[Agda] How can I implement naive sized lambdas?

[Georgi Lyubenov]

Hello!

Sorry if this isn't the right place to ask a question like this!

I want to express named lambda terms in Agda. By doing so in the most
"naive" way possible, I encounter a termination issue when implementing
substitution - when I want to rename a variable to avoid accidental
variable capture, the termination checker is unhappy - I'm trying to do a
recursive call (recursive calling substitution) on the result of a function
(renaming). My first thought is to attach sizes (via either Nats or the
builtin Size type), and show that renaming doesn't "change size". After a
bit of struggling I couldn't manage to do so, and am writing to ask for a
hint or some advice.

This is my initial attempt:

data : Size -> Set where
  v : {i : Size} -> (n : Nat) -> Lambda i
  _app_ : {i j : Size} -> Lambda i -> Lambda j -> Lambda (sizeSuc (sizeMax
i j))
  lam_>_ : {i : Size} -> Nat -> Lambda i -> Lambda (sizeSuc i)

but with this I can't even implement something like

promo : {i j : Size} -> Lambda i -> Lambda (sizeMax i j)

Furthermore I have no idea what return type to even give to substitution!

I am pretty sure I am missing something very fundamental about sizes and
how to use them, but I couldn't find materials to answer my question.

Thanks in advance!

=======
Georgi
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