System F with coercion constraints

Abstract

We present a second-order λ-calculus with coercion constraints that generalizes a previous extension of System F with parametric coercion abstractions by allowing multiple but simultaneous type and coercion abstractions, as well as recursive coercions and equi-recursive types. This enables a uniform presentation of several type system features that had previously been studied separately: type containment, bounded and instance-bounded polymorphism, which are already encodable with parametric coercion abstraction, and ML-style subtyping constraints. Our framework allows for a clear separation of language constructs with and without computational content. We also distinguish coherent coercions that are fully erasable from potentially incoherent coercions that suspend the evaluation-and enable the encoding of GADTs.Technically, type coercions that witness subtyping relations between types are replaced by a more expressive notion of typing coercions that witness subsumption relations between typings, e.g. pairs composed of a typing environment and a type. Our calculus is equipped with full reduction that allows reduction under abstractions-but we also introduce a form of weak reduction as reduction cannot proceed under incoherent type abstractions. Type soundness is proved by adapting the step-indexed semantics technique to full reduction, moving indices inside terms so as to control the reduction steps internally-but this is only detailed in the extended version.

Full Reduction in the Face of Absurdity

“…These issues were well-understood by Cretin and Rémy [2014] and left for future work. We present an improved variant of F cc that solves both problems simultaneously.…”

“…As another instance of calculus based on erasable coercions, it is strongly inspired by the previous work on F cc Cretin and Rémy [2014] and fol- lows the same global structure of judgments. Yet, we do not assume familiarity with F cc 5 .…”

“…All runtime-irrelevant typing constructions, such as subtyping conversion and polymorphism introduction and elimination, are factorized into coercions. To express polymorphism, these coercions are typing coercions (Σ τ ) σ rather than type coercions τ σ: they also affect the typing environment Γ in which the coercion is used, by extending Γ with Σ when typechecking the premise of type τ , as described by Rule TermCoerce This factorization has been explained in previous works of Cretin and Rémy [2014] and is orthogonal to our point of interest in the present paper, namely, the interplay between program types and logical propositions in a programming system. We thus focus our presentation on propositions in general rather than coercions, and propositional truths would naturally extend to many other program logics, such as arithmetic reasoning or general refinement types.…”

“…Previous work by Cretin [2014] and Cretin and Rémy [2014] introduced the calculus F cc , built around consistent coercion abstraction, a mechanism that allows implicit abstraction over coercions and use typing-transforming coercions, provided we prove at their abstraction point that they are instantiable; such coercions are completely erasable -they're not at all present at the term level. This generalizes the traditional ML-style polymorphism in an expressive way, encompassing the type systems of System F, MLF, F <: , F η , and F ω .…”

The Essence of Dependent Object Types

“…those that correspond to a store). Cretin, Scherer and Rémy propose an alternative approach in their F cc and F th systems (Cretin and Rémy, 2014;Scherer and Rémy, 2015). They consider a more general notion of coercion constraints which come in two flavors: consistent constraints, under which reduction is safe, and (potentially) inconsistent constraints, under which reduction is forbidden.…”