[Agda] Univalence via Agda's primTrustMe?

[Jason Gross]

Is it possible to push this even further?  For example, define an alternate
definition of, say, ap, relying on primTrustMe in some fancy way such that
when it gets either univalence or refl, it computes?

-Jason

On Sat, Jan 17, 2015 at 2:33 AM, Alan Jeffrey <ajeffrey at bell-labs.com>
wrote:

> Hi everyone,
>
> In the Agda development of Homotopy Type Theory at
> https://github.com/HoTT/HoTT-Agda/ the univalence axiom is given by three
> postulates (the map from (A ≃ B) to (A ≡ B) and its β and η rules).
>
> I wonder whether these postulates could be replaced by uses of primTrustMe?
>
> As a reminder, primTrustMe is a trusted function which inhabits the type
> (M ≡ N) for any M and N. It is possible to introduce contradictions (e.g. 0
> ≡ 1) in the same way as with postulates, so it has to be handled with care.
> The semantics is as for postulates, but with an extra beta reduction:
>
>   primTrustMe M M → refl
>
> In the attached Agda code, primTrustMe is used to define:
>
>   private
>     trustme : ∀ {ℓ} {A B : Set ℓ} (p : A ≃ B) → (∃ q ∙ ((≡-to-≃ q) ≡ p))
>     trustme p = ⟨ primTrustMe , primTrustMe ⟩
>
> from which we get the map from (A ≃ B) to (A ≡ B) and its β rule:
>
>   ≃-to-≡ : ∀ {ℓ} {A B : Set ℓ} → (A ≃ B) → (A ≡ B)
>   ≃-to-≡ p with trustme p
>   ≃-to-≡ .(≡-to-≃ refl) | ⟨ refl , refl ⟩ = refl
>
>   ≃-to-≡-β : ∀ {ℓ} {A B : Set ℓ} (p : A ≃ B) → (≡-to-≃ (≃-to-≡ p) ≡ p)
>   ≃-to-≡-β p with trustme p
>   ≃-to-≡-β .(≡-to-≃ refl) | ⟨ refl , refl ⟩ = refl
>
> Interestingly, the η rule and the coherence property for β and η then
> become trivial:
>
>   ≃-to-≡-η : ∀ {ℓ} {A B : Set ℓ} (p : A ≡ B) → (≃-to-≡ (≡-to-≃ p) ≡ p)
>   ≃-to-≡-η refl = refl
>
>   ≃-to-≡-τ : ∀ {ℓ} {A B : Set ℓ} (p : A ≡ B) →
>     (cong ≡-to-≃ (≃-to-≡-η p) ≡ ≃-to-≡-β (≡-to-≃ p))
>   ≃-to-≡-τ refl = refl
>
> Note there's some hoop-jumping with private declarations to hide trustme
> from users, because:
>
>   (fst (trustme p)) → refl  (for any p : (A ≃ A))
>
> that is, all proofs of (A ≃ A) would be identified if we were allowed
> unfettered access to trustme. Instead, we only allow (≃-to-≡ p) to reduce
> to refl when (trustme p) reduces to ⟨ refl , refl ⟩, that is not only do we
> have (A ≃ A) but also that p must be the trivial proof that (A ≃ A).
>
> Now, this isn't a conservative extension of HOTT because it introduces
> extra beta reductions that were previously just propositional equalities,
> in particular:
>
>   (≃-to-≡ (≡-to-≃ refl)) → refl
>   (≃-to-≡-β (≡-to-≃ refl)) → refl
>   (≃-to-≡-η refl) → refl
>   (≃-to-≡-τ refl) → refl
>
> So questions... a) Is this re-inventing the wheel? b) Is this sound? c) Is
> there a connection between this and a computational interpretation of
> univalence?
>
> Alan.
>
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