Combining Higher Order Abstract Syntax with Tactical Theorem Proving and (Co)Induction

Ambler, Simon; Crole, Roy L.; Momigliano, Alberto

doi:10.1007/3-540-45685-6_3

Cited by 32 publications

(93 citation statements)

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“…On the other hand, weak HOAS is is easier to define directly inductively. However, as illustrated in this paper and in previous work [19], [7], in a general-purpose setting having strong HOAS (perhaps on top of weak HOAS as in [19], or directly on top of the firstorder syntax as here) is only a matter of some definitional work. Because variables are particular terms, strong HOAS can accommodate weak induction and recursion principles, and in fact in most situations only such weak principles are available due to the need of well-foundedness -Prop.…”

Section: More Related Workmentioning

confidence: 69%

“…Because variables are particular terms, strong HOAS can accommodate weak induction and recursion principles, and in fact in most situations only such weak principles are available due to the need of well-foundedness -Prop. 1 (similar to an axiom postulated in the Theory of Contexts [34], [15] and to a fact proved by Hybrid [7]), as well as our permutative induction for 2-abstractions expressed in Prop. 2, are examples of "weak" principles within strong HOAS.…”

Section: More Related Workmentioning

confidence: 73%

“…The HOAS-tailored framework approach yielded several theorem provers and functional programming environments (some of them already mature and with an extensive case-study record), including several extensions of LF -Twelf [5], Delphin [1], ATS [16], Beluga [53] -and Abella [3], a HOASspecialized prover based on definitional reflection. On the other hand, the Hybrid package [7], written in Isabelle/HOL, is a successful realization of the general-purpose framework approach. Later versions of this system [44], [46], [22] also import the three-level architecture idea from the HOAStailored framework approach.…”

Section: More Related Workmentioning

confidence: 99%

“…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. In weak HOAS, some of the convenience is lost, since substitution of terms for variables is not mere function application, as in 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: 69%

Section: More Related Workmentioning

confidence: 73%

Section: More Related Workmentioning

confidence: 99%

Section: More Related Workmentioning

confidence: 99%

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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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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Induction and Co-induction in Sequent Calculus

Momigliano

Tiu

2004

Lecture Notes in Computer Science

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Abstract. Proof search has been used to specify a wide range of computation systems. In order to build a framework for reasoning about such specifications, we make use of a sequent calculus involving induction and co-induction. These proof principles are based on a proof theoretic (rather than set-theoretic) notion of definition [13,20,25,51]. Definitions are akin to (stratified) logic programs, where the left and right rules for defined atoms allow one to view theories as "closed" or defining fixed points. The use of definitions makes it possible to reason intensionally about syntax, in particular enforcing free equality via unification. We add in a consistent way rules for pre and post fixed points, thus allowing the user to reason inductively and co-inductively about properties of computational system making full use of higher-order abstract syntax. Consistency is guaranteed via cut-elimination, where we give the first, to our knowledge, cut-elimination procedure in the presence of general inductive and co-inductive definitions.

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Combining Higher Order Abstract Syntax with Tactical Theorem Proving and (Co)Induction

Cited by 32 publications

References 20 publications

Strong Normalization for System F by HOAS on Top of FOAS

Strong Normalization for System F by HOAS on Top of FOAS

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

Induction and Co-induction in Sequent Calculus

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