Comments (2)
The problem is that Ω n ..
does not reduce, because Ω
is defined by pattern matching on the natural number.
That's what the error message is trying to say
The target of a constructor must be the datatype applied to its
parameters, Ω n (Sphere n , base n) .fst isn't
when checking the constructor cell in the declaration of Sphere
"The target of a constructor must be the datatype applied to its parameters" means that the result type of a constructor should reduce to "Sphere .."
The other examples work because e.g. (Ω 2 (S² , base)) .fst
normalizes to (λ _ → base) ≡ (λ _ → base)
which is a path type in S²
.
from cubical.
This HIT is not supported by what we did in
https://arxiv.org/abs/1802.01170
or by the schema of
http://www.cs.cmu.edu/~ecavallo/works/popl19.pdf
It is in some sense clear that it works semantically as the types are equivalent to the spheres for a fixed n, but it is not clear how to get it to work type theoretically. A concrete issue is how to even write down the eliminator in general. Given n : ℕ
what do we do in the cell case? We will need to give an equation of the form:
f (cell i_1 ... i_n) =
but you cannot write "..." in type theory.
from cubical.
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from cubical.