[Agda] Agda master is anti-classical, injective type constructors are back

[Matthieu Sozeau]

Hi,

  I'm curious here:

  fzero n : Fin (suc n) =?= fzero m : Fin (suc m) -> n =?= m

How come this is not postponed due to the fact that the types are different?
-- Matthieu

2015-01-17 16:37 GMT+05:30 Andreas Abel <abela at chalmers.se>:
> This concerns only the unreleased Agda master development branch.
>
> Using heterogeneous equality and big forced indices in a data type SING, I
> manage to prove injectivity of type constructor
>
>   SING : (Set -> Set) -> Set
>
> The rest follows Chung-Kil Hur's refutation of excluded middle, as published
> in January 2010 on the Coq and Agda lists.
>
> My development was fathered by my surprise that I could prove
>
>   inj-Fin-≅ : ∀ {n m} {i : Fin n} {j : Fin m} → i ≅ j → n ≡ m
>
> but not by directly matching on the heterogeneous equality (which Agda
> refuses as Fin n = Fin m does not yield n = m), but by first matching on i
> and j, which exposes n and m in the constructor patterns and makes then
> available for unification.
>
>   inj-Fin-≅ {i = zero}  {zero} ≅refl = refl  -- matching on ≅refl succeeds!
>   inj-Fin-≅ {i = zero}  {suc j} ()
>   inj-Fin-≅ {i = suc i} {zero} ()
>   inj-Fin-≅ {i = suc i} {suc j} p = {!p!}   -- Splitting on p succeeds!
>
> This is weird.
>
> A fix would be to not generate any unification constraints for forced
> constructor arguments, for instance
>
>   fzero : (n : Nat) -> Fin (suc n)
>
>   fzero n : Fin (suc n) =?= fzero m : Fin (suc m)
>
> would not simplify to n =?= m : Nat.
>
> This change would break a lot of developments using heterogeneous equality,
> as did the fix of issue 292
> (https://code.google.com/p/agda/issues/detail?id=292).  In particular,
> heterogeneous equality would not imply the equality of the types, but the
> equality of the types has to be given in order to sensibly work with
> heterogeneous equality.  In case of Fin, one would work with telescopes
>
>   {n m : Nat} (p : n == m) {i : Fin n} {j : Fin m} (q : i ~= j)
>
> rather than being able to derive p from q.
>
> This is Conor McBride's proposal for heterogeneous equality, if I am not
> mistaken.
>
> RFC,
> Andreas
>
> --
> Andreas Abel  <><      Du bist der geliebte Mensch.
>
> Department of Computer Science and Engineering
> Chalmers and Gothenburg University, Sweden
>
> andreas.abel at gu.se
> http://www2.tcs.ifi.lmu.de/~abel/
>
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-- 
-- Matthieu