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This paper discusses an application of the higher-order abstract syntax technique to general-purpose theorem proving, yielding shallow embeddings of the binders of formalized languages. Higher-order abstract syntax has been applied with success in specialized logical frameworks which satisfy a closed-world assumption. As more general environments (like Isabelle/HOL or Coq) do not support this closed-world assumption, higher-order abstract syntax may yield exotic terms, that is, datatypes may produce more terms than there should actually be in the language. The work at hand demonstrates how such exotic terms can be eliminated by means of a two-level well-formedness predicate, further preparing the ground for an implementation of structural induction in terms of rule induction, and hence providing fully-fledged syntax analysis. In order to apply and justify well-formedness predicates, the paper develops a proof technique based on a combination of instantiations and reabstractions of higher-order terms. As an application, syntactic principles like the theory of contexts (as introduced by Honsell, Miculan, and Scagnetto) are derived, and adequacy of the predicates is shown, both within a formalization of the π-calculus in Isabelle/HOL.

Higher-order abstract syntax is a natural way to formalize programming languages with binders, like the π-calculus, because α-conversion, instantiations and capture avoidance are delegated to the meta-level of the provers, making tedious substitutions superfluous. However, such formalizations usually lack structural induction, which makes syntax-analysis impossible. Moreover, when applied in logical frameworks with object-logics, like Isabelle/HOL or standard extensions of Coq, exotic terms can be defined, for which important syntactic properties become invalid. The paper presents a formalization of the π-calculus in Isabelle/HOL, using well-formedness predicates which both eliminate exotic terms and yield structural induction. These induction-principles are then used to derive the Theory of Contexts fully within the mechanization. 1 Motivation The π-calculus was introduced to model and analyse mobile systems [18,17]. In it, communication channels and messages belong to the same sort, called names. This simplicity gives the π-calculus the power to encode the λ-calculus [16], as well as higher-order object-oriented and imperative languages [27,26]. Communications are synchronous, that is, a senderāb.P transmits a message b to a recipient ax.Q, in a transitionāb.P | ax.Q τ −→ P | Q{b/x}. Usually, a substitution is applied to describe that b replaces x in Q. This can be tedious for processes with binders, like Q = (νb)Q , where a further substitution is necessary to avoid name-capture: Q{b/x} = α (νb)Q {b /b, b/x}. Higher-order abstract syntax builds upon a functional view, considering binders as abstractions with respect to an underlying λ-calculus to which the replacement of names and capture-avoidance are delegated. The above transition could thus be rewritten asāb.P | ax.f Q (x) τ −→ P | f Q (b), where the function f Q (x) = (νb)f Q (b, x) corresponds to the process Q, and the process f Q (b) represents Q{b/x}. Proofs in the π-calculus, and in particular bisimulation proofs, tend to be very large and tedious, hence machine-assistance is necessary to prevent errors.

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We study the use of the-calculus for semantical descriptions of languages such a s Concurrent Idealised ALGOL (CIA), combining imperative, functional and concurrent features. We rst present a n operational semantics for CIA, given by SOS rules and a contextual form of behavioural equivalence then a-calculus semantics. As behavioural equivalence on-calculus processes we c hoose the standard (weak early) bisimilarity. W e compare the two s e m a n tics, demonstrating that there is a close operational correspondence between them and that the-calculus semantics is sound. This allows for applying the-calculus theory in proving behavioural properties of CIA phrases. We discuss laws and examples which have served as benchmarks to various semantics, and a more complex example involving procedures of higher order.