Coercive subtyping: Theory and implementation

International audienceCoercive subtyping is a useful and powerful framework of subtyping for type theories. The key idea of coercive subtyping is subtyping as abbreviation. In this paper, we give a new and adequate formulation of T[C], the system that extends a type theory T with coercive subtyping based on a set C of basic subtyping judgements, and show that coercive subtyping is a conservative extension and, in a more general sense, a definitional extension. We introduce an intermediate system, the star-calculus T[C]^@?, in which the positions that require coercion insertions are marked, and show that T[C]^@? is a conservative extension of T and that T[C]^@? is equivalent to T[C]. This makes clear what we mean by coercive subtyping being a conservative extension, on the one hand, and amends a technical problem that has led to a gap in the earlier conservativity proof, on the other. We also compare coercive subtyping with the 'ordinary' notion of subtyping - subsumptive subtyping, and show that the former is adequate for type theories with canonical objects while the latter is not. An improved implementation of coercive subtyping is done in the proof assistant Plastic

Coherence and transitivity in coercive subtyping

The aim of this thesis is to study coherence and transitivity in coercive subtyping. Among other things, coherence and transitivity are key aspects for a coercive subtyping system to be consistent and for it to be implemented in a correct way. The thesis consists of three major parts. First, I prove that, for the subtyping rules of some parameterised inductive data types, coherence holds and the normal transitivity rule is admissible. Second, the notion of weak transitivity is introduced. The subtyping rules of a large class of parameterised inductive data types are suitable for weak transitivity, but not compatible with the normal transitivity rule. Third, I present a new formulation of coercive subtyping in order to combine incoherent coercions for the type of dependent pairs. There are two subtyping relations in the system and hence a further understanding of coherence and transitivity is needed. This thesis has the first case study of combining incoherent coercions in a single system. The thesis provides a clearer understanding of the subtyping rules for parameterised inductive data types and explains why the normal transitivity rule is not admissible for some natural subtyping rules. It also demonstrates that coherence and transitivity at type level can sometimes be very difficult issues in coercive subtyping. Besides providing theoretical understanding, the thesis also gives algorithms for implementing the subtyping rules for parameterised inductive data types

Definitional Functoriality for Dependent (Sub)Types

Dependently-typed proof assistant rely crucially on definitional equality, which relates types and terms that are automatically identified in the underlying type theory. This paper extends type theory with definitional functor laws, equations satisfied propositionally by a large class of container-like type constructors $F : \operatorname{Type} \to \operatorname{Type}$, equipped with a $\operatorname{map}_{F} : (A \to B) \to F\ A \to F\ B$, such as lists or trees. Promoting these equations to definitional ones strengthen the theory, enabling slicker proofs and more automation for functorial type constructors. This extension is used to modularly justify a structural form of coercive subtyping, propagating subtyping through type formers in a map-like fashion. We show that the resulting notion of coercive subtyping, thanks to the extra definitional equations, is equivalent to a natural and implicit form of subsumptive subtyping. The key result of decidability of type-checking in a dependent type system with functor laws for lists has been entirely mechanized in Coq

Definitional Extension in Type Theory

When we extend a type system, the relation between the original system and its extension is an important issue we want to know. Conservative extension is a traditional relation we study with. But in some cases, like coercive subtyping, it is not strong enough to capture all the properties, more powerful relation between the systems is required. We bring the idea definitional extension from mathematical logic into type theory. In this paper, we study the notion of definitional extension for type theories and explicate its use, both informally and formally, in the context of coercive subtyping

Copredication in homotopy type theory

This paper applies homotopy type theory to formal semantics of natural languages and proposes a new model for the linguistic phenomenon of copredication. Copredication refers to sentences where two predicates which assume different requirements for their arguments are asserted for one single entity, e.g., "the lunch was delicious but took forever". This paper is particularly concerned with copredication sentences with quantification, i.e., cases where the two predicates impose distinct criteria of quantification and individuation, e.g., "Fred picked up and mastered three books." In our solution developed in homotopy type theory and using the rule of existential closure following Heim analysis of indefinites, common nouns are modeled as identifications of their aspects using HoTT identity types, e.g., the common noun book is modeled as identifications of its physical and informational aspects. The previous treatments of copredication in systems of semantics which are based on simple type theory and dependent type theories make the correct predictions but at the expense of ad hoc extensions (e.g., partial functions, dot types and coercive subtyping). The model proposed here, also predicts the correct results but using a conceptually simpler foundation and no ad hoc extensions

Type-Based Termination, Inflationary Fixed-Points, and Mixed Inductive-Coinductive Types

Type systems certify program properties in a compositional way. From a bigger program one can abstract out a part and certify the properties of the resulting abstract program by just using the type of the part that was abstracted away. Termination and productivity are non-trivial yet desired program properties, and several type systems have been put forward that guarantee termination, compositionally. These type systems are intimately connected to the definition of least and greatest fixed-points by ordinal iteration. While most type systems use conventional iteration, we consider inflationary iteration in this article. We demonstrate how this leads to a more principled type system, with recursion based on well-founded induction. The type system has a prototypical implementation, MiniAgda, and we show in particular how it certifies productivity of corecursive and mixed recursive-corecursive functions.Comment: In Proceedings FICS 2012, arXiv:1202.317

Subtype Universes

We introduce a new concept called a subtype universe, which is a collection of subtypes of a particular type. Amongst other things, subtype universes can model bounded quantification without undecidability. Subtype universes have applications in programming, formalisation and natural language semantics. Our construction builds on coercive subtyping, a system of subtyping that preserves canonicity. We prove Strong Normalisation, Subject Reduction and Logical Consistency for our system via transfer from its parent system UTT[?]. We discuss the interaction between subtype universes and other sorts of universe and compare our construction to previous work on Power types

A Metatheoretic Analysis of Subtype Universes

Subtype universes were initially introduced as an expressive mechanisation of bounded quantification extending a modern type theory. In this paper, we consider a dependent type theory equipped with coercive subtyping and a generalisation of subtype universes. We prove results regarding the metatheoretic properties of subtype universes, such as consistency and strong normalisation. We analyse the causes of undecidability in bounded quantification, and discuss how coherency impacts the metatheoretic properties of theories implementing bounded quantification. We describe the effects of certain choices of subtyping inference rules on the expressiveness of a type theory, and examine various applications in natural language semantics, programming languages, and mathematics formalisation