Type classes and overloading in higher-order logic

Abstract. Types in Higher-Order Logic (HOL) are naturally interpreted as nonempty sets-this intuition is reflected in the type definition rule for the HOL-based systems (including Isabelle/HOL), where a new type can be defined whenever a nonempty set is exhibited. However, in HOL this definition mechanism cannot be applied inside proof contexts. We propose a more expressive type-definition rule that addresses the limitation and we prove its soundness. This higher expressive power opens the opportunity for a HOL tool that relativizes type-based statements to more flexible set-based variants in a principled way. We also address particularities of Isabelle/HOL and show how to perform the relativization in the presence of type classes. MotivationThe proof assistant community is divided in two successful camps. One camp, represented by provers such as Agda [7] According to the HOL school of thought, a main goal is to acquire a sweet spot: keep the logic as simple as possible while obtaining sufficient expressiveness. The notion of sufficient expressiveness is of course debatable, and has been debated. For example, PVS [29] includes dependent types (but excludes polymorphism), HOL-Omega [16] adds first-class type constructors to HOL, and Isabelle/HOL adds ad hoc overloading of polymorphic constants. In this paper, we want to propose a gentler extension of HOL: we do not want to promote new "first-class citizens," but merely to give better credit to an old and venerable HOL citizen: the notion of types emerging from sets.The problem we address in this paper is best illustrated by an example. Let lists : α set → α list set be the constant that takes a set A and returns the set of lists whose elements are in A, and P : α list → bool be another constant (whose definition is not important here). Consider the following statements, where we extend the usual HOL syntax by explicitly quantifying over types at the outermost level:∀α. ∃xs α list . P xs (1) ∀α. ∀A α set . A ̸ = / 0 −→ (∃xs ∈ lists A. P xs)

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“…The rule for internalization of type classes is a result of the work by Haftmann and Wenzel [13,31].…”

Section: Local Axiomatic Type Classesmentioning

confidence: 99%

From Types to Sets by Local Type Definitions in Higher-Order Logic

Kunčar

Popescu

2016

Interactive Theorem Proving

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“…HOL's type system (see [8,23]) essentially coincides with ML's. The notation t :: ν indicates that the term t has type ν.…”

Section: Isabelle/holmentioning

confidence: 89%

Term Rewriting in Logics of Partial Functions

Schmalz¹

2011

Formal Methods and Software Engineering

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Abstract. We devise a theoretical foundation of directed rewriting, a term rewriting strategy for logics of partial functions, inspired by term rewriting in the Rodin platform. We prove that directed rewriting is sound and show how to supply new rewrite rules in a soundness preserving fashion. In the context of Rodin, we show that directed rewriting makes a significant number of conditional rewrite rules unconditional. Our work not only allows us to point out a number of concrete ways of improving directed rewriting in Rodin, but also has applications in other logics of partial functions. Additionally, we give a semantics for the logic of Event-B.

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“…Polymorphic properties can also be associated to type classes, and need to be verified upon instantiation. Type classes do not require any logical extension, but are representable as predicates inside the logic- [48,Sect. 5] explains the mechanism in detail.…”

Section: Introductionmentioning

confidence: 99%

Comprehending Isabelle/HOL’s Consistency

KunăźAr¹,

Popescu²

2017

Programming Languages and Systems

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Abstract. The proof assistant Isabelle/HOL is based on an extension of HigherOrder Logic (HOL) with ad hoc overloading of constants. It turns out that the interaction between the standard HOL type definitions and the Isabelle-specific ad hoc overloading is problematic for the logical consistency. In previous work, we have argued that standard HOL semantics is no longer appropriate for capturing this interaction, and have proved consistency using a nonstandard semantics. The use of an exotic semantics makes that proof hard to digest by the community. In this paper, we prove consistency by proof-theoretic means-following the healthy intuition of definitions as abbreviations, realized in HOLC, a logic that augments HOL with comprehension types. We hope that our new proof settles the Isabelle/HOL consistency problem once and for all. In addition, HOLC offers a framework for justifying the consistency of new deduction schemas that address practical user needs.

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Type classes and overloading in higher-order logic

Cited by 106 publications

References 8 publications

From Types to Sets by Local Type Definitions in Higher-Order Logic

From Types to Sets by Local Type Definitions in Higher-Order Logic

Term Rewriting in Logics of Partial Functions

Comprehending Isabelle/HOL’s Consistency

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