blog/posts/beluga.md

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date: 2022-07-12 title: a few undocumented beluga features tags: [computer, beluga, cool languages] toc: true summary: | some undocumented features i found while looking through the source code of the proof language beluga. ...

several of these are in the examples, just not in the documentation. but some of them i did find while looking through the parser.

unicode

you can call your context variables Γ and it works just fine. you can also use → and ⇒ and ⊢ if they are legible in the font you're using.

block values in substitutions

a lot of the time you have a variable typed like \[Γ, b : block (x : term, t : oft x A[..]) ⊢ ty], but you actually need a \[Γ, x : term, t : oft x A[..] ⊢ ty], with the block flattened out. or vice versa. or you want to substitute the block more conveniently.

for this purpose, there is actually a block literal syntax <a; b; c> usable in substitutions:

let [Γ, b : block (x : term, t : oft x A[..]) ⊢ X] = blah in
[Γ, x : term, t : oft x A[..] ⊢ X[.., <x; t>]]

% but equivalent to this, but clearer (imo)
let [Γ, block (x : term, t : oft x A[..]) ⊢ X[.., b.x, b.t]] = blah in
[Γ, x : term, t : oft x A[..] ⊢ X]

explicit binders before patterns

sometimes in a case expression, the type of the pattern variables is too hard for beluga to work out on its own. in this case you get an error message about a stuck unification problem.

most of the time you can get out of this by writing the types of some variables explicitly. the syntax is like a forall-type:

case [Γ ⊢ #p] of
| {#p : [Γ ⊢ term]}
  [Γ, y : term ⊢ #p[..]] ⇒ ?body

mutual recursion

of course this exists. but it's not mentioned anywhere in the documentation for some reason. the syntax is this:

LF  term : type = …
and elim : type = …;

rec     eval-term : (Γ : ctx) [Γ ⊢ term] → [Γ ⊢ value] = …
and rec eval-elim : (Γ : ctx) [Γ ⊢ elim] → [Γ ⊢ value] = …;

inductive     ReduceTerm : (Γ : ctx) [Γ ⊢ term] → ctype = …
and inductive ReduceElim : (Γ : ctx) [Γ ⊢ elim] → ctype = …;

two recs! which is because you can mix rec/proof, or inductive/coinductive/stratified, within a block.