Nominal Reasoning Techniques in Coq

Aydemir, Brian; Bohannon, Aaron; Weirich, Stephanie

doi:10.1016/j.entcs.2007.01.028

Cited by 27 publications

(31 citation statements)

References 1 publication

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“…However, the use of the notion of support, as opposed to the usual notion of free atoms, is crucial for this work: the bijective set we describe in the next section includes some functions, and for those it is far from obvious what the definition of the set of free atoms should be (the obstacle is to find an appropriate definition for free variables of functions with type, say α 1 ⇒ α 2 , in terms of the free variables for elements of the type α 1 and α 2 ). Contrast this with the definition of permutation for functions given in (2), which is defined in terms of the permutation acting on the domain and co-domain of functions. It will turn out that, albeit slightly unwieldy, Definition 3 coincides exactly with what one intuitively associates with the set of free atoms for the functions we shall use.…”

Section: Atoms Permutations and Supportmentioning

confidence: 90%

“…using in the last clause the permutations operation for functions given in (2). It is not hard to show that phi is a permutation type (routine induction over the structure of phi-terms).…”

Section: Proposition 2 the Relation ≈ Is An Equivalence Relationmentioning

confidence: 99%

“…Unwinding this definition and the permutation operations given in (2), one can often easily calculate the support for "finitary" permutation types such as:…”

Section: Atoms Permutations and Supportmentioning

confidence: 99%

See 2 more Smart Citations

Nominal Techniques in Isabelle/HOL

Urban¹

2008

J Autom Reasoning

140

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This paper describes a formalisation of the lambda-calculus in a HOLbased theorem prover using nominal techniques. Central to the formalisation is an inductive set that is bijective with the alpha-equated lambda-terms. Unlike de-Bruijn indices, however, this inductive set includes names and reasoning about it is very similar to informal reasoning with "pencil and paper". To show this we provide a structural induction principle that requires to prove the lambda-case for fresh binders only. Furthermore, we adapt work by Pitts providing a recursion combinator for the inductive set. The main technical novelty of this work is that it is compatible with the axiom of choice (unlike earlier nominal logic work by Pitts et al); thus we were able to implement all results in Isabelle/HOL and use them to formalise the standard proofs for Church-Rosser, strong-normalisation of beta-reduction, the correctness of the type-inference algorithm W, typical proofs from SOS and much more.

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Section: Atoms Permutations and Supportmentioning

confidence: 90%

“…using in the last clause the permutations operation for functions given in (2). It is not hard to show that phi is a permutation type (routine induction over the structure of phi-terms).…”

Section: Proposition 2 the Relation ≈ Is An Equivalence Relationmentioning

confidence: 99%

See 1 more Smart Citation

Nominal Techniques in Isabelle/HOL

Urban¹

2008

J Autom Reasoning

140

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“…Many programming languages use de Bruijn numbers in their internal representations for machine manipulation during operations such as type checking. The idea of using names for free variables and numbers for bound variables, known as the locally nameless approach [Charguéraud, 2012], is employed for formalizing programming language metatheory [Aydemir et al, 2006[Aydemir et al, , 2008. Also, de Bruijn numbers, combined with explicit substitution, have been introduced in higherorder unification [Dowek et al, 2000] to improve the efficiency of unification.…”

Section: Introductionmentioning

confidence: 99%

Efficiency of a good but not linear nominal unification algorithm

Ma¹,

Siek²,

Christiansen³

et al. 2018

EasyChair Preprints

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We present a nominal unification algorithm that runs in O(n × log(n) × G(n)) time, where G is the functional inverse of Ackermann's function. Nominal unification generates a set of variable assignments if there exists one, that makes terms involving binding operations α-equivalent. We preserve names while using special representations of de Bruijn numbers to enable efficient name management. We use Martelli-Montanari style multi-equation reduction to generate these name management problems from arbitrary unification terms.

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“…In contrast to the approach underlying our work, no separate meta-logic has as yet been developed for reasoning about nominal logic descriptions. Reasoning about specifications written in this logic is instead realized by axiomatizing the primitives of the logic in a rich system such as Coq or Isabelle/HOL [1,37]. This approach has proved successful for many applications.…”

Section: Related Workmentioning

confidence: 99%