[Agda] separate definition of constructors?

[Thorsten Altenkirch]

Thank you - this is a nice hack.

It does solve my problem but the question remains why this hack should be necessary. We should be able to declare the constructors for a collection of inductive types in any order. Would this be difficult to achieve?

Thorsten

On 24/05/2019, 17:18, "Kaposi Ambrus" <akaposi at inf.elte.hu> wrote:

    Hi Thorsten,
    
    There is another workaround discovered recently by Szumi Xie: you can
    reduce any inductive-inductive type to one with only two sorts (using
    essentially the same technique as reducing mutual inductive types to
    an indexed inductive type). Then you can specify the constructors all
    at once. Here is Szumi's implementation of tt-in-tt using cubical:
    https://bitbucket.org/szumixie/tt-in-tt/src/master/Cubical/Syntax.agda
    
    Cheers,
    Ambrus
    
    
    
    On Fri, May 24, 2019 at 4:06 PM Thorsten Altenkirch
    <Thorsten.Altenkirch at nottingham.ac.uk> wrote:
    >
    > Hi,
    >
    >
    >
    > I am trying to port the definition of Type Theory in Type Theory form our paper
    >
    > Type theory in type theory using quotient inductive types. POPL 2016
    >
    > to cubical agda (yes I know inductive families don’t yet work but Andrea is working on it).
    >
    >
    >
    > However, when we faked this we were able to first introduce the point constructors and then the equality constructors but when doing this in cubical agda all the constructors have to appear together. This leads to the old problem that you have to create forward references for contructors which is a bit ugly. E.g.
    >
    >
    >
    > data Con : Set
    >
    > data Ty : (Γ : Con) → Set
    >
    > data Tm : (Γ : Con)(A : Ty Γ) → Set
    >
    > data Tms : (Γ Δ : Con) → Set
    >
    >
    >
    > data Ty where
    >
    >     _[_] : Ty Δ → Tms Γ Δ → Ty Γ
    >
    >
    >
    > data Tms where
    >
    >       id    : Tms Γ Γ
    >
    >       _,_  : (σ : Tms Γ Δ) → Tm Γ (A [ σ ]) → Tms Γ (Δ , A)
    >
    >
    >
    > data Ty where
    >
    >     [id]T : ∀{Γ}{A : Ty Γ} → A [ id ] ≡ A
    >
    >
    >
    > The problem is that Tms uses  _[_] hence I have to declare the point constructors for Ty first, but then the equality for Ty refers to id!
    >
    >
    >
    > This is an old issue (already with inductive-inductive definitions) but it gets worse once we have QIITs. There is a workaround to define a forward definition
    >
    >
    >
    > id'    : Tms Γ Γ
    >
    >
    >
    > and then use id’ in the equation for Ty and define later
    >
    >
    >
    > id' = id
    >
    >
    >
    > but this is a bit ugly. Would it be possible to allow separate definitions of constructors?
    >
    >
    >
    > Thorsten
    >
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