Gluing together proof environments: Canonical extensions of LF type theories featuring locks

© F. Honsell, L. Liquori, P. Maksimovic, I. Scagnetto This work is licensed under the Creative Commons Attribution License.We present two extensions of the LF Constructive Type Theory featuring monadic locks. A lock is a monadic type construct that captures the effect of an external call to an oracle. Such calls are the basic tool for gluing together diverse Type Theories and proof development environments. The oracle can be invoked either to check that a constraint holds or to provide a suitable witness. The systems are presented in the canonical style developed by the CMU School. The first system, CLLF/p,is the canonical version of the system LLF p, presented earlier by the authors. The second system, CLLF p?, features the possibility of invoking the oracle to obtain a witness satisfying a given constraint. We discuss encodings of Fitch-Prawitz Set theory, call-by-value λ-calculi, and systems of Light Linear Logic. Finally, we show how to use Fitch-Prawitz Set Theory to define a type system that types precisely the strongly normalizing terms

Formal foundations for semantic theories of nominalisation

This paper develops the formal foundations of semantic theories dealing with various kinds of nominalisations. It introduces a combination of an event-calculus with a type-free theory which allows a compositional description to be given of such phenomena like Vendler's distinction between perfect and imperfect nominals, iteration of gerunds and Cresswell's notorious non-urrival of'the train examples. Moreover, the approach argued for in this paper allows a semantic explanation to be given for a wide range of grammatical observations such as the behaviour of certain tpes of nominals with respect to their verbal contexts or the distribution of negation in nominals

Refinement Types for Logical Frameworks and Their Interpretation as Proof Irrelevance

Refinement types sharpen systems of simple and dependent types by offering expressive means to more precisely classify well-typed terms. We present a system of refinement types for LF in the style of recent formulations where only canonical forms are well-typed. Both the usual LF rules and the rules for type refinements are bidirectional, leading to a straightforward proof of decidability of typechecking even in the presence of intersection types. Because we insist on canonical forms, structural rules for subtyping can now be derived rather than being assumed as primitive. We illustrate the expressive power of our system with examples and validate its design by demonstrating a precise correspondence with traditional presentations of subtyping. Proof irrelevance provides a mechanism for selectively hiding the identities of terms in type theories. We show that LF refinement types can be interpreted as predicates using proof irrelevance, establishing a uniform relationship between two previously studied concepts in type theory. The interpretation and its correctness proof are surprisingly complex, lending support to the claim that refinement types are a fundamental construct rather than just a convenient surface syntax for certain uses of proof irrelevance

Constrained Type Families

We present an approach to support partiality in type-level computation without compromising expressiveness or type safety. Existing frameworks for type-level computation either require totality or implicitly assume it. For example, type families in Haskell provide a powerful, modular means of defining type-level computation. However, their current design implicitly assumes that type families are total, introducing nonsensical types and significantly complicating the metatheory of type families and their extensions. We propose an alternative design, using qualified types to pair type-level computations with predicates that capture their domains. Our approach naturally captures the intuitive partiality of type families, simplifying their metatheory. As evidence, we present the first complete proof of consistency for a language with closed type families.Comment: Originally presented at ICFP 2017; extended editio

Logical equivalence for subtyping object and recursive types

Subtyping in first order object calculi is studied with respect to the logical semantics obtained by identifying terms that satisfy the same set of predicates, as formalised through an assignment system. It is shown that equality in the full first order $\varsigma$-calculus is modelled by this notion, which in turn is included in a Morris-style contextual equivalence