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We present a general framework for specifying and reasoning about syntax with bindings. Abstract binder types are modeled using a universe of functors on sets, subject to a number of operations that can be used to construct complex binding patterns and binding-aware datatypes, including non-well-founded and infinitely branching types, in a modular fashion. Despite not committing to any syntactic format, the framework is łconcretež enough to provide definitions of the fundamental operators on terms (free variables, alpha-equivalence, and capture-avoiding substitution) and reasoning and definition principles. This work is compatible with classical higher-order logic and has been formalized in the proof assistant Isabelle/HOL.

We present the formalization of a theory of syntax with bindings that has been developed and refined over the last decade to support several large formalization efforts. Terms are defined for an arbitrary number of constructors of varying numbers of inputs, quotiented to alpha-equivalence and sorted according to a binding signature. The theory includes a rich collection of properties of the standard operators on terms, such as substitution and freshness. It also includes induction and recursion principles and support for semantic interpretation, all tailored for smooth interaction with the bindings and the standard operators.

We design and implement Zooid, a domain specific language for certified multiparty communication, embedded in Coq and implemented atop our mechanisation framework of asynchronous multiparty session types (the first of its kind). Zooid provides a fully mechanised metatheory for the semantics of global and local types, and a fully verified end-point process language that faithfully reflects the typelevel behaviours and thus inherits the global types properties such as deadlock freedom, protocol compliance, and liveness guarantees.

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