## Beluga and explicit contexts

In my recent work, I found it useful to pair a term and its context in order to more easily reason about weakening the context. At the prompting of a colleague, I’ve been reading about Beluga, [1] [2], and their support for programming with explicit contexts. The idea seems neat, but I’m not quite sure I understand the motivations or implications.

So it seems Beluga has support for describing what a context contains (schemas), describing in which context a type/term is valid, and referring to the variables in a context by name without explicitly worrying about alpha-renaming. This technique supports reasoning about binders with HOAS in more settings, such as in the presence of open data and dependent types. Since HOAS simplifies reasoning about binders by taking advantage of the underlying language’s implementation of substitutions, this can greatly simplify formalized meta-theory in the presence of advanced features which previously required formalizing binders using more complicated techniques like De Bruijn indices. By including weakening, meta-variables, and parameter variables, Beluga enables meta-theory proofs involving binders to be much more natural, i.e., closer to pen-and-paper proofs.

Obviously this is great for formalized meta-theory. While I have seen how HOAS can simplify life for the meta-theorist, and seen how it fails, I don’t fully understand the strengths and weakness of this work, or how it compares to techniques such as the locally nameless. I’m also not sure if there is more to this work than a better way to handle formalization of binding (which is a fine, useful accomplishment by itself).

If anyone can elaborate on or correct my understanding, please do.