[kmill]

In Lean's parsed `Syntax`, binders are plain identifiers. The way this works is that identifiers can be annotated with the module it was parsed in as well as a "macro scope", which is a number that's used to make identifiers created by macros be distinct from any previously created identifiers (the current macro scope is some global state that's incremented whenever a macro is being expanded) — an identifier with this annotation is called a hygienic identifier, and when identifiers are tested for equality the annotations are tested too. With this system in place, there's nothing special you need to do to elaborate binders (and it also lets you splice together syntaxes without any regard for hygiene!). For example, `fun x => b x` elaborates by (1) adding a variable `x` to the local scope, (2) elaborating `b x` in that scope, and then (3) abstracting `x` to make the lambda. The key here is that `x` is a hygienic identifier, so an `x` that's from a different module or macro scope won't be captured by the binder `x`.

Yes you can define the syntax that's in the article in Lean. A version of this is the Mathlib `notation3` command, but it's for defining notation rather than re-using the function name (e.g. using a union symbol for `Set.iUnion`), and also the syntax is a bit odd: notation3 "⋃ "(...)", "r:60:(scoped f => iUnion f) => r

The ideas in the article are neat, and I'll have to think about whether it's something Lean could adopt in some way... Support for nested binders would be cool too. For example, I might be able to see something like `List.all (x in xs) (y in ys) => x + y < 10` for `List.all (fun x => List.all (fun y => x + y < 10) ys) xs`.

[JoelMcCracken]

ah nice explanation. I've actually read (or tried to) the "macros as a set of scopes" paper that IIUC lean 4's scoping is based upon; I did watch the talk on lean4's macro system. Does it not have some kind of "set of scopes" tracking?

[kmill]

The system used in Lean 4 is explained in https://arxiv.org/abs/2001.10490v7 (Ullrich and Moura, "Beyond Notations: Hygienic Macro Expansion for Theorem Proving Languages").

There's still a set-of-scopes system, but it seems to be pretty different from https://users.cs.utah.edu/plt/scope-sets/ (And to clarify what I wrote previously, each identifier has a list of macro scopes.)

The set-of-scopes in Lean 4 is for correct expansion of macro-creating-macros. The scopes are associated to macro expansions rather than binding forms. Lean's syntax quotations add the current macro scope to every identifier that appears in quotations in the expansion. That way, if there are multiple expansions of the same macro for example, it's not possible for identifiers in the expansion to collide. There's an example in section 3.2 of a macro macro that defines some top-level definitions.

(Section 8 of the paper, related work, suggests that set-of-scopes in Lean and Racket are closer than I'm understanding; I think what's going on is that Lean has a separate withFreshMacroScope primitive that macros can invoke, and syntax quotations participate in macro scopes, whereas Racket seems to give this responsibility to binding forms, and they're responsible for adding macro scopes to their binders and bodies. I'm a bit unclear on this.)