We introduce System F C , which extends System F with support for non-syntactic type equality. There are two main extensions: (i) explicit witnesses for type equalities, and (ii) open, non-parametric type functions, given meaning by toplevel equality axioms. Unlike System F, F C is expressive enough to serve as a target for several different source-language features, including Haskell's newtype, generalised algebraic data types, associated types, functional dependencies, and perhaps more besides. System F with Type Equality Coercions AbstractWe introduce System FC, which extends System F with support for non-syntactic type equality. There are two main extensions: (i) explicit witnesses for type equalities, and (ii) open, non-parametric type functions, given meaning by top-level equality axioms. Unlike System F, FC is expressive enough to serve as a target for several different source-language features, including Haskell's newtype, generalised algebraic data types, associated types, functional dependencies, and perhaps more besides.

We present a general framework HM(X) for type systems with constraints. The framework stays in the tradition of the Hindley/Milner type system. Its type system instances are sound under a standard untyped compositional semantics. We can give a generic type inference algorithm for HM(X) so that, under sufficient conditions on X, type inference will always compute the principal type of a term. We discuss instances of the framework that deal with polymorphic records, equational theories, and subtypes. © 1999 John Wiley & Sons, Inc.

Advanced type system features, such as GADTs, type classes, and type families have proven to be invaluable language extensions for ensuring data invariants and program correctness among others. Unfortunately, they pose a tough problem for type inference, because they introduce local type assumptions.In this article we present a novel constraint-based type inference approach for local type assumptions. Our system, called OutsideIn(X), is parameterised over the particular underlying constraint domain X, in the same way as HM(X). This stratification allows us to use a common metatheory and inference algorithm.Going beyond the general framework, we also give a particular constraint solver for X = type classes + GADTs + type families, a non-trivial challenge in its own right.

Functional dependencies are a popular and useful extension to Haskell style type classes. We give a reformulation of functional dependencies in terms of Constraint Handling Rules (CHRs). In previous work, CHRs have been employed for describing user-programmable type extensions in the context of Haskell style type classes. Here, we make use of CHRs to provide for the first time a concise result that under some sufficient conditions, functional dependencies allow for sound, complete and decidable type inference. The sufficient conditions imposed on functional dependencies can be very limiting. We show how to safely relax these conditions and suggest several sound extensions of functional dependencies. Our results allow for a better understanding of functional dependencies and open up the opportunity for new applications.

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