Computer-Aided Reasoning

Kaufmann, Matt; Manolios, Panagiotis; Moore, J. Strother

doi:10.1007/978-1-4615-4449-4

Cited by 295 publications

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“…The other camp, represented by the HOL family of provers (including HOL4 [2], HOL Light [14], HOL Zero [3] and Isabelle/HOL [26]), mostly sticks to a form of classic set theory typed using simple types with rank 1 polymorphism. (Other successful provers, such as ACL2 [19] and Mizar [12], could be seen as being closer to the HOL camp, although technically they are not based on HOL. )…”

Section: Motivationmentioning

confidence: 99%

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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: Motivationmentioning

confidence: 99%

“…Let us define B = [A] θ,ξ and observe that B ∈ U thanks to (17). Notice that the bottom line of the proof was to show a semantic analog of (⋆): given (17), we obtain (19).…”

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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“…There are two commonly used paradigms for inductive theorem proving: explicit induction and implicit induction. In explicit induction (see, e.g., [9,39,30,11,12,22,31,38]), a concrete induction scheme is computed for each conjecture, and the subsequent reasoning is based on this induction scheme. Here, an induction scheme explicitly gives the base cases and the step cases, where a step case consists of an obligation and one or more hypotheses.…”

Section: Introductionmentioning

confidence: 99%

Rewriting Induction + Linear Arithmetic = Decision Procedure

Falke¹,

Kapur

2012

Lecture Notes in Computer Science

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Abstract. This paper presents new results on the decidability of inductive validity of conjectures. For this, a class of term rewrite systems (TRSs) with built-in linear integer arithmetic is introduced and it is shown how these TRSs can be used in the context of inductive theorem proving. The proof method developed for this couples (implicit) inductive reasoning with a decision procedure for the theory of linear integer arithmetic with (free) constructors. The effectiveness of the new decidability results on a large class of conjectures is demonstrated by an evaluation of the prototype implementation Sail2.

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“…For an introduction to ACL2, see the tutorials in the ACL2 web page [6]. To obtain more background on ACL2, see [5].…”

Section: Introductionmentioning

confidence: 99%

Verifying an Applicative ATP Using Multiset Relations

Martín-Mateos

Alonso

Hidalgo

et al. 2001

Computer Aided Systems Theory — EUROCAST 2001

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Abstract. We present in this paper a formalization of multiset relations in the ACL2 theorem prover [6], and we show how multisets can be used to mechanically prove non-trivial termination properties. Every relation on a set A induces a relation on finite multisets over A; it can be shown that the multiset relation induced by a well-founded relation is also wellfounded [3]. We have carried out a mechanical proof of this property in the ACL2 logic. This allows us to provide well-founded multiset relations in order to prove termination of recursive functions. Once termination is proved, the function definition is admitted as an axiom in the logic and formal mechanized reasoning about it is possible. As a major application of this tool, we show how multisets can be used to prove termination of a tableaux based theorem prover for propositional logic.

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Computer-Aided Reasoning

Abstract: 1. Formal methods (Computer science) 2. Software engineering. 3. Expert systems (Computer science) 1.

Cited by 295 publications

References 0 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

Rewriting Induction + Linear Arithmetic = Decision Procedure

Verifying an Applicative ATP Using Multiset Relations

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