[Agda] algebra hierarchy in library
Matthew Daggitt
matthewdaggitt at gmail.com
Sat Mar 16 11:16:03 CET 2019
Hi Sergei,
(I wonder, why people, - and standard library, - call provers solvers).
>
Unsure.
If this is the only reason, then it occurs that `Is' structure is needed
> only in few cases.
>
I think your examples miss the point. For example take any binary operator.
That binary operator may form many different Semigroups/Monoids/Groups etc.
depending on what the underlying equality is. The `Is` structures allow you
to expose which equality you're using at a particular point, whereas your
suggestion would hide it.
Version II looks more natural to me. But I may be missing something.
> To make sure, I could rewrite a part of the library for Version II and
> demonstrate. And what if it occurs better? It will be late to consider
> for standard.
>
As I mentioned to you in an issue on Github, non-backwards compatible
changes will only be considered where either i) the implementation is
incorrect (clearly not the case here) or ii) there's a compelling reason
why the current version isn't good enough. "Looking more natural"
unfortunately isn't such a reason and as mentioned above Version II doesn't
allow you to expose the underlying equality.
Best,
Matthew
On Sat, Mar 9, 2019 at 6:28 PM Sergei Meshveliani <mechvel at botik.ru> wrote:
> On Sat, 2019-03-09 at 19:08 +0300, Sergei Meshveliani wrote:
> > On Sat, 2019-03-09 at 17:08 +0300, Sergei Meshveliani wrote:
> > > On Sat, 2019-03-09 at 14:35 +0300, Sergei Meshveliani wrote:
> > >
> > > >
> > > > If this is the only reason, then it occurs that `Is' structure is
> needed
> > > > only in few cases.
> > > > For example,
> > > > (1) There does not exist different semigroups that inherit ("are
> over")
> > > > the same Magma. The difference can be in a _proof_ for associativity,
> > > > but I doubt of whether this feature can be taken here in account.
> > > >
> > > > Let people correct me if I mistake in the following statements.
> > > >
> > > > (2) There does not exist different monoids over the same Semigroup.
> > > > (3) There does not exist different commutative monoids over the
> same
> > > > Monoid.
> > > > (4) There does not exist different groups over the same Monoid.
> > > > (5) There does not exist different Abelian groups over the same
> Group.
> > > > (6) There does not exist different rings over the same Semiring.
> > > > (7) There does not exist different commutative rings over the same
> > > > Ring.
> > > [..]
> > >
> > >
> > > Sorry for a silly error. I discover a mistake in (2), (4) (6).
> > >
> > > For example, Nat1 = Nat\0 is a semigroup by _+_, and zero can be
> > > joined in different ways, so that (Nat1 U 0) and (Nat1 U 0') occur
> > > different monoids. They are isomorphic, but they have different
> > > carriers.
> > > A similar effect with carrier may be in (4) and (6).
> > >
> > > So that there remain (1), (3) and (7).
> >
> >
> > No, again an error.
> >
> > A monoid in _this library_ is on the same carrier C as its inherited
> > semigroup. So that to implement a monoid on a given semigroup means to
> > choose any e in C which satisfy the law \forall x (e*x == x*e == x)
> > and to prove this law.
> > And it is proved above that such e is unique.
> >
> > So, I think that similarly, all the points (1) -- (7) are true
> > -- if I am not missing something.
>
>
>
> We need one more step towards truth :-)
>
> Having a Monoid instance, how many different group instances can be
> defined on this monoid (hence, on the same carrier C) ? Possible
> inversion map on C is unique, as shown in one of previous letters. But
> it can be implemented by different algorithms, and this can be used by
> programmers. Algorithms matter in the library. And according to Agda,
> different algorithms for inversion give different groups.
> So that there remain the above statements (1), (3), (7), and may be (2).
>
> --
> SM
>
>
> >
> >
> > > >
> > > > >
> > > > > On Fri, Mar 8, 2019 at 3:00 PM Sergei Meshveliani <
> mechvel at botik.ru>
> > > > > wrote:
> > > > >
> > > > > Dear standard library developers and supporters,
> > > > >
> > > > > can you please answer in (simple words) several questions
> > > > > about the
> > > > > representation of the algebraic hierarchy in standard
> library?
> > > > >
> > > > >
> > > > > 1. Why `Raw' structures?
> > > > >
> > > > > There are classical generic algebraic structures (call them
> > > > > GAS):
> > > > > Magma, Semigroup, Monoid, and so on.
> > > > >
> > > > > Those of them having some new field respectively to
> previous
> > > > > structures
> > > > > are accompanied with the corresponding `Raw' record. For
> > > > > example, Magma
> > > > > is preceded with RawMagma, Monoid with RawMonoid. Each
> `Raw'
> > > > > structure
> > > > > expresses only the signature of the corresponding GAS.
> > > > >
> > > > > What the `Raw' structures serve for?
> > > > >
> > > > >
> > > > > 2. Why putting `Is' structures into a different file?
> > > > >
> > > > > For example, the reader looks into Algebra.agda to find
> what
> > > > > is
> > > > > Semigroup:
> > > > >
> > > > > -------------------------------------------------------
> > > > > record Semigroup c ℓ : Set (suc (c ⊔ ℓ)) where
> > > > > ...
> > > > > field Carrier : Set c
> > > > > _≈_ : Rel Carrier ℓ
> > > > > _∙_ : Op₂ Carrier
> > > > > isSemigroup : IsSemigroup _≈_ _∙_
> > > > >
> > > > > open IsSemigroup isSemigroup public
> > > > > ...
> > > > > magma = record { isMagma = isMagma }
> > > > > ...
