[Agda] Postulated computing quotients are unsound

[Altenkirch Thorsten]

That's a shame.

How can this be fixed? Can we hide things better?

We were first considering to implement quotients directly but then used
the hiding trick. We could now go back to the first alternative but maybe
a more general approach would be to address the issues which make the
current approach unsound.

Thorsten

On 16/05/2012 14:13, "Nils Anders Danielsson" <nad at chalmers.se> wrote:

>Hi,
>
>At the last AIM Thorsten Altenkirch, James Chapman and I experimented
>with postulated, computing quotients, perhaps inspired by Dan Licata's
>"Running Circles Around (In) Your Proof Assistant; or, Quotients that
>Compute":
>
>   
>http://homotopytypetheory.org/2011/04/23/running-circles-around-in-your-pr
>oof-assistant/
>
>We were not sure if this hack was sound or not. Now, thanks to Dan Doel,
>I know that it isn't.
>
>
>Quotients were defined in the following way:
>
>   data Quotient {c ℓ} (A : Setoid c ℓ) : Set (c ⊔ ℓ) where
>     box : (x : Setoid.Carrier A) → Quotient A
>
>To avoid direct pattern matching we didn't export the constructor, but
>only a function [_] = box. We also exported an eliminator that allowed
>you to "pattern match" as long as you proved that you respected the
>relation, as well as the following postulate, which states that [_]
>respects the relation:
>
>   postulate
>     [_]-cong : ∀ {c ℓ} {A : Setoid c ℓ} {a₁ a₂} → let open Setoid A in
>                a₁ ≈ a₂ → _≡_ {A = Quotient A} [ a₁ ] [ a₂ ]
>
>
>Dan Doel pointed out [1] that certain hacks intended to emulate "higher
>inductive types" (as in Dan Licata's post mentioned above) are
>suspicious:
>
>* Even if you hide the constructors the absurd pattern is still
>   available.
>
>* The exported constructors compute to the real ones, also in
>   equalities. For instance, [ true ] ≡ [ false ] computes to box true ≡
>   box false.
>
>
>Given these observations it is easy to prove that the postulate above is
>unsound:
>
>* Quotient Bool by the trivial relation.
>
>* Use [_]-cong to prove that [ true ] and [ false ] are equal. Denote
>   the proof by p.
>
>* Prove that the empty type is inhabited by using an absurd pattern:
>
>     bad : ⊥
>     bad with p
>     ... | ()
>
>
>A similar approach can be used to prove that Dan Licata's encoding of
>the interval is unsound.
>
>
>[1] The #agda IRC channel, 2012-05-15
>     (http://agda.orangesquash.org.uk/2012/May/16.html); Dan Doel's nick
>     is dolio.
>
>-- 
>/NAD
>
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