a result, type-level computation does not have access to kind- level functions or promoted GADTs, the type-level analogues to expression-level features that have been so useful. In this paper, we eliminate such discrepancies by introducing kind equalities to System FC. Our approach is based on dependent type systems with heterogeneous equality and the “Type-in-Type” axiom, yet it preserves the metatheoretic properties of FC. In particular, type checking is simple, decidable and syntax directed. We prove the preservation and progress theorems for the extended language

Sound and complete bidirectional typechecking for higher-rank polymorphism with existentials and indexed types

Bidirectional typechecking, in which terms either synthesize a type or are checked against a known type, has become popular for its applicability to a variety of type systems, its error reporting, and its ease of implementation. Following principles from proof theory, bidirectional typing can be applied to many type constructs. The principles underlying a bidirectional approach to indexed types (generalized algebraic datatypes) are less clear. Building on proof-theoretic treatments of equality, we give a declarative specification of typing based on focalization. This approach permits declarative rules for coverage of pattern matching, as well as support for first-class existential types using a focalized subtyping judgment. We use refinement types to avoid explicitly passing equality proofs in our term syntax, making our calculus similar to languages such as Haskell and OCaml. We also extend the declarative specification with an explicit rules for deducing when a type is principal, permitting us to give a complete declarative specification for a rich type system with significant type inference. We also give a set of algorithmic typing rules, and prove that it is sound and complete with respect to the declarative system. The proof requires a number of technical innovations, including proving soundness and completeness in a mutually recursive fashion.EPSRC grant EP/N02706X/

Aspect-Oriented Programming with Type Classes

We consider the problem of adding aspects to a strongly typed language which supports type classes. We show that type classes as supported by the Glasgow Haskell Compiler can model an AOP style of programming via a simple syntax-directed transformation scheme where AOP programming idioms are mapped to type classes. The drawback of this approach is that we cannot easily advise functions in programs which carry type annotations. We sketch a more principled approach which is free of such problems by combining ideas from intentional type analysis with advanced overloading resolution strategies. Our results show that type-directed static weaving is closely related to type class resolution -- the process of typing and translating type class programs

Type Variables in Patterns

For many years, GHC has implemented an extension to Haskell that allows type variables to be bound in type signatures and patterns, and to scope over terms. This extension was never properly specified. We rectify that oversight here. With the formal specification in hand, the otherwise-labyrinthine path toward a design for binding type variables in patterns becomes blindingly clear. We thus extend ScopedTypeVariables to bind type variables explicitly, obviating the Proxy workaround to the dustbin of history

Type Variables in Patterns

For many years, GHC has implemented an extension to Haskell that allows type variables to be bound in type signatures and patterns, and to scope over terms. This extension was never properly specified. We rectify that oversight here. With the formal specification in hand, the otherwise-labyrinthine path toward a design for binding type variables in patterns becomes blindingly clear. We thus extend ScopedTypeVariables to bind type variables explicitly, obviating the Proxy workaround to the dustbin of history