[Agda] Fwd: Fwd: Question about transport and cubical

[John Leo]

Thanks so much Anders for the detailed explanation! It's extremely helpful.

John

On Thu, Sep 3, 2020 at 11:09 PM Anders Mortberg <andersmortberg at gmail.com>
wrote:

> Yes, John's understanding of that very opaque error message is
> correct. When checking c Agda will have to verify that whenever i1=i1
> (that is "everywhere") then (λ i → e i) is a constant function. This
> is clearly not the case as:
>
> (λ i → e i) /= (λ i → Bool)
>
> This is what the error message is trying to say, but e has been
> unfolded too far and there is some mysterious metavariable _28.
>
>
> Your understanding of what happens when r is i0 is also correct, in
> that case the condition r=i1 is just i0=i1 which is absurd and there
> is nothing to check as anything follows from an absurd assumption.
> This is why a typechecks.
>
>
> In general r is some element of dM(X), i.e. an element of the free De
> Morgan algebra on some subset X of the dimension variables currently
> in context. One way to check if some judgment holds when r=i1 is to
> first convert r to disjunctive normal form and propagate the _=i1 all
> the way down to the atoms. This gives us a big disjunction of
> conjuncts where each conjunct corresponds to a list of substitutions.
> For example if r is (i /\ j) \/ ~ k then r=i1 will reduce to
>
> ((i = i1) /\ (j = i1)) \/ (k = i0)
>
> To check that some judgment J holds in this context restriction
> amounts to checking that it holds either when (i = i1) and (j = i1) or
> when (k = i0). If I write G for the ambient context we hence need to
> check
>
> G, (i = i1) /\ (j = i1) |- J
> G, (k = i0) |- J
>
> which boils down to checking
>
> G |- J(i1/i)(i1/j)
> G |- J(i0/k)
>
> I don't think Cubical Agda actually performs these substitutions as
> it's too expensive to always substitute, but this intuitive algorithm
> can maybe be helpful to understand how one can typecheck cubical
> programs.
>
> --
> Anders
>
> On Fri, Sep 4, 2020 at 1:05 AM John Leo <leo at halfaya.org> wrote:
> >
> > I do have one further point I'd like clarified. Is the check for the
> r=i1 condition for transp done only when r is not known to be i0 or is it
> always done? For example is the check run at all when transport p (defined
> as "transp (λ i → p i) i0") is called? I assume not. For example in the
> following code
> >
> > notnot : (b : Bool) → not (not b) ≡ b
> > notnot true  = refl
> > notnot false = refl
> >
> > e : Bool ≡ Bool
> > e = isoToPath (iso not not notnot notnot)
> >
> > a = transp (λ i → e i) i0 true
> > b = transp (λ _ → Bool) i1 true
> > c = transp (λ i → e i) i1 true
> >
> > I get that "a" and "b" typecheck ("a" evaluates to false and "b" to true
> as expected) but "c" fails to typecheck with the following error, which I
> assume is due to "e" not being definitionally constant. But perhaps I'm
> still confused.
> >
> > primGlue Bool
> > (λ .x →
> >    (λ { (i = i0) → Bool , isoToEquiv (iso not not notnot notnot)
> >       ; (i = i1) → Bool , idEquiv Bool
> >       })
> >    _ .fst)
> > (λ .x →
> >    (λ { (i = i0) → Bool , isoToEquiv (iso not not notnot notnot)
> >       ; (i = i1) → Bool , idEquiv Bool
> >       })
> >    _ .snd)
> > != Bool of type Type
> > when checking that the expression transp (λ i → e i) i1 true has
> > type _28
> >
> >
> >
> > _______________________________________________
> > Agda mailing list
> > Agda at lists.chalmers.se
> > https://lists.chalmers.se/mailman/listinfo/agda
>
-------------- next part --------------
An HTML attachment was scrubbed...
URL: <http://lists.chalmers.se/pipermail/agda/attachments/20200904/a87a7857/attachment.html>