Comprehending Isabelle/HOL’s Consistency

KunăźAr, Ondřej; Popescu, Andrei

doi:10.1007/978-3-662-54434-1_27

Cited by 16 publications

(28 citation statements)

References 33 publications

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“…The next two proofs rely on a sound translation between Isabelle/HOL and an extension of HOL with comprehension types (called HOLC), discussed in [21]. The main idea is that the new rules are manifestly sound when translated to HOLC.…”

Section: Appendixmentioning

confidence: 99%

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

Kunčar

Popescu

2016

Interactive Theorem Proving

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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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Section: Appendixmentioning

confidence: 99%

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

Kunčar

Popescu

2016

Interactive Theorem Proving

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“…Employing a nonstandard semantics, we proved that, after these modifications, any definitional theory is consistent. In more recent work, we gave an alternative syntactic proof, based on translating HOL to a richer logic, HOLC, having comprehension types as first-class citizens [Kunčar and Popescu 2017a]. The current paper improves on these results, by proving properties much stronger than consistency.…”

Section: :6mentioning

confidence: 81%

“…Our translations compile away the constant and type definitions, the latter being significantly more problematic due to the lack of HOL infrastructure for unfolding them. In previous work [Kunčar and Popescu 2017a] we address this infrastructure problem by introducing HOLC, an extension of HOL with comprehension/refinement types. HOL type definitions can be naturally unfolded into HOLC types, yielding a HOL to HOLC translation that was sufficient for showing the consistency of Isabelle/HOL definitions.…”

Section: Discussionmentioning

confidence: 99%

Safety and conservativity of definitions in HOL and Isabelle/HOL

Kunčar

Popescu

2017

Proc. ACM Program. Lang.

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Definitions are traditionally considered to be a safe mechanism for introducing concepts on top of a logic known to be consistent. In contrast to arbitrary axioms, definitions should in principle be treatable as a form of abbreviation, and thus compiled away from the theory without losing provability. In particular, definitions should form a conservative extension of the pure logic. These properties are crucial for modern interactive theorem provers, since they ensure the consistency of the logic, as well as a valid environment for total/certified functional programming. We prove these properties, namely, safety and conservativity, for Higher-Order Logic (HOL), a logic implemented in several mainstream theorem provers and relied upon by thousands of users. Some unique features of HOL, such as the requirement to give non-emptiness proofs when defining new types and the impossibility to unfold type definitions, make the proof of these properties, and also the very formulation of safety, nontrivial. Our study also factors in the essential variation of HOL definitions featured by Isabelle/HOL, a popular member of the HOL-based provers family. The current work improves on recent results which showed a weaker property, consistency of Isabelle/HOL's definitions.

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“…are identifiable as part of that logic, e.g., localized to a given type and/or relativised by a given predicate. This allows our framework to capture most variants of higher-order logic and type theory (including the variant underlying Isabelle/HOL itself [21,22]), and also, we believe, many of the logics surveyed by Buldt [5], including non-classical and fuzzy. But enabling "mass instantiation" that is both formal and painless requires more progress on the agenda we started here: recognizing reusable construction and proof patterns and formalizing them as abstract results.…”

Section: Instances Of the Abstract Resultsmentioning

confidence: 99%