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Nominal Unification
 Theoretical Computer Science
, 2003
"... We present a generalisation of firstorder unification to the practically important case of equations between terms involving binding operations. A substitution of terms for variables solves such an equation if it makes the equated terms #equivalent, i.e. equal up to renaming bound names. For the a ..."
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Cited by 52 (20 self)
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We present a generalisation of firstorder unification to the practically important case of equations between terms involving binding operations. A substitution of terms for variables solves such an equation if it makes the equated terms #equivalent, i.e. equal up to renaming bound names. For the applications we have in mind, we must consider the simple, textual form of substitution in which names occurring in terms may be captured within the scope of binders upon substitution. We are able to take a `nominal' approach to binding in which bound entities are explicitly named (rather than using nameless, de Bruijnstyle representations) and yet get a version of this form of substitution that respects #equivalence and possesses good algorithmic properties. We achieve this by adapting an existing idea and introducing a key new idea. The existing idea is terms involving explicit substitutions of names for names, except that here we only use explicit permutations (bijective substitutions). The key new idea is that the unification algorithm should solve not only equational problems, but also problems about the freshness of names for terms. There is a simple generalisation of the classical firstorder unification algorithm to this setting which retains the latter's pleasant properties: unification problems involving #equivalence and freshness are decidable; and solvable problems possess most general solutions.
Strong Induction Principles in the Locally Nameless Representation of Binders (Preliminary Notes)
"... Abstract. When using the locally nameless representation for binders, proofs by rule induction over an inductively defined relation traditionally involve a weak and strong version of this relation, and a proof that both versions derive the same judgements. In these notes we demonstrate with examples ..."
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Cited by 3 (1 self)
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Abstract. When using the locally nameless representation for binders, proofs by rule induction over an inductively defined relation traditionally involve a weak and strong version of this relation, and a proof that both versions derive the same judgements. In these notes we demonstrate with examples that it is often sufficient to define just the weak version, using the infrastructure provided by the nominal Isabelle package to automatically derive (in a uniform way) a strong induction principle for this weak version. The derived strong induction principle offers a similar convenience in induction proofs as the traditional approach using weak and strong versions of the definition. From our experience, we conjecture that our technique can be used in many rule and structural induction proofs. 1
GFOL: A TermGeneric Logic for Defining λCalculi
, 2006
"... Generic firstorder logic (GFOL) is a firstorder logic parameterized with terms de ned axiomatically (rather than constructively), by requiring them to only provide generic notions of free variable and substitution satisfying reasonable properties. GFOL has a complete Gentzen system generalizing th ..."
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Generic firstorder logic (GFOL) is a firstorder logic parameterized with terms de ned axiomatically (rather than constructively), by requiring them to only provide generic notions of free variable and substitution satisfying reasonable properties. GFOL has a complete Gentzen system generalizing that of FOL. An important fragment of GFOL, called HORN 2, possesses a much simpler Gentzen system, similar to traditional contextbased derivation systems of λcalculi. HORN 2 appears to be sufficient for defining virtually any λcalculi (including polymorphic and typerecursive ones) as theories inside the logic. GFOL endows its theories with a default loose semantics, complete for the specified calculi.