> > > > >
> > > > > Now, one needs to find a declaration for IsSemigroup.
> > > > > And it resides in a different file of
> > > > > Algebra/Structures.agda
> :
> > > > >
> > > > > record IsSemigroup (∙ : Op₂ A) : Set (a ⊔ ℓ) where
> > > > > field
> > > > > isMagma : IsMagma ∙
> > > > > assoc : Associative ∙
> > > > >
> > > > > open IsMagma isMagma public
> > > > > ---------------------------------------------------------
> > > > >
> > > > > And all this implements the meaning of a small sentence:
> > > > > ``Semigroup is Magma in which multiplication _∙_ is
> > > > > associative''.
> > > > >
> > > > > Why not put ``record IsSemigroup'' before ``record
> Semigroup''
> > > > > in the
> > > > > same file Algebra.agda ?
> > > > >
> > > > > Similarly other `Is' GAS decls can join. So that
> Algebra.agda
> > > > > and
> > > > > Algebra/Structures.agda will merge into Algebra.agda in
> > > > > which each
> > > > > GAS will be defined in one place.
> > > > > For example, to see what is a group will need to look into
> one
> > > > > file, not
> > > > > in two files.
> > > > > ?
> > > > >
> > > > >
> > > > > 3. Why Magma declares the fields Carrier and _≈_ by new?
> > > > >
> > > > > Similarly, why other GAS re-declare many fields?
> > > > >
> > > > > In theory, we have
> > > > > ``Magma is a setoid with an operation _∙_ congruent with
> > > > > respect to the
> > > > > equality _≈_''.
> > > > >
> > > > > So, Magma inherits Setoid. And it is natural for its
> > > > > representation in
> > > > > Agda to have setoid somewhere inside it. So, it opens
> this
> > > > > setoid and
> > > > > uses its fields in further definitions. For example, like
> > > > > this:
> > > > >
> > > > > -- Version II
> > > > > ---------------------------------------------------------
> > > > >
> > > > > record IsMagma {α α=} (S : Setoid α α=) (_∙_ : Op₂
> > > > > (Setoid.Carrier S)) :
> > > > >
> Set
> > > > > (α ⊔ α=)
> > > > > where
> > > > > open Setoid S using (_≈_; Carrier)
> > > > > field
> > > > > ∙cong : _∙_ Preserves₂ _≈_ ⟶ _≈_ ⟶ _≈_
> > > > >
> > > > > record Magma α α= : Set (suc (α ⊔ α=))
> > > > > where
> > > > > field setoid : Setoid α α=
> > > > >
> > > > > open Setoid setoid public
> > > > > infixl 7 _∙_
> > > > >
> > > > > field _∙_ : Op₂ Carrier
> > > > > isMagma : IsMagma setoid _∙_
> > > > >
> ------------------------------------------------------------------------
> > > > >
> > > > > And let us call Version I the approach of Standard
> library
> > > > > lib-0.17.
> > > > >
> > > > > Both versions use an `Is' structure, but II does not
> > > > > re-declare fields.
> > > > > Is not II more natural?
> > > > >
> > > > >
> > > > > Another question may be:
> > > > > ``why splitting each GAS into proper structure and `Is'
> > > > > structure?''.
> > > > >
> > > > > My guess is that this approach allows us to express two
> GAS-s
> > > > > that are
> > > > > over the same inherited GAS. For example, a programmer can
> > > > > express a
> > > > > product of two Magmae over the same Setoid:
> > > > >
> ------------------------------------------------------------
> > > > > module _ {α α=} (S : Setoid α α=)
> > > > > where
> > > > > open Setoid S using (Carrier; _≈_)
> > > > > SS = ×-setoid S S
> > > > > open Setoid SS using () renaming (Carrier to CC; _≈_ to
> > > > > _=p_)
> > > > >
> > > > > magmaProduct' :
> > > > > (_*₁_ _*₂_ : Op₂ Carrier) → IsMagma _≈_ _*₁_ →
> > > > > IsMagma _≈_ _*₂_ → Magma α
> α=
> > > > > magmaProduct' _*₁_ _*₂_ insM₁ isM₂ =
> > > > > <define coordinate-wise
> multiplcation
> > > > > on CC;
> > > > > prove ...; return the Magma record
> > > > > >
> > > > >
> ------------------------------------------------------------
> > > > >
> > > > > (is there any other purpose to introduce `Is'
> -structures?).
> > > > >
> > > > > This is equally easy to set both in Version I and Version
> II.
> > > > >
> > > > > But note that both approaches still deviate, a bit, from
> the
> > > > > theory.
> > > > > Because in theory, it is
> > > > > \ (mg1 : Magma_ _) (mg2 : Magma _ _) (HaveCommonSetoid mg1
> > > > > mg2) →
> > > > > product-magma,
> > > > > while magmaProduct' takes certain parts of the two magmae.
> > > > >
> > > > > So, there remain somewhat three and a half questions.
> > > > >
> > > > > Thank you in advance for your possible explanation.
> > > > >
> > > > > ------
> > > > > Sergei
> > > > >
> > > > >
> > > > >
> > > > >
> > > > > _______________________________________________
> > > > > Agda mailing list
> > > > > Agda at lists.chalmers.se
> > > > > https://lists.chalmers.se/mailman/listinfo/agda
> > > >
> > > >
> > > > _______________________________________________
> > > > Agda mailing list
> > > > Agda at lists.chalmers.se
> > > > https://lists.chalmers.se/mailman/listinfo/agda
> > >
> > >
> > >
> >
> >
> >
>
>
> _______________________________________________
> 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/20190316/b7208579/attachment.html>
More information about the Agda
mailing list