[Agda] How can I implement naive sized lambdas?

[Andreas Abel]

Hello Georgi,

you already received some good advice.  In general, I second Andras' 
advice to first define renaming and then (parallel) substitution.  If 
you want to do it in one definition, you can use Conor McBride's trick:

   http://www.cse.chalmers.se/~abela/html/ParallelSubstitution.html

(See http://www.cse.chalmers.se/~abela/projects.html for some more code 
snippets.)

The blog post by Rob Simmons is neat, but I'd say not entirely state of 
the art; you definitely want parallel substitution as the primitive, not 
single substitution as Rob defines it (maybe owned to the fact that he 
comes from a Twelf background).

Cheers,
Andreas

On 2020-03-24 23:28, Georgi Lyubenov wrote:
> 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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