[isabelle] Notation for series coefficients


I am currently thinking about introducing a type 'a egf for exponential generating functions, derived from formal power series ('a fps).

I was wondering how to best express the n-th coefficient of an egf. For polynomials, only the rather unwieldy notation "coeff f n" exists. For formal power series, "f $ n" exists.

I considered overloading the $ notation so that it works for polynomials, formal power series, and exponential generating functions. However: â introducing a typeclass does not work, because this would be a higher-order typeclass. â Vanilla âoverloadingâ works, but the type of "op $" must then be 'a â nat â 'b, which means that type inference cannot deduce the result type of "f $ 0" even when the type of "f" is known, requiring type annotations everywhere.
â Adhoc overloading would probably work, but that is somewhatâ ad-hoc.

Does anybody have other ideas?


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