[Agda] Help with understanding REWRITE

[Andrew M. Pitts]

Dear Jesper,

Aha, so that’s where the underscore comes from! Projection-like analysis was something of which I was unaware until you mentioned it. 

Selectively turning it off would be nice, but for now I can get by with the global --no-projection-like

Very many thanks,

Andy


> On 5 Oct 2020, at 10:03, Jesper Cockx <Jesper at sikanda.be> wrote:
> 
> Hi Andrew,
> 
> This seems to be an unfortunate interaction between --rewriting and an optimization performed by Agda, the detection of "projection-like" functions. Agda detects that the function `Cset` is projection-like, which allows it to erase the first argument. Unfortunately this means the argument is no longer available for the rewrite rule either.
> 
> If you want, you can turn off the optimization using the `--no-projection-like` flag. I'm not sure if there's currently an option to turn it off on a case-by-case basis.
> 
> -- Jesper
> 
> On Sun, Oct 4, 2020 at 3:07 PM Andrew M. Pitts <amp12 at cam.ac.uk> wrote:
> I’m struggling to understand an aspect of --rewriting in Agda (2.6.1) and wonder if someone can help?
> 
> The following artificial example illustrates my problem (which is more complicated and less artificial).
> 
> ---
> {-# OPTIONS --rewriting #-}
> 
> open import Agda.Builtin.Equality
> open import Agda.Builtin.Equality.Rewrite
> 
> data C (S : Set) : Set₁ where
>   mkC :
>     (A : Set)
>     (e : A ≡ S)
>     → ---------
>     C S
> 
> Cset : (S : Set) → C S → Set
> Cset S (mkC A e) = A
> 
> Cequ : (S : Set)(c : C S) → Cset S c ≡ S
> Cequ S (mkC A e) = e
> 
> {-# REWRITE Cequ #-} — this is not accepted!
> 
> ——
> 
> Given S : Set, the datatype C S contains values given by a set A and a proof that A equals S. Can we make that proof definitional rather than propositional, using a REWRITE? Apparently not — if you check the above Agda code, Agda 2.6.1 complains
> 
> -- Cequ is not a legal rewrite rule, since the following variables are
> -- not bound by the left hand side: S
> -- when checking the pragma REWRITE Cequ
> 
> The “left-hand side” it refers to is Cset S c, which certainly contains an occurrence of S. Why is S not “bound by” this left-hand side? I declared the type of Cequ to be
> 
> (S : Set)(c : C S) → Cset S c ≡ S
> 
> bu Agda 2.6.1 reports it to be
> 
> (S : Set) (c : C S) → Cset _ c ≡ S
> 
> Note the underscore. Presumably this is why REWRITE Cequ is not accepted.
> 
> Where did that underscore come from? And is there some way to avoid it and have the REWRITE accepted?
> 
> Andy
> 
> 
> 
> 
> 
> 
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