Inductive Datatypes in HOL — Lessons Learned in Formal-Logic Engineering

Berghofer, Stefan; Wenzel, Markus

doi:10.1007/3-540-48256-3_3

Cited by 50 publications

(39 citation statements)

References 12 publications

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“…on our own. 7 We first formalized everything using the latter alternative, where we just defined an inductive datatype [4] with constructors for atomic constraints, one constructor for building implications of the shape ·∧· · ·∧· −→ · (where the set of premises is conveniently represented as a list), and one constructor for universal quantification which takes a variable and a constraint as argument.…”

Section: Third Problem: Solving Conditional Constraintsmentioning

confidence: 99%

Formalizing Bounded Increase

Thiemann

2013

Interactive Theorem Proving

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Abstract. Bounded increase is a termination technique where it is tried to find an argument x of a recursive function that is increased repeatedly until it reaches a bound b, which might be ensured by a condition x < b. Since the predicates like < may be arbitrary user-defined recursive functions, an induction calculus is utilized to prove conditional constraints. In this paper, we present a full formalization of bounded increase in the theorem prover Isabelle/HOL. It fills one large gap in the pen-andpaper proof, and it includes generalized inference rules for the induction calculus as well as variants of the Babylonian algorithm to compute square roots. These algorithms were required to write executable functions which can certify untrusted termination proofs from termination tools that make use of bounded increase. And indeed, the resulting certifier was already useful: it detected an implementation error that remained undetected since 2007.

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Section: Third Problem: Solving Conditional Constraintsmentioning

confidence: 99%

Formalizing Bounded Increase

Thiemann

2013

Interactive Theorem Proving

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“…4 By contrast, Isabelle/HOL allows first declaring a constant, and at later times overloading it by defining some of its instances, as in the following example: 5 In between the declaration and the instance definitions, arbitrary commands may occur, including type definitions ("typedef") and (co)datatype definitions (which are derived from typedef [7,45]). For example, the following definition introduces a type of polynomials over an arbitrary domain α, where ∀ ∞ is the "for all but finitely many" quantifier: typedef α poly ≡ {f : nat → α | ∀ ∞ n. f n = 0}…”

Section: Introductionmentioning

confidence: 99%

“…To run these examples in Isabelle, one must enclose in overloading blocks the overloaded definitions of constants and add the overloaded attribute to type definitions that depend on overloaded constants; in addition, one must provide nonemptiness proofs for type definitions [47,Sect. 11(3,7)]. Note also that Isabelle uses ⇒ instead of → for function types and :: instead of : for typing.…”

Section: Introductionmentioning

confidence: 99%

“…The technical issues have been addressed [26,28] and are no longer exhibited in Isabelle2016. 7 In addition to taking care of these issues, one of course needs some guarantees that similar issues are not still present in the logic. To address this, in previous work [28] we have proved that the logic is now consistent, employing a semantic argument in terms of a nonstandard semantics for HOL.…”

Section: Introductionmentioning

confidence: 99%

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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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“…Isabelle's datatype package was developed by Berghofer and Wenzel [4], who drew on the work of Melham [22], Gunter [11,12], Paulson [24], and Harrison [14]. The package supports positive recursion through functions and reduces nested recursion through datatypes to mutual recursion, but otherwise allows no nesting.…”

Section: Introductionmentioning

confidence: 99%

Truly Modular (Co)datatypes for Isabelle/HOL

Blanchette

Hölzl

Lochbihler

et al. 2014

Interactive Theorem Proving

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Abstract. We extended Isabelle/HOL with a pair of definitional commands for datatypes and codatatypes. They support mutual and nested (co)recursion through well-behaved type constructors, including mixed recursion-corecursion, and are complemented by syntaxes for introducing primitive (co)recursive functions and by a general proof method for reasoning coinductively. As a case study, we ported Isabelle's Coinductive library to use the new commands, eliminating the need for tedious ad hoc constructions.

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Inductive Datatypes in HOL — Lessons Learned in Formal-Logic Engineering

Cited by 50 publications

References 12 publications

Formalizing Bounded Increase

Formalizing Bounded Increase

Comprehending Isabelle/HOL’s Consistency

Truly Modular (Co)datatypes for Isabelle/HOL

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