[Agda] Associativity of vector concatenation

[Tom Jack]

I guess that you are referring to the 'John Major' equality in Conor
McBride's thesis here? http://strictlypositive.org/thesis.pdf

In section 5.1.3 (page 120), there is the following note:

Observe that eqElim is not the elimination rule which one would expect if
> ≃ was inductively defined. The ‘usual’ rule eliminates over all the
> formable equations, and it is quite useless...


Even if the usual rule is useless, it is what we will have in Agda for an
inductively defined heterogeneous equality :).

It is trivial using the 'useless' rule to give an equivalence between (a ≅
b) and (A , a) ≡ (B , b), where _≡_ is the usual homogeneous equality. So
then _≅_ will be related to your _≡[_]_ as total space to fiber.

On Thu, Oct 27, 2016 at 3:49 PM, Martin Escardo <m.escardo at cs.bham.ac.uk>
wrote:

>
>
> On 27/10/16 23:00, Tom Jack wrote:
> > Does the elimination rule for heterogeneous equality really imply K?
>
> If I am not mistaken, this was proved by the very person who introduced
> heterogeneous equality in the first place, in his PhD thesis.
>
> But I am open to be corrected.
>
> Martin
>
>
> >
> > Take the type as defined in agda stdlib
> > Relation.Binary.HeterogeneousEquality:
> >
> >     data _≅_ {i} {A : Set i} (x : A) : {B : Set i} → B → Set i where
> >        refl : x ≅ x
> >
> > This is the eliminator, right?
> >
> >     elim-≅ : ∀ {i j} {A : Set i} {x : A}
> >       (C : {B : Set i} {y : B} → x ≅ y → Set j)
> >       → C refl
> >       → {B : Set i} {y : B} (p : x ≅ y) → C p
> >     elim-≅ C c refl = c
> >
> > This does not imply K, does it?
> >
> > On Thu, Oct 27, 2016 at 2:30 PM, Martin Escardo <m.escardo at cs.bham.ac.uk
> > <mailto:m.escardo at cs.bham.ac.uk>> wrote:
> >
> >
> >     Often people mention the formulation and proof of associativity of
> >     vector concatenation as something tricky.
> >
> >     The problem, of course, is that for vectors xs, ys, and zs, the
> >     concatenations
> >
> >         xs ++ (ys ++ zs)    of length l + (m + n)
> >
> >     and
> >
> >         (xs ++ ys) ++ zs    of length (l + m) + n
> >
> >     belong to the different (but isomorphic) types
> >
> >         Vec X  (l + (m + n)) and Vec X ((l + m) + n).
> >
> >     One approach considers heterogeneous equality, which is a notion of
> >     equality that allows to compare elements of different types. A
> >     limitation of this equality is that its elimination rule implies the
> K
> >     axiom (any two proofs of equality are equal), which is incompatible
> >     with univalent type theory as in the HoTT Book or as in Cubical Type
> >     Theory, which are type theories some people wish to either use or at
> >     least be compatible with.
> >
> >     Not only can we be compatible with univalent type theory, but also
> >     we can learn from it (even without using univalence or other
> >     features that
> >     don't belong to Martin-Loef Type Theory as represented by Agda).
> >
> >     I would like to advertise this here, as the ideas are of general use,
> >     even if one is not interested in HoTT, and hopefully intuitive and
> >     natural too.
> >
> >     Given a type X and a type family F:X->Set, suppose we have x,y:X,
> >     p:x≡y and a : F x and b : F y. Even though x and y are equal, the
> >     types F x and F y are not the same (they are equal under univalence,
> >     but we don't want to use this). So can't compare a and b for
> equality.
> >
> >     But we can define a notion of equality between the types F x and F y,
> >     dependent on p:x≡y,
> >
> >         a ≡[ p ] b,
> >
> >     by
> >
> >         a ≡[ refl ] b   =   a ≡ b.
> >
> >     In particular, we get
> >
> >     _≡[_]_ : ∀ {X} {m n} → Vec X m → m ≡ n → Vec X n → Set
> >     xs ≡[ refl ] ys = xs ≡ ys
> >
> >     Then vector associativity becomes
> >
> >     ++-assoc : ∀ {X : Set} l m n (xs : Vec X l) (ys : Vec X m) (zs : Vec
> >     X n)
> >             → xs ++ (ys ++ zs)  ≡[ +-assoc l m n ]  (xs ++ ys) ++ zs
> >
> >     where
> >
> >     +-assoc : ∀ l m n → l + (m + n) ≡ (l + m) + n.
> >
> >     Its proof is the same as that of associativity of list concatenation
> >     (where the problem discussed in this message doesn't arise), but
> >     taking care of the extra subscript p for dependent equality, in a way
> >     that doesn't become disruptive or require new ideas, as shown here:
> >
> >     http://www.cs.bham.ac.uk/~mhe/agda/VecConcatAssoc.html
> >     <http://www.cs.bham.ac.uk/~mhe/agda/VecConcatAssoc.html>
> >
> >     The idea is very similar to using heterogeneous equality, as you will
> >     see.
> >
> >     Anyway, I wanted to advertise this idea coming from univalent type
> >     theory, and try to convince you of its simplicity and naturality and
> >     usefulness.
> >
> >     Martin
> >     _______________________________________________
> >     Agda mailing list
> >     Agda at lists.chalmers.se <mailto:Agda at lists.chalmers.se>
> >     https://lists.chalmers.se/mailman/listinfo/agda
> >     <https://lists.chalmers.se/mailman/listinfo/agda>
> >
> >
>
> --
> Martin Escardo
> http://www.cs.bham.ac.uk/~mhe
>
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