问题
I am trying to figure out how type hierarchies work in Agda.
Assuming I define a set type X:
X : Set
and then proceed to constructing an inductive type
data Y : X -> Set where
What is the type of X -> Set? Is it Set or Type?
Thank you!
回答1:
Well, why not ask Agda itself? I'm going to use excellent Agda mode for Emacs. We start with:
module Hierarchy where
postulate
X : Set
data Y : X → Set where
-- empty
We have to load the file using C-c C-l; this typechecks the file, turns ?s into holes, does syntax highlighting and so on.
Now, there is a command "Infer (deduce) type" available via C-c C-d, so let's use that:
> C-c C-d
Expression:
> Y
X → Set
Right, that makes sense. We defined Y : X → Set, so it should come as no surprise. Let's ask again:
> C-c C-d
Expression:
> X → Set
Set₁
So, there you have it: Y : X → Set : Set₁.
While the first part answers the question and shows you how to check this stuff yourself, doing that everytime would be dull, at least. Here's how it works:
To avoid paradoxes, we require
Set i : Set (i + 1)
which gives you the (infinite) hierarchy of Sets. If you had Set : Set (which Agda allows via the --type-in-type flag), you could derive contradiction such as this one.
This also gives us a simple rule for functions:
A : Set i
B : A → Set j
(a : A) → B a : Set (max i j)
Applying this to your example:
X : Set
Set : Set₁
X → Set : Set (max 0 1)
X → Set : Set₁
来源：https://stackoverflow.com/questions/15924509/type-hierarchy-in-agda