[Agda] Congruence and substitutivity rules in heterogeneous equality

[Liang-Ting Chen]

On Mon, Jul 13, 2009 at 10:06 PM, Nils Anders Danielsson
<nad at cs.nott.ac.uk>wrote:

> On 2009-07-13 13:22, Liang-Ting Chen wrote:
>
>> subst (P : A → Set) {x : A} {y : A} → x ≅ y → P x → P y
>>
>
> You could do something like the following:
>
>  ≅⇒≡₁ : ∀ {A B} {x : A} {y : B} → x ≅ y → A ≡₁ B
>  ≅⇒≡₁ refl = refl
>
>  coerce : ∀ {A B} → A ≡₁ B → A → B
>  coerce refl = id
>
>  subst : ∀ {A B} (P : A → Set) {x : A} {y : B}
>         (y≅x : y ≅ x) → P x → P (coerce (≅⇒≡₁ y≅x) y)
>  subst P refl p = p
>
> However, I suspect that you do not really want to prove
> P (coerce (≅⇒≅₁ y≅x) y), but rather something slightly different.

Thanks for it ! How about cong ?

subst looks not possible to drop the annotation before unless it is
evaluated? I noticed in RingSolver.agda
there is an interesting idiom to eliminate the use of subst by adding an
casting constructor.


>
>  Furthermore, it also breaks the heterogeneous equational reasoning for
>> the same reason.
>>
>
> In what way is this broken? The standard library requires homogeneity in
> many places, but not for equational reasoning for _≅_ (in the latest
> version).

Ooops, I got it. I checked the heterogeneous equational reasoning few months
ago,
and now it works properly now.

>
>
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-- 
sincerely,
Liang-Ting
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