Higher-order abstract syntax in Coq

Despeyroux, Joëlle; Felty, Amy P.; Hirschowitz, Andrè

doi:10.1007/bfb0014049

Cited by 102 publications

(107 citation statements)

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“…Both capture object-level bindings by meta-level functional bindings; "weak" refers to the considered functions mapping variables to terms, while "strong" refers to these functions mapping terms to terms. Weak HOAS approaches are taken in [18], [34], [57], [30], including in category-theoretic form (with a denotational-semantics flavor) in [23], [33], [8], [24]. Our work in this paper, the above HOAS-tailored approaches, as well as [19], the work on Hybrid [7], [44], [46], [22], as well as parametric HOAS [17], parametricity-based HOAS [35], 6 and de-Bruijn-mixed-HOAS [32], fall within strong HOAS.…”

Section: More Related Workmentioning

confidence: 99%

“…-(II.b) The general-purpose framework approach [19], [18], [7]. This approach employs a general-purpose setting for developing mathematics, such as ZF Set Theory, Calculus of Constructions, or HOL with Infinity, as the logical framework, with object-level bindings captured again by means of meta-level bindings, here typically functional bindingsthis means that terms with bindings from the object system are denoted using standard functions.…”

Section: Introduction Hoas (Higher-ordermentioning

confidence: 99%

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Strong Normalization for System F by HOAS on Top of FOAS

Popescu

Gunter

Osborn

2010

2010 25th Annual IEEE Symposium on Logic in Computer Science

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Abstract-We present a point of view concerning HOAS (Higher-Order Abstract Syntax) and an extensive exercise in HOAS along this point of view. The point of view is that HOAS can be soundly and fruitfully regarded as a definitional extension on top of FOAS (First-Order Abstract Syntax). As such, HOAS is not only an encoding technique, but also a higher-order view of a first-order reality. A rich collection of concepts and proof principles is developed inside the standard mathematical universe to give technical life to this point of view. The exercise consists of a new proof of Strong Normalization for System F. HOAS makes our proof considerably more direct than previous proofs. The concepts and results presented here have been formalized in the theorem prover Isabelle/HOL. [49] and has ever since been extensively developed in frameworks with a wide variety of features and flavors. We distinguish two main (overlapping) directions in these developments. Keywords--(I) First, the employment of a chosen meta logic as a pure logical framework, used for defining object systems for the purpose of reasoning inside those systems. A standard example is higher-order logic (HOL) as the meta logic and firstorder logic (FOL) as the object system. Thanks to affinities between the mechanisms of these two logics, one obtains an adequate encoding of FOL in HOL by merely declaring in HOL types and constants and stating the FOL axioms and rules as HOL axioms -then the mechanisms for building FOL deductions (including substitution, instantiation, etc.) are already present in the meta logic, HOL.-(II) Second, the employment of the meta-logic to reason about the represented object systems, i.e., to represent not only the object systems, but also (some of) their metatheory. (E.g., cut elimination is a property about Gentzenstyle FOL, not expressible in a standard HOAS-encoding of FOL into HOL.) While direction (I) has been quasi-saturated by the achievement of quasi-maximally convenient logical frameworks (such Edinburgh LF [31] and generic Isabelle [49]), this second direction undergoes these days a period of active research. We distinguish two main approaches here:

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Section: More Related Workmentioning

confidence: 99%

Section: Introduction Hoas (Higher-ordermentioning

confidence: 99%

Strong Normalization for System F by HOAS on Top of FOAS

Popescu

Gunter

Osborn

2010

2010 25th Annual IEEE Symposium on Logic in Computer Science

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“…The predicate abstr identifies those functions which are sufficiently parametric to be realized as the body of a λ-term, and lambda is injective on these. This predicate can be seen as a full HOAS counterpart of the valid 1 judgment in [5], but it must be defined at the de Bruijn level, since a higher-order definition would require a theory of n-ary abstractions, which is the object of current research.…”

Section: Hybrid Infrastructurementioning

confidence: 99%

Multi-level Meta-reasoning with Higher-Order Abstract Syntax

Momigliano

Ambler

2003

Lecture Notes in Computer Science

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Abstract. Combining Higher Order Abstract Syntax (HOAS) and (co)-induction is well known to be problematic. In previous work [1] we have described the implementation of a tool called Hybrid, within Isabelle HOL, which allows object logics to be represented using HOAS, and reasoned about using tactical theorem proving and principles of (co)induction. Moreover, it is definitional, which guarantees consistency within a classical type theory. In this paper we describe how to use it in a multi-level reasoning fashion, similar in spirit to other meta-logics such F Oλ ∆I N and Twelf. By explicitly referencing provability, we solve the problem of reasoning by (co)induction in presence of non-stratifiable hypothetical judgments, which allow very elegant and succinct specifications. We demonstrate the method by formally verifying the correctness of a compiler for (a fragment) of Mini-ML, following [10]. To further exhibit the flexibility of our system, we modify the target language with a notion of non-well-founded closure, inspired by Milner & Tofte [16] and formally verify via co-induction a subject reduction theorem for this modified language.

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“…Despeyroux, Felty, and Hirschowitz [7,6] show that induction principles for a restricted form of second-order abstract syntax can be derived in the Coq proof development system. To keep the definitions monotone, they introduce a separate type for variables and explicit coercions from variables to other types.…”

Section: Related Workmentioning

confidence: 99%