def∇: A Drop In def∇

Thu, 08 Oct 2026 00:00:00 +0000

Yesterday I looked at def∇, a macro for Dyalog APL like self-referencing functions in Lean.

Note

Id like to thank my Lisp friend Armin for pointing out Lisp people have also been doing this since time immemorial, and pointing me to Paul Graham’s alambda from his book On Lisp where he does effectively the same thing in Common Lisp.

(defmacro alambda (parms &body body)
 `(labels ((self ,parms ,@body))
 #'self))

(alambda (n)
 (if (> n 0)
 (cons n (self (- n 1)))))

Unfortunately, after using that version of def∇ in practice in a huge project beyond the toy examples I gave yesterday, I quickly realized its largest limitation: it is not a drop in replacement. It turns the body of the definition into a where helper, so proofs that rely on the original definition can break. The ∇ in the body refers to that helper, which closes over the explicit binders rather than calling the function being defined.

def∇: An APL-Style Self Reference Macro for Lean4

Wed, 07 Oct 2026 00:00:00 +0000

I’ve been writing recursive functions over syntax trees a lot recently. I also like my Lean code to be concise, but as things start piling up, this becomes harder and harder. Take the following function over propositional logic formulas as an example:

inductive Formula where
| atom : String → Formula
| neg : Formula → Formula
| conj : Formula → Formula → Formula

def depth : Formula → Nat
| .atom _ => 1
| .neg c => 1 + depth c
| .conj c d => 1 + max (depth c) (depth d)

depth is recursive and depth is used three times in its own definition. This is fine for small functions, but it can become cumbersome for larger ones. This is an actual function over description logic syntax from a codebase im working on. (You don’t need to understand the details to see thats A LOT of polarize all all just to handle a pretty simple recursive traversal)

DefDel on James Oswald

def∇: A Drop In def∇

def∇: An APL-Style Self Reference Macro for Lean4