C++ has a couzy group of friends among mainstream languages, that share similar age.
Denotational or operational semantics: pick one for your programming language and stick to it. The author (who I generally think is very smart) here is striving for denotational semantics (type level data) and trying to torture the operations into supplying the appropriate result. Operationally `cols (replicate 0 (replicate 3 0))` is 0 not 3. So now you have to bend over backwards and implement custom shape functions that not only return weird answers but have to be special cased AND context sensitive - ie without trying the language I'm 100% sure that
cols (replicate 0 "x")
cols (replicate 0 (replicate k "x"))
cols (replicate 0 "x")
def cols [n] [m] 't (x: [n][m]t) : i64 = m
def replicate5 n x = replicate 5 x
cols (replicate5 0 (replicate5 3 0)) == 5
spoiler alert every single tensor/array/matrix/ML/AI compiler runs into this same problem. there is only one solution: a fixed op set with a fixed number of corresponding shape functions. and then your compiler tries to perform shape inference/propagation. sometimes it works and you can specialize for fixed sizes and sometimes it fails and you get "dynamic" or "unknown" dims in your shapes and you can't do anything. oh well that's life in a universe where the halting problem exists.
Yeah. I was screaming for most of this piece, because this all seems like standard dependently-typed stuff, and ironically enough implementing full dependent types would probably end up being easier than trying to handle this one feature as a special case.