[Agda] [HoTT] Re: Why do we need judgmental equality?

[Thorsten Altenkirch]

On 17/02/2019, 01:25, "homotopytypetheory at googlegroups.com on behalf of Michael Shulman" <homotopytypetheory at googlegroups.com on behalf of shulman at sandiego.edu> wrote:

    However, I don't find it
    arbitrary at all: *negative* types have strict eta, while positive
    types don't.

This is a very good point. However Streams are negative types but for example agda doesn't use eta conversion on them, I think for a good reason. Actually I am not completely sure whether this is undecidable.

E.g. the following equation cannot be proven using refl (it can be proven in cubical agda btw). The corresponding equation for Sigma types holds definitionally.

infix 5 _∷_

record Stream (A : Set) : Set where
  constructor _∷_
  coinductive
  field
    hd : A
    tl : Stream A

open Stream
etaStream : {A : Set}{s : Stream A} → hd s ∷ tl s ≡ s
etaStream = {!refl!}

CCed to the agda list. Maybe somebody can comment on the decidabilty status?




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