Note: This document accompanies the paper “Practical type inference for arbitrary-rank types ” [6]. Prior reading of the main paper is required. 1 Contents

The relation of inclusion between types has been suggested by the practice of programming, as it enriches the polymorphism of functional languages. We propose a simple (and linear) calculus of sequents for subtyping as logical entailment. This allows us to derive a complete and coherent approach to subtyping from a few, logically meaningful, sequents. In particular, transitivity and anti-symmetry will be derived from elementary logical principles, which stresses the power of sequents and Gentzen-style proof methods. Proof techniques based on cut-elimination will be at the core of our results. 1 Introduction 1.1 Motivations, Theories and Models In recent years, several extensions of core functional languages have been proposed to deal with the notion of subtyping; see, for example, [CW85, Mit88, BL90, BCGS91, CMMS91, CG92, PS94, Tiu96, TU96]. These extensions were suggested by the practice of programming in computer science. In particular, they were inspired by the notion of inheritance...

The relation of inclusion between types has been suggested by the practice of programming as it enriches the polymorphism of functional languages. We propose a simple (and linear) sequent calculus for subtyping as logical entailment. This allows us to derive a complete and coherent approach to subtyping from a few, logically meaningful sequents. In particular, transitivity and anti-symmetry will be derived from elementary logical principles. 1 Introduction 1.1 Motivations, theories and models In recent years, several extensions of core functional languages have been proposed to deal with the notion of subtyping; see, for example, [CW85, Mit88, BL90, BCGS91, CMMS91, CG92, PS94, Tiu96, TU96]. These extensions were suggested by the practice of programming in computer science. In particular, they were inspired by the notion of inheritance as used in object-oriented programming languages, or by other concrete implementations of the following form of polymorphism: data living in a type oe, ...

We introduce new modal logical calculi that describe subtyping properties of Cartesian product and disjoint union type constructors as well as mutually-recursive types defined using those type constructors. Basic Logic of Subtyping S extends classical propositional logic by two new binary modalities ⊗ and ⊕. An interpretation of S is a function that maps standard connectives into set-theoretical operations (intersection, union, and complement) and modalities into Cartesian product and disjoint union type constructors. This allows S to capture many subtyping properties of the above type constructors. We also consider logics Sρ and S ω ρ that incorporate into S mutually-recursive types over arbitrary and well-founded universes correspondingly. The main results are completeness of the above three logics with respect to appropriate type universes. In addition, we prove Cut elimination theorem for S and establish decidability of S and S ω ρ.

A type may be a subtype of another type. The intuition about this should be clear: a type is a type of data, some data then may live in a given type as well as in a larger one, up to a simple "transformation". The advantage is that those data may be "seen" or used in different contexts. The formal treatment of this intuition, though, is not so obvious, in particular when data may be programs. In Object Oriented Programming, where the issue of "reusing data" is crucial, there has been a long-lasting discussion on "inheritance" and ... little agreement. There are several ways to understand and formalize inheritance, which depend on the specific programming environment used. Since early work of Cardelli and Wegner, there has been a large amount of papers developing several possible functional approaches to inheritance, as subtyping. Indeed, functional subtyping captures only one point of view on inheritance, yet this notion largely motivated most of that work. Whethe

This paper is devoted to a comprehensive study of polymorphic subtypes with products.

A type may be a subtype of another type. The intuition about this should be clear: a type is a type of data, some data then may live in a given type as well as in a larger one, up to a simple "transformation". The advantage is that those data may be "seen" or used in different contexts. The formal treatment of this intuition, though, is not so obvious, in particular when data may be programs. In Object Oriented Programming, where the issue of "reusing data" is crucial, there has been a long-lasting discussion on "inheritance" and ... little agreement. There are several ways to understand and formalize inheritance, which depend on the specific programming environment used. Since early work of Cardelli and Wegner, there has been a large amount of papers developing several possible functional approaches to inheritance, as subtyping. Indeed, functional subtyping captures only one point of view on inheritance, yet this notion largely motivated most of that work. Whethe

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arbitrary-rank types