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

[Manuel Bärenz]

Yes, that seems a good question to me as well.

Relatedly: I wanted to start improving the docs, but the Github link is
broken:
https://github.com/agda/agda/blob/v2.6.1/doc/user-manual/language/cubical.rst

I reached the link from
https://agda.readthedocs.io/en/v2.6.1/language/cubical.html

On 05.09.20 14:06, Dan Krejsa wrote:
> Hi,
>
> Why isn't transp declared like this:
>
> transp : ∀ {ℓ} (A : I → I → Set ℓ) (r : I) → A i0r → A i1 r
>
> with an intended usage that  'A s i1'  is definitionally independent of s ?
>
> On Fri, Sep 4, 2020 at 6:17 AM John Leo <leo at halfaya.org
> <mailto:leo at halfaya.org>> wrote:
>
>     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 <mailto: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
>         <mailto: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
>         >
>         >
>         >
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