Initial Semantics for higher-order typed syntax

We present an initial semantics result for typed higher-order syntax based on monads and modules over monads. The notion of module generalizes the substitution structure of monads. For a simply typed binding signature S we define a representation of S to be a monad equipped with a morphism of modules for each of its arities. The monad of abstract syntax of S then is the initial object in the category of representations of S, and the constructors are module morphisms. Our theory is implemented and proved correct in the proof assistant Coq, making heavy use of dependent types. The monadic theory gives rise to an implementation of syntax where both terms and variables are intrinsically typed. Terms are implemented as doubly parametrized data types, dependent on a typed set of variables as well as on object types. The implementation is fully constructive. Given any signature, the implementation automatically provides the initial representation, recursion principles, a certified substitution operation and – via initiality – a morphism towards any other representation of S.